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On the Cost and Benefits of Procrastination: Approximation

Algorithms for Stochastic Combinatorial Optimization Problems

(SODA 2004)

Nicole Immorlica, David Karger, Maria Minkoff, Vahab S. Mirrokni

Jaehan Koh (jaehanko@cs.tamu.edu)

Dept. of Computer ScienceTexas A&M University

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Outline

IntroductionPreplanning FrameworkExamplesResultsSummaryHomework

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Introduction

Scenarios Harry is designing a network for the Computer Science

Department at Texas A&M University. In designing, he should make his best guess about the future demands in a network and purchase capacity in accordance with them.

A mobile robot is navigating around a room. Since the information about the environment is unknown or becomes available too late to be useful to the robot, it might be impossible to modify a solution or improve its value once the actual inputs are revealed.

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Planning under uncertainty

Problem data frequently subject to uncertainty May represent information about the future Inputs may be evolve over time

On-line model Assumes no knowledge of the future

Stochastic modeling of uncertainty Given: probability distribution on potential

outcomes Goal: minimize expected cost over all potential

outcomes

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Approaches

Plan aheadFull solution has to be specified before we learn

values of unknown parameters Information becomes available too late to be

useful

Wait-and-SeePossibility to defer some decisions until getting

to know exact inputsTrade-off: decisions made late may be more

expensive

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Approaches (Cont’d)

Trade-offsMake some purchase/allocation decisions early

to reduce cost while deferring others at greater expense to take advantage of additional information.

Problems in which the probolem instance is uncertain.

Min-cost flow, bin packing, vertex cover, shortest path, and the Steiner tree problems.

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Preplanning framework

Stochastic combinatorial optimization problem ground set of elements e E A (randomly selected) problem instance I, which defines

a set of feasible solutions FI, each corresponding to a subset FI 2E.

We can buy certain elements “in advance” at cost ce, then sample a problem instance, then buy other elements at “last-minute” cost ce so as to produce a feasible solution S FI for the problem instance.

Goal: to choose a subset of elements to buy in advance to minimize the expected total cost.

Ee

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Two types of instance prob. distribution

Bounded support distributionNonzero probability to only a polynomial

number of distinct problem instances.

Independent distributionEach element / constraint for the problem

instance active independently with some probability.

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Versions

Scenario-basedBounded number of possible scenariosExplicit probability distribution over problem

instances

Independent events modelRandom instance is defined implicitly by

underlying probabilistic processThe number of possible scenarios can be

exponential in the problem size

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Problems

Min Cost Flow Given a source and sink and a probability distribution on

demand, buy some edges in advance and some after sampling (at greater cost) such that the given amount of demand can be routed from source to sink.

Bin Packing A collection of item is given, each of which will need to

be packed into a bin with some probability. Bins can be purchased in advance at cost 1; after the determination of which items need to be packed, additional bins can be purchased as cost > 1. How many bins should be purchased in advance to minimize the expected total cost?

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Problems (Cont’d)

Vertex CoverA graph is given, along with a probability

distribution over sets of edges that may need to be covered. Vertices can be purchased in advance at cost 1; after determination of which edges need to be covered, additional vertices can be purchased at cost . Which vertices should be purchased in advance?

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Problems (Cont’d)

Cheap PathGiven a graph and a randomly selected pair of

vertices (or one fixed vertex and one random vertex), connect them by a path. We can purchase edge e at cost ce before the pair is known or at cost ce after. We wish to minimize the expected total edge cost.

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Problems (Cont’d)

Steiner TreeA graph is given, along with a probability

distribution over sets of terminals that need to be connected by a Steiner tree. Edge e can be purchased at cost ce in advance or at cost ce after the set of terminal is known.

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Preplanning combinatorial optimization problem

A ground set of elements e E, a probability distribution on instances {I}, a cost function c: E R, a penalty factor 1Each instance I has a corresponding set of feasible solutions FI 2E associated with it.

Suppose a set of elements A E is purchased before sampling the probability distribution. The posterior cost function cA is defined by

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Preplanning combinatorial optimization problem (Cont’d)The objective of a PCO problme: choose a subset of elements A to be purchased in advance so as to minimize the total expected cost of a feasible solution

over a random choice of an instance I.

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The Threshold Property

Theorem 1.An element should be purchased in advance if

and only if the probability it is used in the solution for a randomly chosen instance exceeds 1/.

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Example: Min-cost FlowWe wish to provide capacity on a network sufficient to carry a random amount of flow demand D from a source s to a sink t.We have an option to pre-install some amout of capacity in advance at some cost per unit.We rent additional capacity once the demands become known, but at cost a factor of or larger per unit.The sum of capacity in advance and capacity rented must satisfy a given upper bound of total capacity for each edge.Goal: Over a given probability distribution on demands, minimize the expected cost of installing sufficient capacity in the network so that the network satisfies the demand.

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Example: Min-cost Flow (Cont’d)

Suppose Cap(s-a) = Cap(a-t) = Cap(s-b) = Cap(a-b) = Cap(b-t) = 1 = 2 Pr[D=0] = ¼, Pr[D=1] = ¼, Pr[D=2] = ½

as

bt

1 3

3 11

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Example: Min-cost Flow (Cont’d)

Suppose Cap(s-a) = Cap(a-t) = Cap(s-b) = Cap(a-b) = Cap(b-t) = 1 = 2 Pr[D=0] = 5/12, Pr[D=1] = ¼, Pr[D=2] = 1/3

as

bt

1 3

3 11

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Example: Vertex cover

Classical vertex cover problem Given: a graph G = (V, E) Output: a subset of vertices such that each edge has at

least one endpoint in the set Goal: minimize cardinality of the vertex subset

Stochastic vertex cover Random subset of edges is present Vertices picked in advance cost 1 Additional vertices can be purchased at cost >1 Goal: minimize expected cost of a vertex cover Time-information trade-off

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Our techniques

Merger of Linear Programs Easy way to handle scenario-based stochastic problems

with LP-formulations

Threshold Property Identity elements likely to be needed and buy them in

advance

Probability Aggregation Cluster probability mass to get nearly deterministic

subproblems Justifies buying something in advance

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Vertex Cover with preplanning

Given Graph G = (V, E)

Random instance: a subset of edges to be covered Scenario-based: poly number of possible edge sets

Independent version: each e E present independently with probability p.

Vertices cost 1 in advance; after edge set is sampled Goal: select a subset A of vertices to buy in advance

such that minimize expected cost of a vertex cover

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Idea 1: LP merger

Deterministic problem has LP formulation Write separate LP for each scenario of stochastic

problem Take probability-weighted linear combination of objective

functions

Applications of scenario-based vertex cover VC has LP relaxation Combine LP relaxation of scenarios Round in standard way Result: 4-approximation

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Idea 2: Threshold property

Buy element e in advance Pr[need e] 1/ Advance cost ce Expected buy-later cost cePr[need e]

Application of independent-events vertex cover

Threshold degree k = 1/p Vertex with degree k is adjacent to an active edge with

probability 1/ Not worth buying vertices with degree < k in advance Idea: purchase a subset of high degree vertices in

advance

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k-matching

k-matching is a subset of edges that induces degree k on each vertexVertex v V is tight if its degree in matching is exactly kCan construct a maximal k-matching greedily

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Algorithm for k-matchingAssume 4Set k = 1 / pConstruct some maximal k-matching Mk

Purchase in advance the set of tight vertices At

Solution costPrepurchase cost | At | “Wait-and-See” cost expected size of minimum vertex cover of

active edges not already covered by At

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Bounding 2nd stage cost

Claim: once purchased all the tight vertices, it is optimal to buy nothing else in advance

Vertices in At cover all edges not in k-matching, then e has at least one tight endpoint

Vertices not in At have degree < k

E[cost] a vertex cover in V \ At-induced subgraph is at most OPT

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Bounding prepurchase cost

Restrict attention to instnace induced by Mk Costs no more than OPT

Intuition Vertex of degree k = 1/p is likely to be adjacent to an

active edge Active edges are not likely to be clustered together Can show that a tight vertex is useful for the vertex cover

with sufficiently high probability | At | = O(OPT)

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Steiner network preplanning

Given Undirected graph G = (V, E) with edge costs ce 0 Probability pi of node i becoming active Penalty 1

Goal Buy subset of edges in advance in order to minimize

expected cost of s Steiner tree over active nodes

Approaches: probability aggregation Cluster vertices into groups of 1/ probability Purchase in advance edges of an MST over clusters

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CLUSTER algorithm

The Steiner tree for the ultrametric case An assignment of edge weights satisfying ĉuv max

(ĉu, ĉv).

The basic idea To cluster nodes into components, each containing

clients of total probability mass (1/).

Lemma Algorithm CLUSTER produces an MST with the specified

properties.

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CLUSTER algorithm (Cont’d)

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CLUSTER algorithm (Cont’d)

A hub tree h

h

<T<T

<T<T>T

>T

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Probability Aggregation

p p

ppp

pp

pp

p

p p

ppp

p pp

p p

pp

pp

p

1/

1/1/

1/1/

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ResultsProblem Stochastic

elementsApprox.

guaranteeMin-cost s-t flow

Demand 1 Via LP

Bin packing Items Asymptotic FPAS

Using FPAS for determ. version

Vertex cover Edges 44.37

Scenario-basedIndep. Events

Shortest path Start nodeStart & end

O(1)O(1)

ConnFacLocRent-or-Buy

Steiner tree Terminals 3O(log n)

UltrametricsGeneral case

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SummaryStochastic combinatorial optimization problems in a novel “preplanning” framework

Study of time-information trade-off in problems with uncertain inputs

Algorithms to approximately optimize the choice of what to purchase in advance and what to defer

Open questions Combinatorial algorithms for stochastic min-cost flow Metric Steiner tree preplanning Applying preplanning scheme to other problems:

scheduling, multicommodity network flow, network design

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Homework1. (40 pts) In the paper, the authors argue that we can

reduce the preplanning version of an NP-hard problem to solve a preplanning instance of another optimization problem that has a polynomial-time algorithm. Explain in your own words why this is true and give at least one example.

2. (a) (30 pts) Formulate the Minimum Steiner Tree problem as an optimization problem (b) (30 pts) Reformulate the Minimum Steiner Tree problem in a PCO (Preplanning Combinatorial Optimization) problem version. Explain the difference between (a) and (b).

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Thank you …Any Questions?