scalable package queries in relational database systems · scalable package queries in relational...

Scalable Package Queries in Relational Database Systems

Matteo Brucato Juan Felipe Beltran Azza Abouzied Alexandra Meliou

College of Information and Computer Sciences Computer ScienceUniversity of Massachusetts New York University

Amherst, MA, USA Abu Dhabi, UAE{matteo,ameli}@cs.umass.edu {juanfelipe,azza}@nyu.edu

ABSTRACTTraditional database queries follow a simple model: they define con-straints that each tuple in the result must satisfy. This model is com-putationally efficient, as the database system can evaluate the queryconditions on each tuple individually. However, many practical,real-world problems require a collection of result tuples to satisfyconstraints collectively, rather than individually. In this paper, wepresent package queries, a new query model that extends traditionaldatabase queries to handle complex constraints and preferencesover answer sets. We develop a full-fledged package query system,implemented on top of a traditional database engine. Our workmakes several contributions. First, we design PaQL, a SQL-basedquery language that supports the declarative specification of pack-age queries. We prove that PaQL is at least as expressive as integerlinear programming, and therefore, evaluation of package queriesis in general NP-hard. Second, we present a fundamental evaluationstrategy that combines the capabilities of databases and constraintoptimization solvers to derive solutions to package queries. The coreof our approach is a set of translation rules that transform a packagequery to an integer linear program. Third, we introduce an offlinedata partitioning strategy allowing query evaluation to scale to largedata sizes. Fourth, we introduce SKETCHREFINE, a scalable algo-rithm for package evaluation, with strong approximation guarantees((1)6-factor approximation). Finally, we present extensive exper-iments over real-world and benchmark data. The results demonstratethat SKETCHREFINE is effective at deriving high-quality packageresults, and achieves runtime performance that is an order of mag-nitude faster than directly using ILP solvers over large datasets.

1. INTRODUCTIONTraditional database queries follow a simple model: they define

constraints, in the form of selection predicates, that each tuple in theresult must satisfy. This model is computationally efficient, as thedatabase system can evaluate each tuple individually to determinewhether it satisfies the query conditions. However, many practical,real-world problems require a collection of result tuples to satisfyconstraints collectively, rather than individually.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. To view a copyof this license, visit http://creativecommons.org/licenses/by-nc-nd/4.0/. Forany use beyond those covered by this license, obtain permission by [email protected] of the VLDB Endowment, Vol. 9, No. 7Copyright 2016 VLDB Endowment 2150-8097/16/03.

EXAMPLE 1 (MEAL PLANNER). A dietitian needs to designa daily meal plan for a patient. She wants a set of three gluten-freemeals, between 2,000 and 2,500 calories in total, and with a lowtotal intake of saturated fats.

EXAMPLE 2 (NIGHT SKY). An astrophysicist is looking forrectangular regions of the night sky that may potentially containpreviously unseen quasars. Regions are explored if their overallredshift is within some specified parameters, and ranked accordingto their likelihood of containing a quasar [17].

In these examples, there are some conditions that can be verifiedon individual data items (e.g., gluten content in a meal), while othersneed to be evaluated on a collection of items (e.g., total calories).Similar scenarios arise in a variety of application domains, such asinvestment planning, product bundles, course selection [25], teamformation [2, 21], vacation and travel planning [8], and computa-tional creativity [27]. Despite the clear application need, databasesystems do not currently offer support for these problems, andexisting work has focused on application- and domain-specific ap-proaches [2, 8, 21, 25].

In this paper, we present a domain-independent, database-centricapproach to address these challenges: We introduce a full-fledgedsystem that supports package queries, a new query model that ex-tends traditional database queries to handle complex constraintsand preferences over answer sets. Package queries are defined overtraditional relations, but return packages. A package is a collectionof tuples that (a) individually satisfy base predicates (traditionalselection predicates), and (b) collectively satisfy global predicates(package-specific predicates). Package queries are combinatorial innature: the result of a package query is a (potentially infinite) set ofpackages, and an objective criterion can define a preference rankingamong them.

Extending traditional database functionality to provide supportfor packages, rather than supporting packages at the applicationlevel, is justified by two reasons: First, the features of packagesand the algorithms for constructing them are not unique to eachapplication; therefore, the burden of package support should be liftedoff application developers, and database systems should supportpackage queries like traditional queries. Second, the data usedto construct packages typically resides in a database system, andpackages themselves are structured data objects that should naturallybe stored in and manipulated by a database system.

Our work in this paper addresses three important challenges:

1. Declarative specification of packages. SQL enables the declar-ative specification of properties that result tuples should satisfy. InExample 1, it is easy to specify the exclusion of meals with glutenusing a regular selection predicate in SQL. However, it is difficultto specify global constraints (e.g., total calories of a set of meals

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SQL Formulation ILP Formulation

Figure 1: Traditional database technology is ineffective at pack-age evaluation, and the runtime of the nave SQL formulation(Section 2) of a package query grows exponentially. In contrast,tools such as ILP solvers are more effective.

should be between 2,000 and 2,500 calories). Expressing such aquery in SQL requires either complex self-joins that explode thesize of the query, or recursion, which results in extremely complexqueries that are hard to specify and optimize (Section 2). Our goalis to maintain the declarative power of SQL, while extending itsexpressiveness to allow for the easy specification of packages.

2. Evaluation of package queries. Due to their combinatorialcomplexity, package queries are harder to evaluate than traditionaldatabase queries [9]. Package queries are in fact as hard as integerlinear programs (ILP) (Section 2.2). Existing database technologyis ineffective at evaluating package queries, even if one were toexpress them in SQL. Figure 1 shows the performance of evaluatinga package query expressed as a multi-way self-join query in tradi-tional SQL (described in detail in Section 2). As the cardinality ofthe package increases, so does the number of joins, and the runtimequickly becomes prohibitive: In a small set of 100 tuples from theSloan Digital Sky Survey dataset [28], SQL evaluation takes almost24 hours to construct a package of 7 tuples. Our goal is to extendthe database evaluation engine to take advantage of external tools,such as ILP solvers, which are more effective for combinatorialproblems.

3. Performance and scaling to large datasets. Integer program-ming solvers have two major limitations: they require the entireproblem to fit in main memory, and they fail when the problem istoo complex (e.g., too many variables and/or too many constraints).Our goal is to overcome these limitations through sophisticatedevaluation methods that allow solvers to scale to large data sizes.

In this paper, we address these challenges by designing languageand algorithmic support for package query specification and evalua-tion. Specifically, we make the following contributions.

We present PaQL (Package Query Language), a declarative lan-guage that provides simple extensions to standard SQL to supportconstraints at the package level. We prove that PaQL is at leastas expressive as integer linear programming, which implies thatevaluation of package queries is NP-hard (Section 2). We present a fundamental evaluation strategy DIRECT that com-

bines the capabilities of databases and constraint optimizationsolvers to derive solutions to package queries. The core of ourapproach is a set of translation rules that transform a packagequery to an integer linear program. This translation allows forthe use of highly-optimized external tools for the evaluation ofpackage queries (Section 3). We introduce an offline data partitioning strategy that allows

package query evaluation to scale to large data sizes. The coreof our evaluation strategy SKETCHREFINE lies on separating thepackage computation into multiple stages, each with small sub-problems, which the solver can evaluate efficiently. In the firststage, the algorithm sketches an initial sample package from

a set of representative tuples, while the subsequent stages re-fine the current package by solving an ILP within each partition.SKETCHREFINE guarantees a (1 )6-factor approximation forthe package results compared to DIRECT (Section 4).

We present an extensive experimental evaluation on both real-world data and the TPC-H benchmark (Section 5) that showsthat our query evaluation method SKETCHREFINE: (1) is ableto produce packages an order of magnitude faster than the ILPsolver used directly on the entire problem; (2) scales up to sizesthat the solver cannot manage directly; (3) produces packagesof very good quality in terms of objective value; (4) is robust topartitioning built in anticipation of different workloads.

2. LANGUAGE SUPPORT FOR PACKAGESData management systems do not natively support package

queries. While there are ways to express package queries in SQL,these are cumbersome and inefficient.Specifying packages with self-joins. When packages have strictcardinality (number of tuples), and only in this case, it is possible toexpress package queries using traditional self-joins. For instance,self-joins can express the query of Example 1 as follows:

SELECT * FROM Recipes R1, Recipes R2, Recipes R3WHERE R1.pk < R2.pk AND R2.pk < R3.pk AND

R1.gluten = free AND R2.gluten = free AND R3.gluten = freeAND R1.kcal + R2.kcal + R3.kcal BETWEEN 2.0 AND 2.5

ORDER BY R1.saturated_fat + R2.saturated_fat + R3.saturated_fat

This query is efficient only for constructing packages with verysmall cardinality: larger cardinality requires a larger number ofself-joins, quickly rendering evaluation time prohibitive (Figure 1).The benefit of this specification is that the optimizer can use thetraditional relational algebra operators, and augment its decisionswith package-specific strategies. However, this method does notapply for packages of unbounded cardinality.Using recursion in SQL. More generally, SQL can express packagequeries by generating and testing each possible subset of the inputrelation. This requires recursion to build a powerset table; checkingeach set in the powerset table for the query conditions will yield theresult packages. This approach has three major drawbacks. First,it is not declarative, and the specification is tedious and complex.Second, it is not amenable to optimization in existing systems. Third,it is extremely inefficient to evaluate, because the powerset tablegenerates an exponential number of candidates.

2.1 PaQL: The Package Query LanguageOur goal is to support package specification in a declarative and

intuitive way. In this section, we describe PaQL, a declarative querylanguage that introduces simple extensions to SQL to define packagesemantics and package-level constraints. We first show how PaQLcan express the query of Example 1, as our running example, todemonstrate the new language features:

Q: SELECT PACKAGE(R) AS PFROM Recipes R REPEAT 0WHERE R.gluten = freeSUCH THAT COUNT(P.) = 3 AND

SUM(P.kcal) BETWEEN 2.0 AND 2.5MINIMIZE SUM(P.saturated_fat)

Basic semantics. The new keyword PACKAGE differentiates PaQLfrom traditional SQL queries.

Q1: SELECT * Q2: SELECT PACKAGE(R) AS PFROM Recipes R FROM Recipes R

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The semantics of Q1 and Q2 are fundamentally different: Q1 isa traditional SQL query, with a unique, finite result set (the entireRecipes table), whereas there are infinitely many packages that sat-isfy the package query Q2: all possible multisets of tuples from theinput relation. The result of a package query like Q2 is a set of pack-ages. Each package resembles a relational table containing a collec-tion of tuples (with possible repetitions) from relation Recipes, andtherefore a package result of Q2 follows the schema of Recipes. ThePaQL syntax permits multiple relations in the FROM clause; in thatcase, the packages produced will follow the schema of the join result.

In the remainder of this paper, we focus on package queries with-out joins. This is for two reasons: (1) The join operation is part oftraditional SQL and can occur before package-specific computations.(2) There are important implications in the consideration of joinsthat extend beyond the scope of our work. Specifically, materializingthe join result is not always necessary, but rather, there are space-time trade-offs and system-level solutions that can improve queryperformance in the presence of joins. Section 4.5 offers a high-leveldiscussion of these points, but these extensions are orthogonal tothe techniques we present in this work.

Although semantically valid, a query like Q2 would not occur inpractice, as most application scenarios expect few, or even exactlyone result. We proceed to describe the additional constraints in theexample query Q that restrict the number of package results.

Repetition constraint. The REPEAT 0 statement in query Q spec-ifies that no tuple from the input relation can appear multiple timesin a package result. If this restriction is absent (as in query Q2),tuples can be repeated an unlimited number of times. By allowingno repetitions, Q restricts the package space from infinite to 2n,where n is the size of the input relation. Generalizing, the specifica-tion REPEAT K allows a package to repeat tuples up to K times,resulting in (2+K)n candidate packages.

Base and global predicates. A package query defines two typesof predicates. A base predicate, defined in the WHERE clause,is equivalent to a selection predicate and can be evaluated withstandard SQL: any tuple in the package needs to individually satisfythe base predicate. For example, query Q specifies the base predicate:R.gluten = free. Since base predicates directly filter input tuples,they are specified over the input relation R. Global predicates are thecore of package queries, and they appear in the new SUCH THATclause. Global predicates are higher-order than base predicates: theycannot be evaluated on individual tuples, but on tuple collections.Since they describe package-level constraints, they are specifiedover the package result P, e.g., COUNT(P.) = 3, which limits thequery results to packages of exactly 3 tuples.

The global predicates shown in query Q abbreviate aggregatesthat are in reality subqueries. For example, COUNT(P.) = 3, isan abbreviation for (SELECT COUNT() FROM P) = 3. Usingsubqueries, PaQL can express arbitrarily complex global constraintsamong aggregates over a package.

Objective clause. The objective clause specifies a ranking amongcandidate package results, and appears with either the MINIMIZEor MAXIMIZE keyword. It is a condition on the package-level,and hence it is specified over the package result P, e.g., MINIMIZESUM(P.saturated_fat). Similarly to global predicates, this formis a shorthand for MINIMIZE (SELECT SUM(saturated_fat)FROM P). A PaQL query with an objective clause returns a singleresult: the package that optimizes the value of the objective. Theevaluation methods that we present in this work focus on suchqueries. In prior work [6], we described preliminary techniques forreturning multiple packages in the absence of optimization objec-tives, but a thorough study of such methods is left to future work.

While PaQL can use arbitrary aggregate functions in the globalpredicates and the objective clause, in this work, we assume thatpackage queries are limited to linear functions. We defer the studyof non-linear functions and UDFs to future work.

2.2 Expressiveness and complexity of PaQLPackage queries are at least as hard as integer linear programs, as

the following theoretical results establish.

THEOREM 1 (EXPRESSIVENESS OF PAQL). Every integerlinear program can be expressed as a package query in PaQL.

We include the proofs of our theoretical results in the full version ofthe paper [5]. At a high level, the proof employs a reduction froman integer linear program to a PaQL query. The reduction maps thelinear constraints and objective into the corresponding PaQL clauses,using the constraint coefficients to generate the input relation forthe package query. As a direct consequence of Theorem 1, wealso obtain the following result about the complexity of evaluatingpackage queries.

COROLLARY 2 (COMPLEXITY OF PACKAGE QUERIES).Package queries are NP-hard.

In Section 3, we extend the result of Theorem 1 to also show thatevery PaQL query that does not contain non-linear functions can beexpressed as an integer linear program, through a set of translationrules. This transformation is the first step in package evaluation, but,due to the limitations of ILP solvers, it is not efficient or scalablein practice. To make package evaluation practical, we developSKETCHREFINE, a technique that augments the ILP transformationwith a partitioning mechanism, allowing package evaluation to scaleto large datasets (Section 4).

3. ILP FORMULATIONIn this section, we present an ILP formulation for package queries.

This formulation is at the core of our evaluation methods DIRECTand SKETCHREFINE. The results presented in this section areinspired by the translation rules employed by Tiresias [22] to answerhow-to queries. However, there are several important differencesbetween how-to and package queries, which we discuss extensivelyin the overview of the related work (Section 6).

3.1 PaQL to ILP TranslationLet R indicate the input relation, n = |R| the number of tuples

in R, R.attr an attribute of R, P a package, f a linear aggregatefunction (such as COUNT and SUM), {,} a constraintinequality, and v R a constant. For each tuple ti from R, 1 i n,the ILP problem includes a nonnegative integer variable xi (xi 0),indicating the number of times ti is included in an answer package.We also use x = x1,x2, . . . ,xn to denote the vector of all integervariables. A PaQL query is formulated as an ILP problem using thefollowing translation rules:

1. Repetition constraint. The REPEAT keyword, expressible inthe FROM clause, restricts the domain that the variables can takeon. Specifically, REPEAT K implies 0 xi K+1.

2. Base predicate. Let be a base predicate, e.g., R.gluten= free,and R the relation containing tuples from R satisfying . Weencode by setting xi = 0 for every tuple ti 6 R.

3. Global predicate. Each global predicate in the SUCH THATclause takes the form f (P) v. For each such predicate, wederive a linear function f (x) over the integer variables. A cardi-nality constraint f (P) = COUNT(P.) is linearly translated into

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f (x) = i xi. A summation constraint f (P) = SUM(P.attr) islinearly translated into f (x) = i(ti.attr)xi. We further illustratethe translation with two non-trivial examples: AVG(P.attr) v is translated as

i(ti.attr)xi/i xi v i(ti.attr v)xi 0

(SELECT COUNT() FROM P WHERE P.carbs > 0) (SELECT COUNT() FROM P WHERE P.protein 5) istranslated as

Rc := {ti R | ti.carbs> 0}Rp := {ti R | ti.protein 5}1Rc(ti) := 1 if ti Rc; 0 otherwise.1Rp(ti) := 1 if ti Rp; 0 otherwise.i(1Rc(ti)1Rp(ti))xi 0

General Boolean expressions over the global predicates can beencoded into a linear program with the help of Boolean variablesand linear transformation tricks found in the literature [4].

4. Objective clause. We encode MAXIMIZE f (P) as max f (x),where f (x) is the encoding of f (P). Similarly MINIMIZE f (P)is encoded as min f (x). If the query does not include an objectiveclause, we add the vacuous objective maxi 0 xi.

We call the relations R, Rc, and Rp described above base re-lations. This formulation, together with Theorem 1, shows thatsingle-relation package queries with linear constraints correspondexactly to ILP problems.

3.2 Query Evaluation with DIRECTUsing the ILP formulation, we develop our basic evaluation

method for package queries, called DIRECT. We later extend thistechnique to our main algorithm, SKETCHREFINE, which supportsefficient package evaluation in large data sets (Section 4).

Package evaluation with DIRECT employs three simple steps:

1. ILP formulation. We transforms a PaQL query to an ILP prob-lem using the rules described in Section 3.1.

2. Base relations. We compute the base relations, such as R, Rc,and Rp, with a series of standard SQL queries, one for each, or bysimply scanning R once and populating these relations simulta-neously. After this phase, all variables xi such that xi = 0 can beeliminated from the ILP problem because the corresponding tupleti cannot appear in any package solution. This can significantlyreduce the size of the problem.

3. ILP execution. We employ an off-the-shelf ILP solver, as a blackbox, to get a solution xi for all the integer variables xi of theproblem. Each xi informs the number of times tuple ti should beincluded in the answer package.

The DIRECT algorithm has two crucial drawbacks. First, it isonly applicable if the input relation is small enough to fit entirelyin main memory: ILP solvers, such as IBMs CPLEX, require theentire problem to be loaded in memory before execution. Second,even for problems that fit in main memory, this approach may faildue to the complexity of the integer problem. In fact, integer linearprogramming is a notoriously hard problem, and modern ILP solversuse algorithms, such as branch-and-cut [24], that often perform wellin practice, but can choke even on small problem sizes due totheir exponential worst-case complexity [7]. This may result inunreasonable performance due to solvers using too many resources(main memory, virtual memory, CPU time), eventually thrashingthe entire system.

Algorithm 1 Scalable Package Query Evaluation1: procedure SKETCHREFINE(P:Partitioning, Q:Package Query)2: pS SKETCH(P, Q)3: if failure then4: return infeasible5: else6: p REFINE(pS, P, Q)7: if failure then8: return infeasible9: else

10: return p

4. SCALABLE PACKAGE EVALUATIONIn this section, we present SKETCHREFINE, an approximate

divide-and-conquer evaluation technique for efficiently answeringpackage queries on large datasets. SKETCHREFINE smartly decom-poses a query into smaller queries, formulates them as ILP problems,and employs an ILP solver as a black-box evaluation method to an-swer each individual query. By breaking down the problem intosmaller subproblems, the algorithm avoids the drawbacks of theDIRECT approach. Further, we prove that SKETCHREFINE is guar-anteed to always produce feasible packages with an approximateobjective value (Section 4.3).

The algorithm is based on an important observation: similar tu-ples are likely to be interchangeable within packages. A group ofsimilar tuples can therefore be compressed to a single representa-tive tuple for the entire group. SKETCHREFINE sketches an initialanswer package using only the set of representative tuples, which issubstantially smaller than the original dataset. This initial solution isthen refined by evaluating a subproblem for each group, iterativelyreplacing the representative tuples in the current package solutionwith original tuples from the dataset. Figure 2 provides a high-levelillustration of the three main steps of SKETCHREFINE:

1. Offline partitioning (Section 4.1). The algorithm assumes a par-titioning of the data into groups of similar tuples. This partitioningis performed offline (not at query time), and our experimentsshow that SKETCHREFINE remains very effective even withpartitionings that do not match the query workload (Section 5.2.3).In our implementation, we partition data using k-dimensionalquad trees [11], but other partitioning schemes are possible.

2. Sketch (Section 4.2.1). SKETCHREFINE sketches an initialpackage by evaluating the package query only over the set ofrepresentative tuples.

3. Refine (Section 4.2.2). Finally, SKETCHREFINE transformsthe initial package into a complete package by replacing eachrepresentative tuple with some of the original tuples from thesame group, one group at a time.

SKETCHREFINE always constructs approximate feasible packages,i.e., packages that satisfy all the query constraints, but with a pos-sibly sub-optimal objective value that is guaranteed to be withincertain approximation bounds (Section 4.3). SKETCHREFINE maysuffer from false infeasibility, which happens when the algorithmreports a feasible query to be infeasible. The probability of falseinfeasibility is, however, low and bounded (Section 4.4).

In the subsequent discussion, we use R to denote the input rela-tion of n tuples, ti R, 1 i n. R is partitioned into m groupsG1, . . . ,Gm. Each group G j, 1 j m, has a representative tuplet j, which may not always appear in R. We denote the partitionedspace with P= {(G j, t j) | 1 j m}. We refer to packages thatcontain some representative tuples as sketch packages and packageswith only original tuples as complete packages (or simply packages).

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G4(b) Initial query usingrepresentative tuples

(c) Initial package (e) Skipping G2 (g) Refinement query for group G4

(h) Final approximate package

REFINEPARTITION SKETCH

(d) Refinement query for group G1

(f) Refinement query for group G3

(a) Original tuples

Multiplicity of representativetuples in the initial package

Representative and original tuples selected during previous steps, shown by hatching lines, are aggregated and used to modify later refinement queries

Figure 2: The original tuples (a) are partitioned into four groups and a representative is constructed for each group (b). The initialsketch package (c) contains only representative tuples, with possible repetitions up the size of each group. The refine query for groupG1 (d) involves the original tuples from G1 and the aggregated solutions to all other groups (G2, G3, and G4). Group G2 can beskipped (e) because no representatives could be picked from it. Any solution to previously refined groups are used while refining thesolution for the remaining groups (f and g). The final approximate package (h) contains only original tuples.

We denote a complete package with p and a sketch package withpS, where S P is the set of groups that are yet to be refined totransform pS into a complete answer package p.

4.1 Offline PartitioningSKETCHREFINE relies on an offline partitioning of the input

relation R into groups of similar tuples. Partitioning is based on aset of k numerical partitioning attributes, A, from the input relationR, and uses two parameters: a size threshold and a radius limit.

DEFINITION 1 (SIZE THRESHOLD, ). The size threshold ,1 n, restricts the size of each partitioning group G j , 1 j m,to a maximum of original tuples, i.e., |G j| .

DEFINITION 2 (RADIUS LIMIT, ). The radius r j 0 of agroup G j is the greatest absolute distance between the representa-tive tuple of G j , t j , and every original tuple of the group, across allpartitioning attributes:

r j = maxt jG jattrA

|t j.attr t j.attr|

The radius limit , 0, requires that for every partitioning groupG j, 1 j m, r j .

The size threshold, , affects the number of clusters, m, as smallerclusters (lower ) implies more of them (larger m), especially onskewed datasets. As we discuss later (Section 4.2), for best re-sponse time of SKETCHREFINE, should be set so that both mand are small. Our experiments show that a proper setting canyield to an order of magnitude improvement in query response time(Section 5.2.2). The radius limit, , should be set according to adesired approximation guarantee (Section 4.3). Note that the samepartitioning can be used to support a multitude of queries over thesame dataset. In our experiments, we show that a single partitioningperforms consistently well across different queries.Partitioning method. Different methods can be used for parti-tioning. Our implementation is based on k-dimensional quad-treeindexing [11]. The method recursively partitions a relation intogroups until all the groups satisfy the size threshold and meet theradius limit. First, relation R is augmented with an extra group IDcolumn gid, such that ti.gid= j iff tuple ti is assigned to group G j.The procedure initially creates a single group G1 that includes allthe original tuples from relation R, by initializing ti.gid= 1 for alltuples. Then, it recursively proceeds as follows: The procedure computes the sizes and radii of the current groups

via a query that groups tuples by their gid value. The same group-by query also computes the centroid tuple of each group. The

centroid is computed by averaging the tuples in the group on eachof the partitioning attributes A.

If group G j has more tuples than the size threshold, or a radiuslarger than the radius limit, the tuples in group G j are partitionedinto 2k subgroups (k = |A|). The groups centroid is used as thepivot point to generate sub-quadrants: tuples that reside in thesame sub-quadrant are grouped together.

Our method recursively executes two SQL queries on each subgroupthat violates the size or the radius condition. In the last iteration, thelast group-by query computes the centroids for each group. Theseare the representative tuples, t j, 1 j m, and are stored in a newrepresentative relation R(gid,attr1, . . . ,attrk).

Alternative partitioning approaches. We experimented with dif-ferent clustering algorithms, such as k-means [15], hierarchicalclustering [20] and DBSCAN [10], using off-the-shelf libraries suchas Scikit-learn [26]. Existing clustering algorithms present variousproblems: First, they tend to vary substantially in the properties ofthe generated clusters. In particular, none of the existing clusteringtechniques can natively generate clusters that satisfy the size thresh-old and radius limit . In fact, most of the clustering algorithmstake as input the number of clusters to generate, without offering anymeans to restrict the size of each cluster nor their radius. Second,existing implementations only support in-memory cluster computa-tion, and DBMS-oriented implementations usually need complexand inefficient queries. On the other hand, space partitioning tech-niques from multi-dimensional indexing, such as k-d trees [3] andquad trees [11], can be more easily adapted to satisfy the size andradius conditions, and to work within the database: our partitioningmethod works directly on the input table via simple SQL queries.

One-time cost. Partitioning is an expensive procedure. To avoidpaying its cost at query time, the dataset is partitioned in advanceand used to answer a workload of package queries. For a knownworkload, our experiments show that partitioning the dataset on theunion of all query attributes provides the best performance in termsof query evaluation time and approximation error for the computedanswer package (Section 5.2.3). We also demonstrate that our queryevaluation approach is robust to a wide range of partition sizes, andto imperfect partitions that cover more or fewer attributes than thoseused in a particular query. This means that, even without a knownworkload, a partitioning performed on all of the data attributes stillprovides good performance.

The radius limit is necessary for the theoretical guarantee of theapproximation bounds (Section 4.3). However, we show empiricallythat partitioning satisfying the size threshold alone produces satisfac-

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tory answers while reducing the offline partitioning cost: Meetinga size threshold requires fewer partitioning iterations than meetinga radius limit especially if the dataset is sparse across the attributedomains (Section 5).Dynamic partitioning. In our implementation of SKETCHREFINE,the choice of partitioning is static. Our technique also works with adynamic approach to partitioning: by maintaining the entire hierar-chical structure of the quad-tree index, one can traverse the indexat query time to generate the coarsest partitioning that satisfies therequired radius condition. However, our empirical results show thatthis approach incurs unnecessary overhead, as static partitioningalready performs extremely well in practice (Section 5).

4.2 Query Evaluation with SKETCHREFINEDuring query evaluation, SKETCHREFINE first sketches a package

solution using the representative tuples (SKETCH), and then it refinesit by replacing representative tuples with original tuples (REFINE).We describe these steps using the example query Q from Section 2.1.

4.2.1 SKETCHUsing the representative relation R (Section 4.1), the SKETCH

procedure constructs and evaluates a sketch query, Q[R]. The resultis an initial sketch package, pS, containing representative tuples thatsatisfy the same constraints as the original query Q:

Q[R]: SELECT PACKAGE(R) AS pSFROM R

WHERE R.gluten = freeSUCH THAT

COUNT(pS.) = 3 ANDSUM(pS.kcal) BETWEEN 2.0 AND 2.5 AND(SELECT COUNT() FROM pS WHERE gid = 1) |G1|AND . . .(SELECT COUNT() FROM pS WHERE gid = m) |Gm|

MINIMIZE SUM(pS.saturated_fat)

The new global constraints, highlighted in bold, ensure that everyrepresentative tuple does not appear in pS more times than thesize of its group, G j. This accounts for the repetition constraintREPEAT 0 in the original query. Generalizing, with REPEAT K,each t j can be repeated up to |G j|(1+K) times. These constraintsare simply omitted from Q[R] if the original query does not containa repetition constraint.

Since the representative relation R contains exactly m representa-tive tuples, the ILP problem corresponding to this query has onlym variables. This is typically small enough for the black box ILPsolver to manage directly, and thus we can solve this package queryusing the DIRECT method (Section 3.2). If m is too large, we cansolve this query recursively with SKETCHREFINE: the set of mrepresentatives is further partitioned into smaller groups until thesubproblems reach a size that can be efficiently solved directly.

The SKETCH procedure fails if the sketch query Q[R] is infeasi-ble, in which case SKETCHREFINE reports the original query Q asinfeasible (Algorithm 1). This may constitute false infeasibility, ifQ is actually feasible. In Section 4.4, we show that the probabilityof false infeasibility is low and bounded, and we present simplemethods to avoid this outcome.

4.2.2 REFINEUsing the sketched solution over the representative tuples, the

REFINE procedure iteratively replaces the representative tuples withtuples from the original relation R, until no more representatives arepresent in the package. The algorithm refines the sketch package pS,one group at a time: For a group G j with representative t j pS, the

algorithm derives package p j from pS by eliminating all instancesof t j; it then seeks to replace the eliminated representatives withactual tuples, by issuing a refine query, Q[G j], on group G j:

Q[G j]: SELECT PACKAGE(G j) AS p jFROM G j REPEAT 0WHERE G j.gluten = freeSUCH THAT

COUNT(p j.) + COUNT(p j.) = 3 ANDSUM(p j.kcal) + SUM(p j.kcal) BETWEEN 2.0 AND 2.5

MINIMIZE SUM(p j.saturated_fat)

The query derives a set of tuples p j, as a replacement for the oc-currences of the representatives of G j in pS. The global constraintsin Q[G j] ensure that the combination of tuples in p j and p j satisfythe original query Q. Thus, this step produces the new refined sketchpackage pS = p j p j, where S

= S\{(G j, t j)}.Since G j has at most tuples, the ILP problem corresponding to

Q[G j] has at most variables. This is typically small enough for theblack box ILP solver to solve directly, and thus we can solve thispackage query using the DIRECT method (Section 3.2). Similarly tothe sketch query, if is too large, we can solve this query recursivelywith SKETCHREFINE: the tuples in group G j are further partitionedinto smaller groups until the subproblems reach a size that can beefficiently solved directly.

Ideally, the REFINE step will only process each group with repre-sentatives in the initial sketch package once. However, the order ofrefinement matters as each refinement step is greedy: it selects tuplesto replace the representatives of a single group, without consideringthe effects of this choice on other groups. As a result, a particularrefinement step may render the query infeasible (no tuples from theremaining groups can satisfy the constraints). When this occurs,REFINE employs a greedy backtracking strategy that reconsidersgroups in a different order.Greedy backtracking. REFINE activates backtracking when it en-counters an infeasible refine query, Q[G j]. Backtracking greedilyprioritizes the infeasible groups. This choice is motivated by a sim-ple heuristic: if the refinement on G j fails, it is likely due to choicesmade by previous refinements; therefore, by prioritizing G j, wereduce the impact of other groups on the feasibility of Q[G j]. Thisheuristic does not affect the approximation guarantees (Section 4.3).

Algorithm 2 details the REFINE procedure. The algorithm log-ically traverses a search tree (which is only constructed as newbranches are created and new nodes visited), where each node corre-sponds to a unique sketch package pS. The traversal starts from theroot, corresponding to the initial sketch package, where no groupshave been refined (S = P), and finishes at the first encounteredleaf, corresponding to a complete package (S= /0). The algorithmterminates as soon as it encounters a complete package, which itreturns (line 3). The algorithm assumes a (initially random) re-finement order for all groups in S, and places them in a priorityqueue (line 6). During refinement, this group order can change byprioritizing groups with infeasible refinements (line 24).Run time complexity. In the best case, all refine queries are feasibleand the algorithm never backtracks. In this case, the algorithmmakes up to m calls to the ILP solver to solve problems of size up to, one for each refining group. In the worst case, SKETCHREFINEtries every group ordering leading to an exponential number of callsto the ILP solver. Our experiments show that the best case is themost common and backtracking occurs infrequently.

4.3 Approximation GuaranteesSKETCHREFINE provides strong theoretical guarantees. We

prove that for a desired approximation parameter , we can de-rive a radius limit for the offline partitioning that guarantees

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Algorithm 2 Greedy Backtracking Refinement1: procedure REFINE(pS, P, Q)2: if S= /0 then . Base case: all groups already refined3: return pS4: F /0 . Failed groups5: . Arrange S in some initial order (e.g., random)6: U priorityQueue(S)7: while U 6= /0 do8: (G j, t j) dequeue(U)9: . Skip groups that have no representative in pS

10: if t j / pS then11: continue12: p

S Refine pS on group G j13: if Q[G j] is infeasible then14: if S 6= P then . If pS is not the initial package15: . Greedily backtrack with non-refinable group16: F F{(G j, t j)}17: return failure(F)18: else19: . Greedily recurse with refinable group20: p REFINE(p

S ,P,Q)

21: if failure(F) then22: F FF23: . Greedily prioritize non-refinable groups24: prioritize(U,F)25: else26: return p27: . None of the groups in S can be refined (invariant: F = S)28: return failure(F)

that SKETCHREFINE will produce a package with objective value(1 )6-factor close to the objective value of the solution generatedby DIRECT for the same query.

THEOREM 3 (APPROXIMATION BOUNDS). For any feasiblepackage query with a maximization (minimization, resp.) objec-tive and approximation parameter , 0 < 1 ( 0, resp.), anydatabase instance, any set of partitioning attributes A, supersetof the numerical query attributes, any size threshold , and radiuslimit:

= min1 jmattrA

|t j.attr|, where = ( = 1+ , resp.) (1)

The package produced by SKETCHREFINE (if any) is guaranteed tohave objective value (1 )6OPT ( (1+ )6OPT , resp.), whereOPT is the objective value of the DIRECT solution.

At a high level, the proof includes two steps: In the first step,we prove that the initial sketch package is a (1 )3-approximationwith respect to DIRECT; In the second step, we prove that the finalpackage produced by SKETCHREFINE is a (1 )3-approximationwith respect to the initial sketch package.

In general, enforcing this radius limit, especially with lower values, may cause the resulting partitions to become excessivelysmall. While still obeying the approximation guarantees, this couldincrease the number of resulting partitions and thus degrade therunning time performance of SKETCHREFINE. In the worst case,SKETCHREFINE could degenerate into DIRECT. This is an im-portant trade-off between running time and quality that we haveobserved in our experimental data as well, and it is a very commoncharacteristic of most approximation schemes [30].

4.4 The Case of False InfeasibilityFor a feasible query Q, false negatives, or false infeasibility, may

happen in two cases: (1) when the sketch query Q[R] is infeasible;

(2) when greedy backtracking fails (possibly due to suboptimalpartitioning). In both cases, SKETCHREFINE would (incorrectly)report a feasible package query as infeasible. False negatives are,however, extremely rare, as the following theorem establishes.

THEOREM 4 (FALSE INFEASIBILITY). For any feasible pack-age query, any database instance, any set of partitioning attributesA that is a superset of the query attributes, any size threshold , andany radius limit , SKETCHREFINE finds a feasible package withhigh probability that inversely depends on query selectivity.

The selectivity of a package query denotes the probability of arandom package being infeasible (thus, lower selectivity implieshigher probability of a random package being feasible). To provethis result, we show that if a random package is feasible, thenwith high probability the sketch query and all refine queries arefeasible. Thus, lower selectivity implies higher probability thatSKETCHREFINE successfully terminates with a feasible package.

We discuss four potential ways to deal with false infeasibility,which we plan to explore in future work:

1. Hybrid sketch query. A simple method to avoid false infeasi-bility in the SKETCH step is to merge the sketch query Q[R] withone of the refine queries. This hybrid sketch query would selectoriginal tuples for one of the groups, and at the same time selectrepresentative tuples for the remaining groups. Groups can be triedin any order (e.g., in random order), until one of the hybrid sketchqueries is feasible.

2. Further partitioning. In some cases, the centroid of a group maynot be a very good representative (e.g., when the data is skewed).This can sometimes result in false negatives. Further partitioning byreducing the size threshold may eliminate the problem.3. Dropping partitioning attributes. A more principled approachconsists of projecting the partitioning onto fewer dimensionsby reducing the number of partitioning attributes. In doing so,some groups merge, increasing the chance that previously infeasiblegroups become feasible. The choice of attributes to remove fromthe partitioning can be guided by the last infeasible ILP problem.Most ILP solvers provide functionality to identify a minimal set ofinfeasible constraints: removing any constraint from the set makesthe problem feasible.1 Removing the attributes that participate inthese constraints from the partitioning can increase the odds ofdiscovering a feasible solution.

4. Iterative group merging. In a brute-force approach, we canmerge groups iteratively, until the sub-queries become feasible. Inthe worst case, this process reduces the problem to the originalproblem (i.e., with no partitioning), and thus it is guaranteed to finda solution to any feasible query, at the cost of performance.

4.5 DiscussionSKETCHREFINE is an evaluation strategy for package queries

with three important advantages. First, it scales naturally to verylarge datasets, by breaking down the problem into smaller, manage-able subproblems, whose solutions can be iteratively combined toform the final result. Second, it provides flexible approximationswith strong theoretical guarantees on the quality of the packageresults. Third, while our current implementation of SKETCHREFINEemploys ILP solvers to evaluate the generated subproblems, ouralgorithm can use any other black box solution for package queries,even solutions that work entirely in main memory, and whose effi-ciency drastically degrades with larger problem sizes. We plan toexplore these alternatives in our future work.1This set is usually referred to as irreducible infeasible set (IIS).

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Parallelizing SKETCHREFINE. Its data partitioning and problemdivision strategies give SKETCHREFINE great potential for par-allelization. However, the proper parallelization strategy is non-obvious, and is a nontrivial part of future work. A simple paralleliza-tion strategy could perform refinement on several groups in parallel.However, since refinements make local decisions, this process ismore likely to reach infeasibility, requiring costly backtracking stepsand resulting in wasted computation. Alternatively, parallelizationmay focus on the backtracking process, using additional resourcesto evaluate different group orderings in parallel.

Handling joins. In this paper, we focused on package queriesover single relations. Handling joins poses interesting implications,orthogonal to the evaluation techniques that we presented in thiswork. In the presence of joins, the system can simply evaluateand materialize the join result before applying the package-specifictransformations. However, the materialization of the join result isnot always necessary: DIRECT generates variables through a singlesequential scan of the join result, and thus the join tuples can bepipelined into the ILP generation without being materialized. How-ever, not materializing the join results means that some of the jointuples will need to be recomputed to populate the solution package.Therefore, there is a space-time trade-off in the consideration ofmaterializing the join. Further, this trade-off can be improved withhybrid, system-level solutions, such as storing the record IDs ofjoining tuples to enable faster access during package generation.These considerations are well-beyond our current scope, and areorthogonal to the techniques that we present in this work.

5. EXPERIMENTAL EVALUATIONIn this section, we present an extensive experimental evaluation

of our techniques for package query execution, both on real-worldand on benchmark data. Our results show the following propertiesof our methods: (1) SKETCHREFINE evaluates package queries anorder of magnitude faster than DIRECT; (2) SKETCHREFINE scalesup to sizes that DIRECT cannot handle directly; (3) SKETCHREFINEproduces packages of high quality (similar objective value as thepackages returned by DIRECT); (4) the performance of SKETCHRE-FINE is robust to partitioning on different sets of attributes as longas a querys attributes are mostly covered. This makes offline parti-tioning effective for entire query workloads.

5.1 Experimental SetupSoftware. We implemented our package evaluation system as alayer on top of a traditional relational DBMS. The data itself residesin the database, and the system interacts with the DBMS via SQLwhen it needs to perform operations on the data. We use PostgreSQLv9.3.9 for our experiments. The core components of our evaluationmodule are implemented in Python 2.7. The PaQL parser is gener-ated in C++ from a context-free grammar, using GNU Bison [12].We represent a package in the relational model as a standard rela-tion with schema equivalent to the schema of the input relation. Apackage is materialized into the DBMS only when necessary (forexample, to compute its objective value).

We employ IBMs CPLEX [16] v12.6.1 as our black-box ILPsolver. When the algorithm needs to solve an ILP problem, the cor-responding data is retrieved from the DBMS and passed to CPLEXusing tuple iterator APIs to avoid having more than one copy of thesame data stored in main memory at any time. We used the samesettings for all solver executions: we set its working memory to512MB; we instructed CPLEX to store exceeding data used duringthe solve procedure on disk in a compressed format, rather than

TPC-H query Q1 Q2 Q3 Q4 Q5 Q6 Q7Max # of tuples 6M 6M 6M 6M 240k 11.8M 6M

Figure 3: Size of the tables used in the TPC-H benchmark.

Dataset Dataset size Size threshold Partitioning timeGalaxy 5.5M tuples 550k tuples 348 sec.TPC-H 17.5M tuples 1.8M tuples 1672 sec.

Figure 4: Partitioning time for the two datasets, using the work-load attributes and with no radius condition.

using the operating systems virtual memory, which, as per the doc-umentation, may degrade the solvers performance; we instructedCPLEX to emphasize optimality versus feasibility to dampen the ef-fect of internal heuristics that the solver may employ on particularlyhard problems; we enabled CPLEXs memory emphasis parameter,which instructs the solver to conserve memory where possible; weset a solving time limit of one hour; we also made sure that theoperating system would kill the solver process whenever it uses theentire available main memory.

Environment. We run all experiments on a ProLiant DL160 G6server equipped with two twelve-core Intel Xeon X5650 CPUs at2.66GHz each, with 15GB or RAM, with a single 7200 RPM 500GBhard drive, running CentOS release 6.5.

Datasets and queries. We demonstrate the performance of ourquery evaluation methods using both real-world and benchmarkdata. The real-world dataset consists of approximately 5.5 milliontuples extracted from the Galaxy view of the Sloan Digital SkySurvey (SDSS) [28], data release 12. For the benchmark datasetswe used TPC-H [29], with table sizes up to 11.8 million tuples.

For each of the two datasets, we constructed a set of seven pack-age queries, by adapting existing SQL queries originally designedfor each of the two datasets. For the Galaxy dataset, we adaptedsome of the real-world sample SQL queries available directly fromthe SDSS website.2 For the TPC-H dataset, we adapted seven of theSQL query templates provided with the benchmark that containedenough numerical attributes. We performed query specificationmanually, by transforming SQL aggregates into global predicates orobjective criteria whenever possible, selection predicates into globalpredicates, and by adding cardinality bounds. We did not includeany base predicates in our package queries because they can alwaysbe pre-processed by running a standard SQL query over the inputdataset (Section 3), and thus eliminated beforehand. For the Galaxyqueries, we synthesized the global constraint bounds by multiplyingthe original selection constraint bound by the expected size of thefeasible packages. For the TPC-H queries, we generated global con-straint bounds uniformly at random by multiplying random valuesin the value range of a specific attribute by the expected size ofthe feasible packages. The original TPC-H SQL queries involveattributes across different relations and compute various group-byaggregates. In order to transform these queries into single-relationpackage queries, we processed the original TPC-H tables to producea single pre-joined table, obtained with full outer joins, containingall attributes needed by all the TPC-H package queries in our bench-mark. This table contained approximately 17.5 million tuples. Foreach TPC-H package query, we then extracted the subset of tupleshaving non-NULL values on all the query attributes. The size ofeach resulting table is reported in Figure 3.

Comparisons. We compare DIRECT with SKETCHREFINE. Bothmethods use the ILP formulation (Section 3) to transform package

2http://cas.sdss.org/dr12/en/help/docs/realquery.aspx

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Figure 6: Scalability on the TPC-H benchmark. SKETCHREFINE uses an offline partitioning computed on the full dataset, using theworkload attributes, = 10% of the dataset size, and no radius condition. DIRECT scales up to millions of tuples in all queries. Theresponse time of SKETCHREFINE is about an order of magnitude less than DIRECT, and its approximation ratio is generally verylow, even though the partitioning is constructed without radius condition.

queries into ILP problems: DIRECT translates and solves the orig-inal query; SKETCHREFINE translates and solves the sub-queries(Section 4), and uses hybrid sketch query (Section 4.4) as the onlystrategy to cope with infeasible initial queries.

Metrics. We evaluate methods on their efficiency and effectiveness.Response time: We measure response time as wall-clock time togenerate an answer package. This includes the time taken to translatethe PaQL query into one or several ILP problems, the time taken toload the problems into the solver, and the time taken by the solver toproduce a solution. We exclude the time to materialize the packagesolution to the database and to compute its objective value.

Approximation ratio: Recall that SKETCHREFINE is always guar-anteed to return an approximate answer with respect to DIRECT(Section 4.3). In order to assess the quality of a package returned bySKETCHREFINE, we compare its objective value with the objectivevalue of the package returned by DIRECT on the same query. UsingOb jS and Ob jD to denote the objective values of SKETCHREFINEand DIRECT, respectively, we compute the empirical approximationratio Ob jDOb jS for maximization queries, and

Ob jSOb jD for minimization

queries. An approximation ratio of one indicates that SKETCHRE-FINE produces a solution with same objective value as the solutionproduced by the solver on the entire problem. Typically, the ap-proximation ratio is greater than or equal to one. However, sincethe solver employs several approximations and heuristics, valueslower than one, which means that SKETCHREFINE produces a betterpackage than DIRECT, are possible in practice.

5.2 Results and DiscussionWe evaluate three fundamental aspects of our algorithms: (1) their

query response time and approximation ratio with increasing datasetsizes; (2) the impact of varying partitioning size thresholds () on

SKETCHREFINEs performance; (3) the impact of the attributes usedin offline partitioning on query runtime.

5.2.1 Query performance as data set size increasesIn our first set of experiments, we evaluate the scalability of our

methods on input relations of increasing size. First, we partitionedeach dataset using the union of all package query attributes in theworkload: we refer to these partitioning attributes as the workloadattributes. We did not enforce a radius condition () during parti-tioning for two reasons: (1) to show that an offline partitioning canbe used to answer efficiently and effectively both maximization andminimization queries, even though they would normally require dif-ferent radii; (2) to demonstrate the effectiveness of SKETCHREFINEin practice, even without having theoretical guarantees in place.

We perform offline partitioning setting the partition size threshold to 10% of the dataset size. Figure 4 reports the partitioning timesfor the two datasets. We derive the partitionings for the smaller datasizes (less than 100% of the dataset) in the experiments, by randomlyremoving tuples from the original partitions. This operation isguaranteed to maintain the size condition.

Figures 5 and 6 report our scalability results on the Galaxy andTPC-H benchmarks, respectively. The figures display the queryruntimes in seconds on a logarithmic scale, averaged across 10 runsfor each datapoint. At the bottom of each figure, we also report themean and median approximation ratios across all dataset sizes. Thegraph for Q2 on the galaxy dataset does not report approximationratios, because DIRECT evaluation fails to produce a solution forthis query across all data sizes. We observe that DIRECT can scaleup to millions of tuples in three of the seven Galaxy queries, andin all of the TPC-H queries. Its run-time performance degrades, asexpected, when data size increases, but even for very large datasetsDIRECT is usually able to answer the package queries in less than

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Figure 8: Impact of partition size threshold on the TPC-H benchmark, using the full datasets. Partitioning is performed at eachvalue of using all the workload attributes, and with no radius condition. The baseline DIRECT and the approximation ratios areonly shown when DIRECT is successful. The results show that has a major impact on the running time of SKETCHREFINE, butalmost no impact on the approximation ratio. DIRECT can be an order of magnitude faster than DIRECT with proper tuning of .

a few minutes. However, DIRECT has high failure rate for some ofthe Galaxy queries, indicated by the missing data points in somegraphs (queries Q2, Q3, Q6 and Q7 in Figure 5). This happens whenCPLEX uses the entire available main memory while solving thecorresponding ILP problems. For some queries, such as Q3 and Q7,this occurs with bigger dataset sizes. However, for queries Q2 andQ6, DIRECT even fails on small data. This is a clear demonstrationof one of the major limitations of ILP solvers: they can fail evenwhen the dataset can fit in main memory, due to the complexity of theinteger problem. In contrast, our scalable SKETCHREFINE algorithmis able to perform well on all dataset sizes and across all queries.SKETCHREFINE consistently performs about an order of magnitudefaster than DIRECT across all queries, both on real-world data andbenchmark data. Its running time is consistently below one or twominutes, even when constructing packages from millions of tuples.

Both the mean and median approximation ratios are very low,usually all close to one or two. This shows that the substantialgain in running time of SKETCHREFINE over DIRECT does notcompromise the quality of the resulting packages. Our results in-dicate that the overhead of partitioning with a radius condition isoften unnecessary in practice. Since the approximation ratio is notenforced, SKETCHREFINE can potentially produce bad solutions,but this happens rarely. In our experiments, this only occurred withquery Q2 from the TPC-H benchmark.

5.2.2 Effect of varying partition size thresholdThe size of each partition, controlled by the partition size thresh-

old , is an important factor that can impact the performance ofSKETCHREFINE: Larger partitions imply fewer but larger subprob-lems, and smaller partitions imply more but smaller subproblems.Both cases can significantly impact the performance of SKETCHRE-FINE. In our second set of experiments, we vary , which is used

during partitioning to enforce the size condition (Section 4.1), tostudy its effects on the query response time and the approximationratio of SKETCHREFINE. In all cases, along the lines of the previousexperiments, we do not enforce a radius condition. Figures 7 and 8show the results obtained on the Galaxy and TPC-H benchmarks,using 30% and 100% of the original data, respectively. We vary from higher values corresponding to fewer but larger partitions,on the left-hand size of the x-axis, to lower values, correspondingto more but smaller partitions. When DIRECT is able to producea solution, we also report its running time (horizontal line) as abaseline for comparison.

Our results show that the partition size threshold has a majorimpact on the execution time of SKETCHREFINE, with extremevalues of (either too low or too high) often resulting in slowerrunning times than DIRECT. With bigger partitions, on the left-handside of the x-axis, SKETCHREFINE takes about the same time asDIRECT because both algorithms solve problems of comparable size.When the size of each partition starts to decrease, moving from left toright on the x-axis, the response time of SKETCHREFINE decreasesrapidly, reaching about an order of magnitude improvement withrespect to DIRECT. Most of the queries show that there is a sweetspot at which the response time is the lowest: when all partitionsare small, and there are not too many of them. The point is consistentacross different queries, showing that it only depends on the inputdata size (refer to Figure 3 for the different TPC-H data sizes). Afterthat point, although the partitions become smaller, the number ofpartitions starts to increase significantly. This increase has twonegative effects: it increases the number of representative tuples,and thus the size and complexity of the initial sketch query, andit increases the number of groups that REFINE may need to refineto construct the final package. This causes the running time ofSKETCHREFINE, on the right-hand side of the x-axis, to increase

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again and reach or surpass the running time of DIRECT. The meanand median approximation ratios are in all cases very close to one,indicating that SKETCHREFINE retains very good quality regardlessof the partition size threshold.

5.2.3 Effect of varying partitioning coverageIn our final set of experiments, we study the impact of offline

partitioning on the query response time and the approximation ratioof SKETCHREFINE. We define the partitioning coverage as the ratiobetween the number of partitioning attributes and the number ofquery attributes. For each query, we test partitionings created using:(a) exactly the query attributes (coverage = 1), (b) proper subsets ofthe query attributes (coverage < 1), and (c) proper supersets of thequery attributes (coverage > 1).

For each query, we report the effect of the partitioning coverageon query runtime as the ratio of a query response time over the samequerys response time when coverage is one: a higher ratio (> 1)indicates slower response time and a lower ratio (< 1) indicates afaster response time. Figure 9 reports the results on the Galaxy andthe TPC-H datasets. The Galaxy dataset has many more numericalattributes than the TPC-H dataset, allowing us to experiment withhigher values of coverage. The response time of SKETCHREFINEimproves on both datasets when the offline partitioning covers asuperset of the query attributes, whereas it tends to increase whenit only considers a subset of the query attributes. The mean andmedian approximation ratios are consistently low, indicating thatthe quality of the packages returned by SKETCHREFINE remainsunaffected by the partitioning coverage. These results demonstratethat SKETCHREFINE is robust to imperfect partitionings, whichdo not cater precisely to the query attributes. Moreover, using apartitioning over a superset of a querys attributes typically leads tobetter performance. This means that partitioning can be performedoffline using the union of the attributes of an anticipated workload,or even using all the attributes of a relation.

6. RELATED WORKPackage recommendations. Package or set-based recommenda-tion systems are closely related to package queries. A packagerecommendation system presents users with interesting sets of itemsthat satisfy some global conditions. These systems are usuallydriven by specific application scenarios. For instance, in the Cours-eRank [25] system, the items to be recommended are universitycourses, and the types of constraints are course-specific (e.g., pre-requisites, incompatibilities, etc.). Satellite packages [1] are setsof items, such as smartphone accessories, that are compatible with

a central item, such as a smartphone. Other related problems inthe area of package recommendations are team formation [21, 2],and recommendation of vacation and travel packages [8]. Queriesexpressible in these frameworks are also expressible in PaQL, butthe opposite does not hold. The complexity of set-based packagerecommendation problems is studied in [9], where the authors showthat computing top-k packages with a conjunctive query language isharder than NP-complete.

Semantic window queries and Searchlight. Packages are alsorelated to semantic windows [17]. A semantic window defines acontiguous subset of a grid-partitioned space with certain globalproperties. For instance, astronomers can partition the night sky intoa grid, and look for regions of the sky whose overal brightness isabove a specific threshold. If the grid cells are precomputed andstored into an input relation, these queries can be expressed in PaQLby adding a global constraint (besides the brightness requirement)that ensures that all cells in a package must form a contiguous regionin the grid space. Packages, however, are more general than semanticwindows because they allow regions to be non-contiguous, or tocontain gaps. Moreover, package queries also allow optimizationcriteria, which are not expressible in semantic window queries.

A recent extension to methods for answering semantic windowqueries is Searchlight [18], which expresses these queries in the formof constraint programs. Searchlight uses in-memory synopses toquickly estimate aggregate values of contiguous regions. However,it does not support synopses for non-contiguous regions, and thus itcannot solve arbitrary package queries. Searchlight has several othermajor differences with our work: (1) it computes optimal solutionsby enumerating the feasible ones and retaining the optimal, whereasour methods do not require enumeration; (2) Searchlight assumesthat the solver implements redundant and arbitrary data access pathswhile solving the problems, whereas our approach decouples dataaccess from the solving procedure; (3) Searchlight does not providea declarative query language such as PaQL; (4) unlike SKETCHRE-FINE, Searchlight does not allow solvers to scale up to a very largenumber of variables. At the time of this submission, Searchlight hasnot been made available by the authors and thus we could not run acomparison for the types of queries that it can express.

How-to queries. Package queries are related to how-to queries [22],as they both use an ILP formulation to translate the original queries.However, there are several major differences between packagequeries and how-to queries: package queries specify tuple collec-tions, whereas how-to queries specify updates to underlying datasets;package queries allow a tuple to appear multiple times in a pack-age result, while how-to queries do not model repetitions; PaQL isSQL-based whereas how-to queries use a variant of Datalog; PaQLsupports arbitrary Boolean formulas in the SUCH THAT clause,whereas how-to queries can only express conjunctive conditions.

Constraint query languages. The principal idea of constraintquery languages (CQL) [19] is that a tuple can be generalized asa conjunction of constraints over variables. This principle is verygeneral and creates connections between declarative database lan-guages and constraint programming. However, prior work focusedon expressing constraints over tuple values, rather than over setsof tuples. In this light, PaQL follows a similar approach to CQLby embedding in a declarative query language methods that handlehigher-order constraints. However, our package query engine designallows for the direct use of ILP solvers as black box components, au-tomatically transforming problems and solutions from one domainto the other. In contrast, CQL needs to appropriately adapt the algo-rithms themselves between the two domains, and existing literaturedoes not provide this adaptation for the constraint types in PaQL.

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ILP approximations. There exists a large body of research inapproximation algorithms for problems that can be modeled asinteger linear programs. A typical approach is linear programmingrelaxation [30] in which the integrality constraints are dropped andvariables are free to take on real values. These methods are usuallycoupled with rounding techniques that transform the real solutionsto integer solutions with provable approximation bounds. None ofthese methods, however, can solve package queries on a large scalebecause they all assume that the LP solver is used on the entireproblem. Another common approach to approximate a solution toan ILP problem is the primal-dual method [13]. All primal-dualalgorithms, however, need to keep track of all primal and dualvariables and the coefficient matrix, which means that none of thesemethods can be employed on large datasets. On the other hand,rounding techniques and primal-dual algorithms could potentiallybenefit from the SKETCHREFINE algorithm to break down theircomplexity on very large datasets.Approximations to subclasses of package queries. Like pack-age queries, optimization under parametric aggregation constraints(OPAC) queries [14] can construct sets of tuples that collectivelysatisfy summation constraints. However, existing solutions to OPACqueries have several shortcomings: (1) they do not handle tuple repe-titions; (2) they only address multi-attribute knapsack queries, a sub-class of package queries where all global constraints are of the formSUM() c, with a MAXIMIZE SUM() objective criterion; (3) theymay return infeasible packages; (4) they are conceptually differentfrom SKETCHREFINE, as they generate approximate solutions ina pre-processing step, and packages are simply retrieved at querytime using a multi-dimensional index. In contrast, SKETCHREFINEdoes not require pre-computation of packages. Package queriesalso encompass submodular optimization queries, whose recentapproximate solutions use greedy distributed algorithms [23].

7. CONCLUSIONS AND FUTURE WORKIn this paper, we introduced a complete system that supports

the specification and efficient evaluation of package queries. Wepresented PaQL, a declarative extension to SQL, and theoreticallyestablished its expressiveness, and we developed a flexible approxi-mation method, with strong theoretical guarantees, for the evaluationof PaQL queries on large-scale datasets. Our experiments on real-world and benchmark data demonstrate that our scalable evaluationstrategy is effective and efficient over varied data sizes and queryworkloads, and remains robust under suboptimal conditions andparameter settings.

In our future work, we plan to extend our evaluation methods tolarger classes of package queries, including multi-relation and non-linear queries, and we intend to investigate parallelization strategiesfor SKETCHREFINE. Package queries pose interesting challengeson usability aspects as well: Our goal is to develop interaction andlearning methods that let users identify their ideal packages withouthaving to specify a full and precise PaQL query.Acknowledgements. This material is based upon work supportedby the National Science Foundation under grants IIS-1420941, IIS-1421322, and IIS-1453543.

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https://www.gnu.org/software/bison/http://www.ibm.com/software/commerce/optimization/cplex-optimizer/http://www.ibm.com/software/commerce/optimization/cplex-optimizer/http://www.sdss.org/http://www.tpc.org/tpch/

IntroductionLanguage support for packagesPaQL: The Package Query LanguageExpressiveness and complexity of PaQL

ILP FormulationPaQL to ILP TranslationQuery Evaluation with Direct

Scalable Package EvaluationOffline PartitioningQuery Evaluation with SketchRefineSketchRefine

Approximation GuaranteesThe Case of False InfeasibilityDiscussion

Experimental EvaluationExperimental SetupResults and DiscussionQuery performance as data set size increasesEffect of varying partition size thresholdEffect of varying partitioning coverage

Related WorkConclusions and Future WorkReferences

scalable package queries in relational database systems · scalable package queries in relational...

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