A Comparison of Structural CSP Decomposition Methods

We compare tractable classes of constraint satisfaction problems (CSPs). We first give a uniform presentation of the major structural CSP decomposition methods. We then introduce a new class of tractable CSPs based on the concept of hypertree decomposition recently developed in Database Theory. We introduce a framework for comparing parametric decomposition-based methods according to tractabi l i ty cri teria and compare the most relevant methods. We show that the method of hypertree decomposition dominates the others in the case of general (nonbinary) CSPs. 1 Cons t ra in t Sat is fact ion Prob lems An instance of a constraint satisfaction problem (CSP) (also constraint network) is a tr iple / = (Var,U,C), where Var is a finite set of variables, U is a finite domain of values, and is a finite set of constraints. Each constraint is a pair , where is a list of variables of length rni called the constraint scope, and is an -ary relation over [/, called the constraint relation. (The tuples of indicate the allowed combinations of simultaneous values for the variables Si). A solution to a CSP instance is a substi tut ion V : , such that for each The problem of deciding whether a CSP instance has any solution is called constraint satisfiability (CS). (This definit ion is taken almost verbatim from [Jeavons et a/., 1997].) Many problems in Computer Science and Mathematics can be formulated as CSPs. For example, the famous problem of graph three-colorability (3COL), is elegantly formulated as a CSP. Constraint Satisfiability is an NPcomplete problem. It is well-known [Bibel, 1988; Gyssens et al, 1994; Dechter, 1992] that the CS problem is equivalent to various database problems, e.g., to the problem of evaluating Boolean conjunctive queries over a relational database [Maier, 1986], or to the equivalent problem of evaluating join dependencies on a given database. This paper is organized as follows. In Section 2 we discuss tractabi l i ty of CSPs due to restricted structure. In Section 3 we briefly review well-known CSP decomposit ion methods. In Section 4 we describe the new method of hypertree decompositions. In Section 5 we explain our comparison criteria and in Section 6 we present the comparison results for general CSPs. The case of binary CSPs is briefly discussed in Section 7. 2 Trac tab le classes of CSPs Much effort ,has been spent by both the AI and the database communities to indentify tractable classes of CSPs. Both communities have obtained deep and useful results in this direction. The various successful approaches to obtain tractable CSP classes can be divided into two main groups [Pearson and Jeavons, 1997]: 1 . T r a c t a b i l i t y d u e t o r e s t r i c t e d s t r u c t u r e . This includes all tractable classes of CSPs that are identif ied solely on the base of the structure of the constraint scopes {S\,... Sq), independently of the actual constraint relations r1,..., rq. 2 . T r a c t a b i l i t y d u e to r e s t r i c t e d c o n s t r a i n t s . This includes all classes that are tractable due to particular properties of the constraint relations r1,..., rq. The present paper deals w i th t ractabi l i ty due to restricted structure. The structure of a CSP is best represented by its associated hypergraph and by the corresponding primal graph, defined as follows. To any CSP instance / = (Var, U,C), we associate a hypergraph — (V,H), where V = Var, and , where var(S) denotes the set of variables in the scope S of the constraint C. Since in this paper we always deal w i th hypergraphs corresponding to CSPs instances, the vertices of any hypergraph 'H = (V, H) can be viewed as the variables of some constraint satisfaction problem. Thus, we wi l l often use the term variable as a synonym for vertex, when referring to elements of V. Let = (V,H) be the constraint hypergraph of a CSP instance 1. The primal graph of / is a graph G = (V,E), having the same set of variables (vertices) as and an edge connecting any pair of variables 394 CONSTRAINT SATISFACTION such that for some h H. Note that if all constraints of a CSP are binary, then its associated hypergraph is identical to its pr imal graph. The most basic and most fundamental structural property considered in the context of CSPs (and conjunctive queries) is acyclicity. It was recognized independently in AI and in database theory that acyclic CSPs are polynom i a l ^ solvable. / is an acyclic CSP iff its primal graph G is chordal (i.e., any cycle of length greater than 3 has a chord) and the set of its maximal cliques coincide wi th edges (HI) [Beeri et a/., 1983]. A join tree JT(H) for a hypergraph H is a tree whose nodes are the edges of H such that whenever the same vertex X _ V occurs in two edges A and of H, t h e n and are connected in , , and X occurs in each node on the unique path l inking and in Acyclic hypergraphs can be characterized in terms of jo in trees: A hypergraph H is acyclic iff it has a jo in tree [Bernstein and Goodman, 1981; Beeri et a/., 1983; Maier, 1986]. Acyclic CSP satisfiability is not only tractable but also highly parallelizable. In fact, this problem is complete for the low complexity class L O G C F L [Gott lob et a/., 1998]. Many CSPs arising in practice are not acyclic but are in some sense or another close to acyclic CSPs. In fact, the hypergraphs associated wi th many natural ly arising CSPs contain either few cycles or small cycles, or can be transformed to acyclic CSPs by simple operations (such as, e.g., lumping together small groups of vertices). Consequently, CSP research in AI and in database theory concentrated on identi fying, defining, and studying suitable classes of nearly acyclic CSPs, or, equivalently, methods for decomposing cyclic CSPs into acyclic CSPs. 3 Decompos i t i on M e t h o d s In order to study and compare various decomposition methods, we find it useful to introduce a general formal framework for this notion. For a hypergraph H = (V, H), let edges(H) = H. Moreover, for any set of edges H' H, let var(H') = and for the hypergraph 'H, let var(H) — W.l.o.g., we assume that var(H) = V, i.e., every variable in V occurs in at least one edge of 7i, and hence, any hypergraph can be simply represented by the set of its edges. Moreover, we assume w.l.o.g. that all hypergraphs under consideration are both connected, i.e., their pr imal graph consists of a single connected component, and reduced, i.e., no hyperedge is contained in any other hyperedge. A l l our definitions and results easily extend to general hypergraphs. Let US be the set of all (reduced and connected) hypergraphs. A decomposition method (short: DM) D associates to any hypergraph a parameter Dwidth(H), called the D-width of H. The decomposition method D ensures that , for fixed K, every CSP instance / whose hypergraph has Dwid th < K: is polynornially solvable, i.e., it is solvable in t ime, where denotes the size of /. For any k > 0, the k-tractable class C(D,k) of D is defined by C{D,k) = { H | D w i d t h . . Thus, C{D,k) collects the set of CSP instances which, for fixed k, are polynornially solvable by using the strategy D. Typically, the polynomial above depends on the parameter k. In particular, for each D there exists a function / such that, for each A:, each instance can be transformed in t ime into an equivalent acyclic CSP instance (from where it follows that all problems in C(D,k) are polynornially solvable). Every DM D is complete w.r.t. W, i.e., US = I. Note that , by our definitions, it holds that JD-widthCW) = min A l l major tractable classes based on restricted structure fit into this framework. In particular, we shall compare the following decomposition methods: • B i c o n n e c t e d C o m p o n e n t s (short: BICOMP) [Freuder, 1985]. Any graph G = (V,E) can be decomposed into a pair (T, X), where T is a tree, and the labeling function X associates to each vertex of T a biconnected component of G (a component which remains connected after any one-vertex removal). The biconnected width of a hypergraph H, denoted by BIC0MPw id th (H) , is the maximum number of vertices over the biconnected components of the primal graph of H.. • Cyc l e C u t s e t (short: CUTSET) [Dechter, 1992]. A cycle cutset of a hypergraph H is a set S var(H) such that the subhypergraph of H induced by var(H) -~ S is acyclic. The CUTSET width of H is the minimum cardinality over al l its possible cycle cutsets. • T ree C l u s t e r i n g (short: TCLUSTER) [Dechter and Pearl, 1989]. The tree clustering method is based on a tr iangulation algori thm which transforms the primal graph G = (V, E) of any CSP instance I into a chordal graph . The maximal cliques of are then used to build the constraint scopes of an acyclic CSP /' equivalent to /. The tree-clustering width (short: TCLUSTER width) of is 1 if is an acyclic hypergraph; otherwise it is equal to the maximum cardinality over the cliques of the chordal graph • T r e e w i d t h (TREEWIDTH) [Robertson and Seymour, 1986]. We omit a formal definition of graph treewidth here. The TREEWIDTH of a hypergraph H is the treewidth of its pr imal graph plus one. As pointed out below, TREEWIDTH and TCLUSTER are two equivalent methods. • H i n g e D e c o m p o s i t i o n s (short: HINGE) [Gyssens et a/., 1994; Gyssens and Paredaens, 1984], Let H be a hypergraph, and let V var('H) be a set of variables and X,Y ' var(H). X is [V]-adjacent to Y if there exists an edge h _ edges(H) such that {X,Y} {h V). A [V]-path from X to Y is a sequence X = . , . . . , = Y of variables such that: Xi is [V]-adjacent to . , for each i [0...l-1]. A set W var(H) of variables is [V']-connected if W there is a [V]-path from X to Y. A [V]-component is a maximal [V]-connected non-empty set of variables W (var(H) — V). For any GOTTLOB, LEONE, A N D SCARCELLO 395 [V1-component C, let edges(C) = edges Let H HS and let H be either edges(H) or a proper subset of edges(H) containing at least two edges. Let C\, .., be the connected -components of H. Then, H is a hinge if, for i = l , . . . , m , there exists an edge H such that var(edges \ A hinge is minimal if it does not contain any other hinge. (Our definition of hinge is equivalent to the original one in [Gyssens et a/., 1994; Gyssens and Paredaens, 1984].) A hinge-decomposition of H is a tree T such that all the following conditions hold: (1) the vertices of T are min i mal hinges of H; (2) each edge in edges(H) is contained in at least one vertex of T; (3) two adjacent vertices A and D of T share precisely one edge L edges(H)\ moreover, L consists exactly of the variables shared by A and . (4) the variables of H shared by two vertices of T are entirely contained wi th in each vertex on their connecting path in T. The size (i.e., the cardinali ty) of the largest vertex of T is called the degree of cyclicity of H. This is precisely what we call here the HINGE width of H. It was shown in [Gyssens and Paredaens, 1984] that for any CSP instance /, the HINGE wid th of H1 is the cardinali ty of the largest minimal hinge of • H i n g e D e c o m p o s i t i o n + T ree C l u s t e r i n g (short: [Gyssens et a/., 1994]. It has been shown [Gyssens et a/., 1994] that the minimal hinges of a hypergraph can be further decomposed by means of the tr iangulat ion technique of the above-described tree-clustering method. This leads to the method. Let T = (N, E) be a hinge tree of a hypergraph H. For any hinge H N\ let w(H) be the min imum between the cardinali ty of H and the TCLUSTER width of the hypergraph (var(H), H). T h e w i d t h o f T i s . Define t h e w i d t h o f H as the min imum HINGE w id th over all its hinge decompositions. For each of the above decomposition methods D it was shown that for any fixed k, given a CSP instance /, deciding whether a hypergraph has D-width is feasible in polynomial t ime and that solving CSPs whose associated hypergraph is of width A: can be done in polynomial t ime. In part icular, D consists of two phases. Given a CSP instance /, the (A:-bounded) D-width w of along w i th a corresponding decomposition is first computed. Explo i t ing this decomposition, I is then solved in t ime (for most methods this phase consists of the solution of an acyclic CSP instance equivalent to I). The cost of the first phase is independent on the constraint relations of /; in fact, it is , where