Clique inference process for solving Max-CSP

https://doi.org/10.1016/j.ejor.2007.06.069Get rights and content

Abstract

Few works on problems CSP, Max-CSP and weighted CSP was carried out in the field of Combinatorial Optimization, whereas this field contains many algorithmic tools which can be used for the resolution of these problems.

In this paper, we introduce the binary clique concept: clique which expresses incompatibilities between values of two CSP variables. We propose a linear formulation for any binary clique and we present several ways to exploit it in order to compute lower bounds or to solve Max-CSP. We also show that the binary clique concept can be exploited in the weighted CSP framework.

The obtained theoretical and experimental results are very interesting and they open new prospects to exploit the Combinatorial Optimization algorithmic tools for the resolution of CSP, Max-CSP and weighted CSP.

Introduction

Constraint Satisfaction Problem (CSP) consists in assigning values to variables in order to satisfy a set of constraints. If the problem is over-constrained, such assignment does not exist and it may be of interest to find an assignment which minimize the number of violated constraints, this problem is referred Max-CSP (Maximum Constraint Satisfaction Problem). As for much other Combinatorial Optimization problems, most exact algorithms for solving Max-CSP follow a Branch and Bound (B&B) schema. The efficiency of a B&B largely depends on the quality of the lower bound computed at each node of the search. Most efficient techniques used to compute lower bound for Max-CSP are based on local consistency techniques.

In the community of Operational Research, many lower bounds are developed for solving Combinatorial Optimization problems such as those based on the continuous relaxation or the Lagrangean relaxation [7] techniques. In this work, we study the links between local consistency techniques and the techniques of continuous and Lagrangean relaxations in order to develop new lower bounds for the binary Max-CSP and to improve the quality of some classical lower bounds. We exploit the clique concept. A clique is a complete graph i.e. graph whose vertices are pairwise adjacent. We are interested in binary cliques which express incompatibilities between the values of two variables of the Max-CSP. We propose a linear formulation for any binary clique and show how it can be used to propose a new linear models useful for solving Max-CSP. We show that Integer Linear Programming (ILP) can constitute an interesting alternative to solve Max-CSP with sparse constraint graph. We also show that the lower bounds based on the Directed Arc Consistency [15], Reversible Directed Arc Consistency [9] and Weighted Arc Consistency [2] counts can be also computed by solving relaxation of a linear system containing a reduced number of binary cliques. Finally, we propose a clique inference process which leads to construct linear systems useful for computing new lower bounds. The clique inference process is introduced in the PFC-MPRDAC [10] algorithm and the obtained algorithm is called PFC-MPRDAC + CBB (CBB for Clique Based Bound). The carried out experiments have shown that PFC-MPRDAC + CBB leads to obtain very encouraging results.

The remainder of this paper is organized as follows: in Section 2, we present some definitions and some classical lower bounds of the Max-CSP. In Section 3 we give the linear formulation for binary clique and we announce the theorem which enables us to present different ways of computing lower bounds. Section 4 presents a new equivalent linear formulation for the Max-CSP. Section 5 shows how the lower bounds computed by the local consistency techniques based on the Directed Arc Consistency [15], Reversible Directed Arc Consistency [9] and Weighted Arc Consistency [2] counts can be also computed by solving a simple Integer Linear Program with a reduced number of clique constraints. Section 6 is devoted to the presentation of a clique inference process. We present the carried out experiments in Section 7, and Section 8 shows that our work can be extended for weighted CSP. We conclude in the Section 9.

Section snippets

Background

Formally, a binary CSP is defined by a triplet (X,D,C) where X={X1,X2,, Xn} is a set of n variables, D={D1,D2,,Dn} is a set of n domains where Di is a set of di possible values for Xi (did, where d is the size of the greatest domain), C is a set of e constraints in which a constraint Cij involves variables Xi and Xj and assigns costs to assignments to variables Xi and Xj (namely, Cij:Di×Dj{0,1}2). A pair (vk,vl) satisfies Cij if

Clique based linear models for Max-CSP

A CSP is usually represented by a constraint graph G whose vertices represent the variables and the edges the constraints. The complementary of the micro-structure Cμ of the constraint graph is a graph whose vertices represent the values of the variables and the edges represent the not permitted pairs of values. A union of subsets, Ei1Di1,Ei2Di2,,EimDim:1i1<i2<<imn, forms a clique of Cμ if and only if for each couple (j1,j2){i1,i2,,im}2 such that Ej1 and Ej2 we have Cj1j2C and Cj1j

A clique inference schema

A simple answer to the above questions consists in defining a complete clique set Γ and using integer linear programming to solve the associated linear system IP(Γ). A complete clique set can be simply constructed by associating for each incompatible pair (vk,vl)Di×Dj of each constraint Cij the binary clique which contains only the values vk and vl (xik+xjl1+ηij). The disadvantage of this is that the information represented by each clique is poor. The clique xik+xjl1+ηij informs us only

Using cliques to define partial linear systems

This section shows that the clique inference schema presented in the previous section can be exploited to construct a partial clique set useful for computing lower bounds better than those based on the Directed Arc Consistency [15], Reversible Directed Arc Consistency [9] and Weighted Arc Consistency [2] counts.

Consider the set Γ0={Γij(Di),Γji(Dj), CijC} and the linear system:IP(Γ0)minCijCηijs.t.ψ(Di,ϕij(Di))1+ηijCijC,ψ(ϕji(Dj),Dj)1+ηijCijC,xS.The cliques Γij(Di) and Γji(Dj) are

A clique inference process

This section presents a process constructing more interesting partial linear systems than those presented in the previous section.

The process generates p clique sets Γ1,Γ2,,Γp in an iterative way. Each of these sets contains one clique for each constraint. They are generated in such way that LR(1,Γ1)<LR(1,Γ2)<<LR(1,Γp-1)=LR(1,Γp), where LR(1,Γq)(1qp) is the value of the Lagrangean relaxation which we obtain if we dualize the clique constraints of IP(Γq) by using a vector of Lagrangean

Experimental results

This section presents the carried out experiments on different classes of over-constrained binary random CSP. Each class is characterized by n,d,e,t where n is the number of variables, d the number of values per variable, e the number of constraints and t is the constraint tightness defined as the number of forbidden value pairs in each relation. We have experimented on the following problem classes: (1)10,10,45,t,(2)15,10,50,t,(3)40,5,55,t. The problems of the first class are highly

Future work

In this section we show on two simple examples that the binary clique concept can be exploited to compute a stronger lower bound than those computed by AC, FDAC, and EDAC and can be extended to the general concept of weighted clique useful for solving weighted CSP.

Example 3

Fig. 7 shows an example of Max-CSP with four variables (X={X1,X2, X3,X4}) and six constraints (C={C12,C13,C14,C23,C24,C34}). The edges on graph represent the incompatible pairs. It is an arc consistent problem. Thus, none of AC AC

Conclusion

In this paper, we have shown that the concept of clique can be exploited in order to solve Max-CSP. We have proposed a linear formulation for binary cliques and shown that this formulation can be used to construct linear models useful for computing lower bounds or for the exact resolution of Max-CSP. We have proposed a clique inference process which leads to constructs partial linear systems exploited to compute lower bounds. The clique inference process is introduced in the PFC-MPRDAC

References (15)

There are more references available in the full text version of this article.

Cited by (1)

1

Supported by the French Electricity Board (EDF).

View full text