Exploiting Tree Decomposition and Soft Local Consistency in Weighted CSP Simon de Givry and Thomas Schiex Gerard Verfaillie INRA, Toulouse, France ONERA - DCSD, Toulouse, France degivry,tschiex @toulouse.inra.fr [email protected] { } Abstract proved practical time and space complexity inside the origi- nal bounds. Note however that in the context of tree-search, Several recent approaches for processing graphical these bounds are obtained at the cost of restricted freedom models (constraint and Bayesian networks) simultane- in variable assignment ordering (see (Bacchus, Dalmao, & ously exploit graph decomposition and local consis- tency enforcing. Graph decomposition exploits the Pitassi 2003) on this topic). problem structure and offers space and time complex- In this paper, we are specifically interested in optimiza- ity bounds while hard information propagation provides tion problems in the framework of valued and weighted practical improvements of space and time behavior in- constraint networks (Schiex, Fargier, & Verfaillie 1995). side these theoretical bounds. Weighted constraint networks (WCN) provide a very general Concurrently, the extension of local consistency to model with several applications in domains such as resource weighted constraint networks has led to important im- allocation, combinatorial auctions and bioinformatics. provements in branch and bound based solvers. In- In a first part, we show that, with a limited weakening of deed, soft local consistencies give incrementally com- existing theoretical complexity bounds, an alternative com- puted strong lower bounds providing inexpensive yet bination of tree decomposition and branch and bound can be powerful pruning and better informed heuristics. defined. In a second part, we exploit the extension of local In this paper, we consider combinations of tree decom- consistency to WCN.This extension has lead to increasingly position based approaches and soft local consistency en- forcing for solving weighted constraint problems. The efficient branch and bound algorithms using increasingly intricacy of weighted information processing leads to strong local consistency properties (Cooper & Schiex 2004; different approaches, with different theoretical proper- Larrosa & Schiex 2004; 2003).They offer incrementally ties. It appears that the most promising combination maintained strong bounds and also contribute to better in- sacrifices a bit of theory for improved practical effi- formed variable and value ordering. Introduced in tree de- ciency. composition based approaches, they should enhance their practical time and space complexities and also provide better guidance. Introduction Taken together, these two parts show the increased intri- Graphical model processing is a central problem in AI. In cacy of weighted information processing, leading to differ- the last years, in order to solve satisfaction, optimization or ent possible combinations, each having different theoretical counting problems, several algorithms have been proposed properties and practical efficiencies. that simultaneously exploit a decomposition of the graph of the problem and the propagation of hard information using Preliminaries local consistency enforcing. This includes algorithms such as Recursive Conditioning (RC) (Darwiche 2001), Back- A weighted binary CSP (WCSP) is a triplet (X, D, W). track bounded by Tree Decomposition (BTD) (Terrioux X = 1,...,n is a set of n variables. Each variable i X {has a finite} domain D D of values than can be & Jegou 2003), AND-OR tree and graph search (Mari- ∈ i ∈ nescu & Dechter 2005b; 2005a), all related to Pseudo-tree assigned to it. The maximum domain size is d. W is a set of search (Freuder & Quinn 1985). soft constraints (or cost functions). A binary soft constraint W W is a function W : D D [0,k] where k is a Implicitly or not, all these algorithms exploit the proper- ij ∈ ij i × j → ties of Tree Decompositions (Bodlaender 2005) which cap- given maximum integer cost corresponding to a completely ture structural independence information, in order to ob- forbidden assignment (expressing hard constraints). If they tain theoretical bounds on the time and space complexity do not exist, we add to W one unary cost function for every of the algorithms. The combination with hard informa- variable such that Wi : Di [0,k] and a zero arity con- straint W (a constant cost payed→ by any assignment). All tion propagation provides additional pruning leading to im- ∅ these additional cost functions have initial value 0, leaving Copyright c 2006, American Association for Artificial Intelli- the semantics of the problem unchanged. gence (www.aaai.org). All rights reserved. The problem is then to find a complete assignment 22 with a minimum cost: min W + (a1,a2,...,an) i Di ∅ n ∈ { Wi(ai)+ Wij(ai,aj) , an optimization i=1 Wij W } problem with an associated∈ NP-complete decision problem. The constraint graph of a WCSP is a graph G =(X, E) with one vertex for each variable and one edge (i, j) for every binary constraint Wij W . A tree decomposi- tion of this graph is defined by∈ a tree (C, T). The set of nodes of the tree is C = C ,...,C where C is a { 1 m} e set of variables (Ce X) called a cluster. T is a set of edges connecting clusters⊂ and forming a tree (a con- nected acyclic graph). The set of clusters C must cover all the variables ( C = X) and all the constraints Ce C e ( (i, j) E, C ∈C s.t. i, j C ). Furthermore, if a ∀ ∈ ∃ e∈ ∈ e variable i appears in two clusters Ce and Cg, i must also ap- pear in all the clusters Cf on the unique path from Ce to Cg in (C, T). The rationale behind this definition is that acyclic prob- lems can be solved with a good theoretical complexity. Thus, a tree decomposition decomposes a problem in sub- problems (clusters) organized in an acyclic graph and in such a way that the variables that relate two adjacent clusters (variables whose removal disconnects the subproblems) are obtained by intersecting the two clusters. This is illustrated on Figure 1 where the graph of a fre- quency assignment problem is covered by clusters defining Figure 1: The constraint graph of preprocessed RLFAP a tree decomposition. Because of the usual emphasis on SCEN-06 problem covered by clusters and the associated purely random problems, one may think that such nice de- tree decomposition using C1 as root. Problem P5 is outlined compositions are exceptional but the emphasis on modular- with separator S5 = 57, 72 . The constraint W57,72 inside S is not part of P . P{ has a} separator S = 69 . ity in design problems and the (spatial or temporal) locality 5 5 6 6 { } of many problems easily induce such structures. The tree-width of a tree decomposition (C, T) is equal to the variables of a cluster C are assigned before any of the max C 1, denoted w in the sequel. The tree- e Ce C e remaining variables in its son clusters and consider a cur- width w∈ {|of G|}is − the minimum tree-width over all tree de- ∗ rent assignment A. Then, for any cluster C Sons(C ), compositions of G (also called the induced-width of G). If f e and for the current assignment A of the separator∈ S , the a root C C is chosen, the maximum number of variables f f r subproblem P under assignment A (denoted P /A ) can appearing∈ in a path starting from C is called the tree-height f f f f r be solved independently from the rest of the problem. If of the decomposition. Finding a minimum tree-width or a memory allows, the optimal cost of P /A may be recorded minimum tree-height decomposition is NP-hard. f f which means it will never be solved again for the same as- signment of S . Exploiting tree decompositions f If d is the maximum domain size, h the decomposition For a given WCSP, we consider a rooted tree decomposition tree-height and w the tree-width, this idea applied recur- (C, T) with an arbitrary root C . We denote by Father(C ) 1 e sively with no recording guarantees that a cluster C will (resp. Sons(C )) the parent (resp. set of sons) of C in T . e e e never be visited more than d to the power of the number The separator of C is the set S = C Father(C ). The set e e e e of variables in the path from the root to C (proper vari- of proper variables of C is V = C ∩S . Note that the V e e e e e e ables excluded). This gives a guaranteed bound on the family defines a partition of X. For a\ given variable i X, number of visited nodes which is dominated by O(dh). we denote by [i] the index of the cluster such that i V∈ . [i] This covers Pseudo-Tree or AND-OR tree search, also ap- The essential property of tree decomposition is∈ that as- plied to WCSP in (Larrosa, Meseguer, & Sanchez 2002; signing separates the initial problem in two subproblems Se Marinescu & Dechter 2005b). which can then be solved independently. The first subprob- lem, denoted Pe, is defined by the variables of Ce and all With recording, a cluster Ce will never be visited more its descendant clusters in T and by all the cost functions in- than the number of assignments of Se. Given the number of volving at least one proper variable of these clusters. The variables in Ve, this means that the number of nodes is dom- remaining constraints, together with the variables they in- inated by O(dw+1), w +1being the size of the largest clus- volve, define the remaining subproblem. ter. This gives algorithms such as RC with full recording, This property has been exploited by many related algo- BTD and AND-OR graph search, which can be considered rithms.