cbls thuan

Constraint-based local searchfor solving problems on graphs

Phan Thuan DO

School of Information and Communication TechnologyHanoi University of Technology, Hanoi-Vietnam

a joined work withQuang Dung PHAM & Yves DEVILLE

Department of Computing Science and EngineeringUniversite catholique de Louvain

B-1348 Louvain-la-Neuve - Belgium

22 octobre 2010

(thuandp at soict.hut.edu.vn) Constraint-based local search 22 octobre 2010 1 / 16

Outline

1 Introduction

2 Modeling approach

3 Some applied problemsThe Routing for Network Covering (RNC) problemThe quorumcast routing (QR) problemThe Edge-Disjoint Paths (EDP) problemThe edge-weighted k-cardinality tree (KCT) problemThe resource constrained shortest path (RCSP) problemThe Routing and Wavelength Assignment with delay side constraint(RWA-D) problemMixed Chinese postman problem (MCPP)


Outline

1 Introduction

2 Modeling approach



Context

Consider constrained paths finding where vertices and edges can berepeated on the paths (COI - Constrained Optimum Itineraryproblem)

Problem of routing for network covering in the context of naturaldisasters rescue (AROUND project at the MSI/IFI)Other traditional applications : Chinese postman problem (maildelivery, newspaper delivery, trash pick-up) and its variants, WindyPostman problem

The LS(Graph) framework [PHAM et al. 2010] allows to model inhigh-level and solve constrained paths finding by constrained-basedlocal search where paths are elementary.


Related work

Existing techniques for solving COI problems :

Mixed Integer ProgrammingApproximation

HOWEVER

These techniques are specific, and sophisticated,Need high programming efforts for solving related or similar problems,Difficult to extend when other side constraints need to be added.


Objectives

Design of a generic framework for modeling and solving some COIs bylocal search

Generic framework implemented in COMET : extending the LS(Graph)framework [PHAM et al. 2010]

Flexibility :

Easy to state some constraints and objective functions on graphsEasy to extend a model when side constraints need to be addedEasy to explore different neighborhoods

Separation of concerns : the model and the search are independentOpeness : possibility to design and implement new constraints andobjective functions on graphs


Outline

1 Introduction

2 Modeling approach



Local search

N is the neighborhood function

S is the selection operator


COP Neighborhood

In the LS(Graph) framework [PHAM et al. 2010], rooted spanningtree (RST) has been used to represent a path (elementary path) andits neighbors

a. current tree b. neighboring trees


COP Neighborhood


Modeling an itinerary

It is required to represent paths where vertices and edges can berepeated (Itinerary)Simple idea : We propose to use a sequence of rooted spanning treesfor representing an itineraryThe sequence 〈tr1, tr2〉 induces the itinerary〈s, 3, 4, 8, 7, 6, 5, 4, 8, 10, 12, t〉


COI Neighborhood (first neighborhood)

current solution 〈tr1, tr2〉 replaceEdge(tr1, (3, 4), (1, 5))

〈s, 3, 4, 8, 7, 6, 5, 4, 8, 10, 12, t〉〈s, 3, 1, 5, 4, 8, 10, 12, t〉

Drawback : Roots of all RSTs do not change, thus we cannot reachgood solutions that do not contain these vertices


COI Neighborhood (second neighborhood)

Consider new actions over RST

cs(tr , s ′) which changes the source node of tr by a new source s ′

cr(tr , t ′) which changes the root of tr by a new root t ′

current solution 〈tr1, tr2〉 cr(tr1, 11), cs(tr2, 11)

〈s, 3, 4, 8, 7, 6, 5, 4, 8, 10, 12, t〉〈s, 3, 4, 8, 7, 11, 8, 10, 12, t〉


Abstractions

We extend the LS(Graph) framework by constructing abstractionsallowing to model flexibly and solve COI problems by local search :

VarItinerary(g,s,t,k) represents dynamic itinerary from s to t

on the graph g

NBVisitedEdgesItinerary(I,S) : graph objective represents thenumber of edges of S visited by the list of itineraries I

ItineraryCost(I) : the cost of the itinerary I

Arithmetic operators +, -, * between graph objectives

Basic relation operators : <=, >=, == for stating graph constraints


Abstractions

Example : stating constraints of the RNC problem

1. LSGraphSolver ls() ;2. VarItinerary I[1..m](ls,d,d,g,k) ;// m is the number of itineraries of the solution

3. GraphConstraintSystem gcs(ls) ;4. forall(i in I.rng()){5. ItineraryCost c(I[i]) ;6. gcs.post(c <= L) ;7. }

8. NBVisitedEdgesItinerary nb(I,S) ;9. gcs.post(nb == S.getSize()) ;

10. gcs.close() ;


Outline

1 Introduction

2 Modeling approach



The Routing for Network Covering (RNC) problemQuang Dung PHAM, Phan Thuan DO, Yves DEVILLE & Tuong Vinh HO. SOICT2010

Given an undirected weighted graph G = (V ,E ) representing atransportation network and a vertex d ∈ V representing the depot. c is alength function defined on edges : for each edge e = (u, v) ∈ E , c(e) (orc(u, v)) is the cost of e.The cost of a walk I (denoted by c(I )) is defined to be the sum of lengthsof all edges of that path :

c(I ) =∑

e∈E(I )

c(e)

Given an undirected weighted graph G = (V ,E ), a vertex d ∈ Vrepresenting the depot, a set S of edges of G (S ⊆ E ) and a value L, theRNC problems consists of finding a minimal cardinality set of pathsstarting from and terminating at d whose cost are less than or equal to Lthat covers all edges of S .


The quorumcast routing (QR) problem

The quorumcast routing (QR) problem arises in distributed applications.Given a weighted undirected graph G = (V ,E ), a source node s ∈ V , anintegral value q and a set S ⊆ V , the quorumcast routing problem consistsof finding a minimum tree of G spanning s and q nodes Q of S .

The Edge-Disjoint Paths (EDP) problem

We are given an undirected graph G = (V ,E ) and a setT = {〈si , ti 〉 | si 6= ti ∈ V } representing a list of commodities (]T = k).The EDP problem consists of finding a maximal cardinality set of mutuallyedge-disjoint paths from si to ti on G (〈si , ti 〉 ∈ T ). This problem appearsin many applications such as real-time communication, VLSI-design,routing and admission control in modern networks.


The edge-weighted k-cardinality tree (KCT) problem

Given an undirected weighted graph G = (V ,E ) and an integral value k ,the KCT problem consists of finding a subtree of G having exactly k edgessuch that the sum of weights of edges is minimal. This problem appears invarious applications such as oil-field leasing, facility layout, open pitmining, matrix decomposition, quorum-cast routing andtelecommunications.

Mixed Chinese postman problem (MCPP)

Given a weighted mixed graph which contains both arc (directed edge)and edge. MCPP consists of finding a minimum cost closed walk whichvisits each arc, edge at least once.


The resource constrained shortest path (RCSP) problem

The resource constrained shortest path problem (RCSP) is the problem offinding the shortest path between two vertices on a network satisfying theconstraints over resources consumed along the path. There are somevariations of this problem, but we first consider a simplified versionintroduced and evaluated in [?] over instances from the OR-Library [?].Given a directed graph G = (V ,E ), each arc e is associated with a lengthc(e) ≥ 0 and a vector r(e) ≥ 0 of resources consumed in traversing thearc e. Given a source node s, a destination node t and two vectors L, U ofresources corresponding to the minimum and maximum amount that canbe used on the chosen path (i.e., a lower and an upper limit on theresources consumed on the path). The length of a path P is defined asf (P) =

∑e∈P c(e). The resources consumed in traversing P is defined as

r(P) =∑

e∈P r(e) The formulation of RCSP is then given by :

min f (P) (1)s.t. L ≤ r(P) (2)

r(P) ≤ U (3)P is a path from s to t on G (4)


The Routing and Wavelength Assignment with delay side constraint(RWA-D) problem

The WDM (Wavelength Division Multiplexing) optical networks providethe high bandwidth of communications. The routing and wavelengthassignment (RWA) problem appears as essential problem on the WDMoptical networks. The RWA problem can be described as follows. Given aset of requests for all-optical connections, the RWA problem consists offinding routes from source nodes to their respective destination nodes andassigning wavelengths to these routes. A condition the must be satisfied isthat two non edge-disjoint routes must be assigned different wavelengths.Normally, the number of available wavelength is limited and the number ofrequests is high. Two variants of the problem have been considered andstudied extensively in the literature : the minRWA problem aims atminimizing the number of wavelength used for satisfying all requests andthe maxRWA aims at maximizing the number of requests with a givennumber of wavelengths. Both of two variants are NP-Hard.


cbls thuan

Technology

local search22 octobre

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local search generic

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paths coi

constrained paths nding

applied problemsthe

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