New Structural Decomposition Techniques for Constraint Satisfaction Problems

Transcription

1 113 New Structural Decompoition Technique for Contraint Satifaction Problem Yaling Zheng and Berthe Y. Choueiry Contraint Sytem Laboratory, Univerity of Nebraka-Lincoln Abtract. We propoe four new tructural decompoition technique for Contraint Satifaction Problem. We compare thee four technique both theoretically and experimentally with hinge decompoition and hypertree decompoition. Our experiment how that one of our technique offer the bet trade-off between the computational cot of the decompoition and the width of the reulting decompoition tree. 1 Introduction Many important practical problem uch a cheduling, reource allocation, and product configuration can be modeled a a Contraint Satifaction Problem (CSP), which conit of a et of variable, the domain of thee variable, and a et of contraint over thee variable retricting allowed combination of value for variable. Although CSP are in NP-complete in general, decompoition technique borrowed from the area of databae have been ued to characterize tractable clae of CSP [1 4]. The baic principle i to decompoe the CSP into ub-problem that are organized in a tree tructure. The ubproblem are then olved indepently, and the olution are propagated in a backtrack-free manner along the tree [5] to yield a olution to the initial CSP, a decribed by Dechter and Pearl [1]. We propoe new decompoition technique and poition them in the context of the hierarchy pecified by Gottlob et al. [4], which unifie main decompoition trategie and compare them in term of generality. The main technique are biconnected decompoition (BICOMP) [6], hinge decompoition (HINGE) [2, 3], tree clutering (TCLUSTER) [1], hinge decompoition combined with tree clutering (HINGE TCLUSTER ) [2], and hypertree decompoition (HYPERTREE) [7]. Thee technique can be further characterized by their computational complexity and the width of the tree they generate (which i the ize of the larget ub-problem in the tree). Among the above method, HYPERTREE i the mot general and yield tree with the mallet poible width. However, it remain cotly in practice even though it complexity i polynomial [8] (ee experiment in Section 8). HINGE i a more efficient but le general trategy than HYPERTREE. In thi paper, we generalize HINGE into HINGE +, and introduce CUT a a variation of HINGE. Further, we propoe a new technique, TRAVERSE, which we combine with CUT to yield a B. Falting et al. (Ed.): CSCLP 2004, LNAI 3419, pp , c Springer-Verlag Berlin Heidelberg 2005

2 114 Yaling Zheng and Berthe Y. Choueiry new technique CaT. In ummary, HINGE + generalize HINGE, and CaT generalize CUT. We evaluate our new technique theoretically and empirically on randomly generated hypergraph. Our experiment how that CaT provide the bet trade-off between the width of the generated tree and the computational cot of the decompoition. Thi paper i organized a follow. Section 2 review the preliminarie of CSP. Section 3 introduce HINGE +. Section 4 decribe CUT, which i a variation of HINGE +. Section 5 introduce a new technique called TRAVERSE. Section 6 combine CUT and TRAVERSE into CaT. Section 7 etablihe the formal relationhip among thee technique, and alo with repect to HINGE and HYPERTREE. Section 8 demontrate the effectivene of CaT on randomly generated problem. Finally, Section 9 conclude the paper. 2 Background A CSP i defined a a tuple P =(V, D, C), where V i a et of variable, D i a et of value domain for the variable, and C i a et of contraint that retrict the acceptable combination of value to variable. Every contraint C i Ci a relation over a et S i Vof variable, and pecifie the et of allowed tuple a a ubet of the Carteian product of the domain of S i.wedenotetheetof variable involved in contraint C i by Scope(C i ), and the union of the cope of a et of contraint {C i } by Var({C i }). A olution to the CSP i an aignment of value to all variable uch that all the contraint are imultaneouly atified. The CSP can be repreented by it aociated contraint hypergraph. The contraint hypergraph of a CSP P =(V, D, C) igivenbyh =(V, S), where S i a et of hyperedge correponding to the cope of the contraint in the CSP. Figure 1 how the hypergraph H cg of a CSP with 22 variable and 16 contraint. The primal graph of a contraint hypergraph H =(V, S) i a graph G =(V,E), where E i a et of edge relating any 2 variable that appear in the cope of a contraint in the CSP. Figure 2 how the primal graph of H cg. Further, we ay that a hypergraph i connected when it correponding primal graph i connected. Each connected component of the primal graph define a connected component of the hypergraph. Acyclic CSP are thoe CSP whoe aociated contraint hypergraph i acyclic. A contraint hypergraph H i acyclic iff it primal graph G i chordal (i.e., every cycle of length at leat 4 ha an edge connecting 2 non-adjacent ver Fig. 1. A contraint hypergraph H cg. Fig. 2. The primal graph of H cg.

3 New Structural Decompoition Technique for Contraint Satifaction Fig. 3. AjointreeofH cg. tice) and conformal (i.e., there i a one-to-one mapping between each maximal clique of the primal graph and the cope of the contraint) [9]. The contraint hypergraph H cg hown in Figure 1 i not acyclic. Following [10], a join tree JT(H) for a contraint hypergraph H i a tree whoe node are the edge of H uch that whenever the ame vertex X V appear in 2 hyperedge 1 and 2 S,then 1 and 2 are connected, and X appear in each node on the unique path linking 1 and 2 in JT(H). In other word, the et of node in which X appear include a (connected) ubtree of JT(H). The width d of a join tree i the maximum number of hyperedge in all the node of the join tree. Figure 3 how a join tree of H cg of width d=2. The principle of tructural decompoition technique i to compute an equivalent join tree for a given contraint hypergraph. Each node in thi tree i a ub-problem for which we find all olution, then, while applying directional arc-conitency to the join tree, we can olve the CSP in a backtrack-free manner [1, 2]. The complexity of olving the ub-problem i O( S l d d log l), where l i the maximum ize of a contraint in S and d the width of the join tree [2]. Gottlob et al. [4] defined a et of criteria for comparing decompoition method, where C(D i,k) i a cla of CSP for which there exit a decompoition of width k by the decompoition method D i that can be olved in polynomial time. Thee criteria are a follow (taken verbatim from [4]): 1. Generalization. D 2 generalize D 1 if there exit a contant δ 0 uch that, for each level k, C(D 1,k) C(D 2,k+δ) hold. In practical term, thi mean that whenever a cla C of contraint i tractable according to method D 1, it i alo tractable according to D Beating. D 2 beat D 1 if there exit an integer k uch that C(D 2,k) C(D 1,m) for any m. Intuitively, thi mean that ome clae of problem are tractable according to D 2 but not according to D Strong Generalization. D 2 trongly generalize D 1 if D 2 generalize D 1 and D 2 beat D 1. Thi mean that D 2 i really the more powerful method given that, whenever D 1 guarantee polynomial runtime for contraint olving, then D 2 alo guarantee tractable contraint olving. However, there are clae of contraint that can be olved in polynomial time by uing D 2 but are not tractable according to D Strongly Incomparable. D 1 and D 2 are trongly incomparable if both D 1 beat D 2 and D 2 beat D 1. Figure 4 how the hierarchy developed by Gottlob et al. [4] baed on the above compariion criteria. Whenever two decompoition method are not related by a directed path, they are trongly incomparable.

4 116 Yaling Zheng and Berthe Y. Choueiry HINGE TCLUSTER [Gyen et al., 1994] HINGE [Gyen et al., 1994] HYPERTREE [Gottlob et al., 2002] TCLUSTER w * [Dechter & Pearl, 1989] TREEWIDTH [Roberton & Seymour, 1986] HYPERCUTSET [Gottlob et al., 2000] BICOMP [Freuder, 1985] CUTSET [Dechter, 1987] D 1 D 2 indicate that D 2 i trongly more general than D 1 Fig. 4. The hierarchy of contraint tractability of [4]. 3 Hinge + Decompoition (HINGE + ) In thi ection, we introduce HINGE + a an improvement of HINGE. A pecified by Gyen et al. [2], HINGE decompoe the contraint hypergraph into a join tree where each node (called 1-hinge) i a et of hyperedge and 2 node that are adjacent in the tree hare exactly one hyperedge. Figure 5 how a decompoition of H cg of Figure 1 by HINGE where d = 12. The reulting decompoition guarantee a et of propertie (i.e., inheritance, decompoition, and ineparability) that they define. They alo attempted to generalize their approach to k-hinge, where a k-hinge i a node in the join tree connected to other node with at mot k hyperedge. However, they howed that their algorithm for 1- hinge cannot be generalized to achieve a correct reult. The width of the join tree of Figure 5 i particularly high. We noticed that by allowing the node of the tree to connect through more than 1 hyperedge (a uggeted by k-hinge of Jeavon et al. [3]), we can obtain a finer decompoition uch a the one hown in Figure 6. We introduce 3 important definition, which we will ue to define HINGE +, our improvment on HINGE: Definition 1. Remain-hg(F, S). Given a connected contraint hypergraph H = (V, S) and a et of hyperedge F S, we define H r =(V r, S r ),denotedremainhg(f, S), a the remaining contraint hypergraph obtained after removing F from S. More formally: V r = V\Var(F ) and S r = h S h \ Var(F ) Fig. 5. Applying HINGE to H cg Fig. 6. A finer decompoition than that of Figure 5.

5 New Structural Decompoition Technique for Contraint Satifaction 117 Definition 2. i-cut. Given a connected contraint hypergraph H =(V, S) where S i +1,ani-cut of H i a et of hyperedge F uch that: 1. F S and F = i; and 2. Remain-hg(F, S) ha at leat 2 component. Definition 3. Max-Size(F, H). Given an i-cut F of a contraint hypergraph H =(V, S), Max-Size(F, H) i the larget number of hyperedge in a connected component in Remain-hg(F, H). Given a contraint hypergraph H, HINGE continuouly find 1-cut (connecting 1-hinge). We improve HINGE by finding 1-cut through k-cut, where k i a pecified maximum cut-ize. The difficulty here i to chooe among the i-cut for agiveni (1< i k), a there may be more than one poible choice. We olve thi problem by chooing the i-cut that yield the minimum value of Max-Size. Now we define the join tree reulting from HINGE + : Definition 4. k-hinge + -tree. Given a contraint hypergraph H =(V, S), akhinge + -tree of H i a tree, T =(N,A), withnoden and labeled arc A, uch that: 1. For each tree node, p S; 2. For each hyperedge h S, there exit a tree node p uch that h p; 3. For 2 adjacent tree node p 1 and p 2, there exit an i-cut C (1 i k) uch that Var(p 1 ) Var(p 2 ) = Var(C); and 4. For each variable Y V,theet{p N Y Var(p)} induce a connected ubtree of T. Given a contraint hypergraph H and a contant number k, which i the maximum cut ize, HINGE + (ee Algorithm 1) return a k-hinge + -tree by finding 1-cut through k-cut. The wort cae of the algorithm occur when there are no i-cut 1 i (k 1). In thi cae, line 11 loop at mot S k time, and each loop can be performed in O( V S ) time. Therefore, the wort-cae time complexity of HINGE + i O( V S k+1 ). Since k i ued to limit the cut ize, Algorithm 1 remain polynomial. Figure 7 how a 2-hinge + -tree for H cg Fig. 7. Applying HINGE + to H cg with k =2.

6 118 Yaling Zheng and Berthe Y. Choueiry Input: A hypergraph H =(V, S) and a maximum cut-ize k. Output: Ank-hinge + -tree T for (V, S). 1 i 1; 2 S cut ; 3 N i {S}; 4 Mark every hyperedge in S a unchoen ; 5 foreach j from 1 to k tep by 1 do 6 Mark the node in N i a j-non-minimal; 7 while not all node of N i are marked j-minimal do 8 Chooe a j-non-minimal node F in N i; 9 j-combination all combination of j unchoen hyperedge in F ; 10 j-cut ; 11 foreach j-combination X j-combination do 12 Γ {G X G i a connected component in Remain-hg(X, F )}; 13 if ( Γ > 1) and( C q {S cut (S cut S cut) and (S cut F )}, Γ p Γ uch that C q Γ p) then 14 j-cut j-cut {X}; 15 if j-cut then 16 chooe a j-cut C with mallet Max-Size(j-cut, F ); 17 Mark the hyperedge in C a choen ; 18 S cut S cut {C}; 19 Γ {G C G i a connected component in Remain-hg(C, F )}; 20 N i+1 (N i \{F}) Γ ; 21 Mark C a a j-cut of every element in Γ ; 22 Let γ: {FN 1,...,FN q} Γ uch that FN i γ(fn i) ; 23 A i+1 (A i \{({F, F },C) ({F, F },C) A i}) {({γ(fn),fn},c) ({F, FN},C) A i} {({Γ 0,Γ y},c) Γ 0 i an arbitrary choen element from Γ, Γ y Γ and Γ y Γ 0}; 24 Mark all the new node added to N i+1 a j-non-minimal; ele 25 Mark F a j-minimal; 26 i i +1; 27 T (N i,a i); Algorithm 1: HINGE +. 4 Cut Decompoition (CUT) In thi ection, we introduce CUT a a variation of HINGE +. The arc incident to every node in the equivalent join tree of a contraint hypergraph obtained by CUT are labeled by at mot 2 ditinct cut. For HINGE +, the arc incident to a given node in an equivalent join tree of a contraint hypergraph obtained by HINGE + can be labeled by more than 2 ditinct cut. For example, in the join

7 New Structural Decompoition Technique for Contraint Satifaction 119 tree of Figure 7, the arc incident to the node { 4, 5, 6, 11, 12 } are labeled with three different cut, namely { 4, 5 }, { 6, 12 },and{ 11 }. The algorithm of CUT i obtained by replacing the condition in line 13 with the following one: 1. Γ > 1; 2. For C q {S cut (S cut S cut )and(s cut F )}, thereexitγ p Γ uch that C q Γ p ;and 3. For every 2 et of hyperedge C i and C j S cut,ifc i C j,andc i Γ i,c j Γ j,thenγ i Γ j. The above condition guarantee that no more than 2 cut label the arc incident to a node in the join tree obtained by CUT. (Thi feature allow u to further travere each tree node from one cut to another cut and i exploited in Section 5.) The complexity of CUT i the ame a that of HINGE +. Figure 8 how the reult of applying CUT (the maximum cut ize k i 2) to the contraint hypergraph H cg hown in Figure Fig. 8. Applying CUT to H cg Travere Decompoition (TRAVERSE) In thi ection, we introduce a imple weep-like decompoition technique called TRAVERSE. We decribe two variation of TRAVERSE: TRAVERSE-I and TRAVERSE-II. TRAVERSE-I take a contraint hypergraph and one et of hyperedge in it, and weep through the hypergraph from the et of hyperedge to generate an equivalent join tree of the contraint hypergraph. TRAVERSE-II take a contraint hypergraph and 2 et of hyperedge from the hypergraph and weep through the contraint hypergraph from the firt et of hyperedge to the econd et of hyperedge to generate an equivalent join tree of the contraint hypergraph. For convenience, we firt introduce the definition of Neighbor(F, S) that will be ued in Algorithm 2 and Algorithm 3. Definition 5. Neighboring hyperedge. The neighboring hyperedge of a et of hyperedge F in a contraint hypergraph H = (V, S) with F S, denoted Neighbor(F, S), i a et given by: {e e F, e F, and Var({e}) Var(F ) }. (1) Given a contraint hypergraph H =(V, S) and a et of hyperedge F S, TRAVERSE-I return a unique join tree obtained by Algorithm 2 via weeping through the contraint hypergraph tarting from the hyperedge in F. We

8 120 Yaling Zheng and Berthe Y. Choueiry Input: a contraint hypergraph H =(V, S) and a et of hyperedge F S. Output: an equivalent join tree T for H. 1 N ; A ; 2 Mark any hyperedge e S a unviited ; 3 F v {e Var({e}) Var(F )}; 4 N N {F v}; 5 F jv F v; 6 Mark any hyperedge in F jv a viited ; 7 while not all hyperedge in S are viited do 8 F Neighbor(F jv, theetofall unviited hyperedge); 9 F v {e Var(e) Var(F ) }; 10 N N {F v}; 11 A A {(F jv,f v)}; 12 F jv F v; 13 Mark every hyperedge in F jv a viited ; T (N,A); Algorithm 2: TRAVERSE-I. denote Travere-I(H,F) the reult obtained by applying Algorithm 2 with F on H. The loop in line 7 of Algorithm 2 execute at mot S time, and each execution can be performed in O( V S ) time. Therefore, the wort-cae time complexity of TRAVERSE-I i O( V S 2 ). Figure 9 how the join tree computed by TRAVERSE-I tarting from { 1 } in H cg. Becaue it weep through the contraint hypergraph, TRAVERSE alway compute a join tree that i a connected chain, provided the contraint hypergraph i connected. The reult of the decompoition dep on F, the tarting et of hyperedge. If we travere H cg of Figure 1 tarting from { 6,, 12 }, Algorithm 2 would yield a join tree of width d = 10. Starting from { 1 }, the width i d = 3 (ee Figure 9). Our goal i to combine CUT with TRAVERSE to improve the k-hinge + - tree computed by CUT (Section 6). To thi, we introduce TRAVERSE-II (Algorithm 3), which allow u to weep the contraint hypergraph between 2 cut. TRAVERSE-II take a contraint hypergraph and 2 et of hyperedge, and then weep through the contraint hypergraph from the firt et of hyperedge to the econd et of hyperedge to generate an equivalent join tree of thi contraint hypergraph. We denote Travere-II(H,C 1,C 2 ) the reult of applying TRAVERSE-II to H from C 1 to C 2. Figure 10 how the join tree obtained by applying TRAVERSE-II to H cg from { 1 } to {, 16 }. The loop in line 7 of Algorithm 3 execute at mot S time, and each iteration can be performed in O( V S ) time. Therefore, the complexity of TRAVERSE-II i O( V S 2 ) Fig. 9. Applying TRAVERSE-I to H cg from { 1}. Fig. 10. Applying TRAVERSE-II to H cg from { 1} to {, 16}.

9 New Structural Decompoition Technique for Contraint Satifaction 121 Input: a contraint hypergraph H =(V, S), a et of hyperedge C 1 and another et of hyperedge C 2. Output: an equivalent join tree T for H. 1 N ; A ; 2 Mark any hyperedge e S a unviited ; 3 F d {e Var(e) Var(C 2)}; 4 F v {e Var(e) Var(C 1)}; 5 N N {F v}; 6 Mark any hyperedge in F jv a viited ; 7 while (F v F d ) and (not all hyperedge in S are viited ) do 8 F Neighbor(F jv \ F d,theetofall unviited hyperedge F d ); 9 F v {e Var(e) Var(F )}; 10 N N {F v}; 11 A A {(F jv,f v)}; 12 F jv F v; 13 Mark every hyperedge in F jv a viited ; T (N,A); Algorithm 3: TRAVERSE-II. 6 Cut-and-Travere Decompoition (CaT) In thi ection, we introduce CaT, which combine CUT with TRAVERSE. The algorithm of CaT i given in Algorithm 4. Given a contraint hypergraph H =(V, S) and a maximum cut ize k, Algorithm 4 firt applie CUT to H and generate a k-hinge + -tree in which the arc incident to any tree node are labeled with at mot 2 cut. Thi tep can be implemented in O( V S k+1 ) time. Then, Algorithm 4 applie either TRAVERSE-I or TRAVERSE-II to every tree node in the k-hinge + -tree and generate a et of ub-join tree. Finally, the algorithm combine thee ub-join tree into 1 join tree. The travere proce can be performed in O( V S 2 ) time. Therefore, the complexity of CaT i O( V S k+1 + V S 2 ). Since k 1, the complexity of CaT i O( V S k+1 ). Note that the HYPERTREE algorithm compute an optimal hypertree of H that ha a width within a given bound d; the algorithm return failure if no uch decompoition exit [10]. In CaT, the contant k retrict the maximum cut ize but doe not retrict the width of the generated join tree. Figure 11 and Figure 12 how the equivalent join tree of H cg computed by CaT and HYPERTREE. In thi cae, the width of the join tree obtained by CaT and HYPERTREE are both equal to 2. 7 Characterization In thi ection, we compare our 4 technique with HINGE and HYPERTREE in term of the criteria propoed by Gottlob et al. [4]. Then, we integrate our reult into their hierarchy hown in Figure 4. Finally, we ummarize the complexity of all ix technique.

10 122 Yaling Zheng and Berthe Y. Choueiry Input: A hypergraph H =(V, S) and a maximum cut-ize k. Output: An equivalent join tree T for H. Cut H into a tree with tree node P 1,..., P m by CUT; N ; A ; foreach i from 1 to m do witch the number of cut labeling the arc incident to P i; do cae 0 (N i,a i) Travere-I(P i, any hyperedge in P i) cae 1 /* C i the only cut labeling the arc incident to P i */ (N i,a i) Travere-I(P i, C) cae 2 /* C 1 and C 2 are the cut labeling the arc incident to P i */ if the width of Travere-II(P i, C 1, C 2) the width of Travere-II(P i, C 2, C 1) then (N i,a i) Travere-II(P i, C 1, C 2) ele (N i,a i) Travere-II(P i, C 2, C 1) N N {N i}; A A {A i}; T (N,A); Algorithm 4: CaT. Firt, we introduce two pecial clae of contraint hypergraph borrowed from [4]: Circle(n) (ee Figure 13) and book(n) (ee Figure 14). Thee graph are defined a follow. For any n 3, Circle(n) i a contraint hypergraph having n hyperedge {h 1,...,h n } uch that: h i = {X i,x i+1 } for 1 i n 1and h n = {X n,x 1 }. For any n>0, book(n) i a contraint hypergraph with 2n +2 vertice and 3n + 1 hyperedge that form n quare (page of the book) with exactly one common edge {X, Y }. The hyperedge are defined a follow: b 0 = {X, Y }; b 3i+1 = {X, X i } for 1 i n; b 3i+2 = {X i,y i } for 1 i n; and b 3i+3 = {Y i,y} for 1 i n. Theorem 1. HINGE + trongly generalize HINGE. Proof. (HINGE + beat HINGE.) Conider the graph Circle(n) for ome n 3. It i eay to ee that the HINGE width of Circle(n) i n, while it HINGE + width (with a maximum cut ize of 2) i no greater than 4. Hence, n 3 {Circle(n)} C(HINGE +, 4), while n 3 {Circle(n)} C(HINGE, k)holdforeveryk>0.

11 New Structural Decompoition Technique for Contraint Satifaction 123 {0, 1, 2, 3} { 1, 3 } {1, 4, 3, 11} { 3, 4 } {4, 5, 6, 11, 12, 13} { 5, 11 } {13, 14, 22} { 17, 12 } {6, 7, 13, 14} { 6, 22 } {7, 8, 9, 14, 15, 16}{ 7, 13 } {9 10, 16, 17} { 8, 14 } {10, 17, 18, 21} {, 16 } {10, 17, 18, 20}{, 15 } {10, 17, 18, 19} {, 10 } Fig. 11. Applying CaT to H cg. x 1 x x n Fig. 13. Circle(n). Fig. 12. Applying HYPERTREE to H cg. X X 1 X 4 X 2 X Y 1 Y 3 4 Y 2 Y Fig. 14. Book(4). Y 3 Therefore, HINGE + beat HINGE. (HINGE + generalize HINGE.) It i eay to ee that HINGE i a pecial cae of HINGE + when the maximum cut ize i 1. Thu, for I C(HINGE, k), I C(HINGE +, k) hold. Theorem 2. HYPERTREE generalize HINGE +. Proof. It i obviou that I C(HINGE +, k), I C(HYPERTREE, k) hold. Theorem 3. CaT generalize CUT. Proof. The firt phae of CaT i CUT. The econd phae of CaT further decompoe each tree node of the join tree obtained by CUT. It i eay to ee that I C(CUT, k), I C(CaT, k) hold. Theorem 4. HYPERTREE generalize CaT. Proof. It i obviou that I C(CaT, k), I C(HYPERTREE, k) hold. Theorem 5. HYPERTREE trongly generalize TRAVERSE. Proof. (HYPERTREE generalize TRAVERSE.) It i obviou that I C(TRAVERSE,k), I C(HYPERTREE, k) hold. (HYPERTREE beat TRA- VERSE.) Conider the graph book(n) for ome n 1, it i eay to ee that the TRAVERSE width of book(n) i greater than n 2, while it HYPERTREE width i 2. Hence, n 1 {book(n)} C(HYPERTREE, 2), while n 1 {book(n)} C(TRAVERSE, k) for every k>0. Theorem 6. HINGE and TRAVERSE are trongly incomparable. Proof. (HINGE beat TRAVERSE.) Conider the graph book(n) for ome n 1, it i eay to ee that the TRAVERSE width of book(n) i greater than

12 124 Yaling Zheng and Berthe Y. Choueiry n 2, while it HINGE width i 4. Hence, n 1 {book(n)} C(HINGE+,4), while n 1 {book(n)} C(HINGE, k) for every k > 0. (TRAVERSE beat HINGE.) Conider the graph Circle(n) for ome n 3. It i eay to ee that the HINGE width of Circle(n) i n while it TRAVERSE width (from an arbitrary choen hyperedge) i 2. Hence, n 3 {Circle(n)} C(TRAVERSE, 2), while n 3 {Circle(n)} C(HINGE, k) holdforeveryk>0. Therefore, TRA- VERSE beat HINGE. Theorem 7. CUT beat TRAVERSE. Proof. Conider the graph book(n) for ome n 1, it i eay to ee that the TRA- VERSE width of book(n) i greater than n 2, while it CUT width i 4. Hence, n 1 {book(n)} C(CUT, 4), while n 1 {book(n)} C(TRAVERSE,k)for every k>0. Theorem 8. CaT beat TRAVERSE. Proof. Conider the graph book(n) for ome n 1, It i eay to ee that the TRAVERSE width of book(n) i greater than n 2 while it CaT width (with the maximum cut ize being 2) i 2. Hence, n 1 {book(n)} C(CaT, 2), while {book(n)} C(TRAVERSE, k) for every k>0. n 1 Theorem 9. HINGE + beat TRAVERSE. Proof. Conider the graph book(n) for ome n 1, it i eay to ee that the TRAVERSE width of book(n) i greater than n 2, while it HINGE+ width i 4. Hence, n 1 {book(n)} C(HINGE+, 4), while n 1 {book(n)} C(TRAVERSE,k) for every k>0. Theorem 10. CUT beat HINGE. Proof. Conider the graph Circle(n) for ome n 3. It i eay to ee that the HINGE width of Circle(n) i n, while it CUT width (with maximum cut ize being 2) i 2. Hence, n 3 {Circle(n)} C(CUT, 2), while n 3 {Circle(n)} C(HINGE, k) holdforeveryk>0. Therefore, CUT beat HINGE. The above theorem implied that CaT beat HINGE and HYPERTREE generalize CUT. The relationhip between HINGE + and CUT and between HINGE + and CaT are till need to be invetigated. Figure 15 ummarize the main relationhip tudied above. The olid directed edge from D 1 to D 2 indicate that D 2 trongly generalize D 1. The dotted directed edge from D 1 to D 2 indicate D 2 generalize D 1. Note that the picture i incomplete. Table 1 ummarize the complexity of the technique hown in Figure Preliminary Experiment In order to ae empirically the above technique, we compared their performance on randomly generated hypergraph in term of two criteria: the CPU time for computing the decompoition and the width of the reulting join tree.

13 New Structural Decompoition Technique for Contraint Satifaction 125 Table 1. Complexity of decompoition method. Technique Complexity HYPERTREE Normal form: opt-d-decomp [7] O( S 2d V 2 ) Reduced normal form [8] Bet cae: O( S d V + S 2 V ) HINGE O( V S 2 ) HINGE + O( V S k+1 ) CUT O( V S k+1 ) TRAVERSE O( V S 2 ) CaT O( V S k+1 ) Solving the CSP after decompoition O( S l d d log l) V : number of variable (i.e., vertice). S : number of contraint (i.e., hyperedge). d: width of the join tree reulting from a decompoition. k: maximum cut-ize. l: maximum ize of a contraint in S. HYPERTREE [Gottlob et al., 2002] D 1 D 2 indicate that TRAVERSE CaT CUT HINGE + HINGE [Gyen et al., 1994] D 2 i trongly more general than D 1 D 1 D 2 indicate that D 2 i more general than D 1 Fig. 15. Illutrating the relationhip between the variou tudied technique. For HYPERTREE, we ued the algorithm of Harvey and Ghoe [8], which improve on the opt-k-decomp algorithm of Gottlob et al. [10]. By tarting with k=1 and incrementing it value by 1 until it find decompoition, the algorithm we ued guarantee an optimal decompoition. We generated random hypergraph etting the number of contraint to 10, 11, 12, and 13. In each intance, we choe the arity of the contraint randomly in {2, 3, 4}. Table 2 ummarize the contraint hypergraph ued in the experiment. We et the maximum cut ize k=2 for HINGE +, CUT, and CaT. Figure 16 and Figure 17 how, for a fixed number of contraint, the average CPU time and average width of the generated join tree. Figure 16 and Figure 17 how the average CPU time and average width of different decompoition technique. Table 3 average thee reult over all 4000 intance generated. From thee experiment, we have the following obervation: For CPU time, TRAVERSE < HINGE < CUT CaT HINGE + HYPERTREE.

14 126 Yaling Zheng and Berthe Y. Choueiry Table 2. Contraint hypergraph ued in the experiment. #contraint #variable # intance 10 {16, 17,..., 25} 1000 (100 intance for each fixed number of variable) 11 {18, 19,..., 27} 1000 (100 intance for each fixed number of variable) 12 {20, 21,..., 29} 1000 (100 intance for each fixed number of variable) 13 {22, 23,..., 31} 1000 (100 intance for each fixed number of variable) CPU time (mec) HYPERTREE CPU time (mec) # Contraint HINGE+ CaT CUT HINGE TRAVERSE # Contraint Width TRAVERSE HINGE CUT HINGE+ CaT HYPERTREE # Contraint Fig. 16. Average CPU time. Fig. 17. Average width. TRAVERSE i the quicket technique followed by HINGE then CaT, HINGE +, and CUT, which have comparable value for the CPU time. All technique are ignificantly quicker than HYPERTREE. Indeed, the computationally cot of HYPERTREE i prohibitively high although it wort-cae time complexity i polynomial. For width, HYPERTREE CaT < HINGE + CUT HINGE < TRAVERSE. The join tree obtained with TRAVERSE ha the larget width. The average width of the join tree generated by HINGE + and CUT are maller than that of the join tree generated by HINGE. However, the difference of thee value are within 4%. The width of the join tree generated by CaT and HY- PERTREE differ by only 4%, which i negligible. Alo, they are ignificantly maller than thoe generated by the remaining technique. In ummary, CaT offer the bet trade-off between the CPU time and the width of the computed join tree among the decompoition method teted. 9 Concluion In thi paper, we propoed two main new tructural decompoition: HINGE + and CaT. HINGE + trongly generalize HINGE of Gyen et al. [2]. CaT i built by combining CUT (a variation of HINGE + ) and TRAVERSE (a weep-like

15 New Structural Decompoition Technique for Contraint Satifaction 127 Table 3. Average reult over all 4000 intance. Comparion criteria HINGE HINGE + CUT TRAVERSE CaT HYPERTREE Average CPU time [mec] Width decompoition technique). We compared thee technique among themelve and with HINGE and HYPERTREE both theoretically and experimentally. Our experiment howed that the CaT offer the bet trade-off between cot and quality of the reulting decompoition. In the future, we plan to addre the following iue: (1) Compare our technique with the remaining technique hown in Figure 4; and (2) Perform experiment on pecial type of graph (e.g., mall-world graph and clutered graph) and real-world problem (e.g., the one ued in [11]). Acknowledgment Thi work i upported by CAREER Award # from the National Science Foundation. The experiment were conducted utilizing the Reearch Computing Facility of the Univerity of Nebraka-Lincoln. Deb Derrick provided invaluable editorial help. Reference 1. Dechter, R., Pearl, J.: Tree Clutering for Contraint Network. Artificial Intelligence 38 (1989) Gyen, M., Jeavon, P.G., Cohen, D.A.: Decompoing Contraint Satifaction Problem Uing Databae Technique. Artificial Intelligence 66 (1994) Jeavon, P.G., Cohen, D.A., Gyen, M.: A Structural Decompoition for Hypergraph. Contemporary Mathematic 178 (1994) Gottlob, G., Leone, N., Scarcello, F.: A Comparion of Structural CSP Decompoition Method. Artificial Intelligence 124 (2000) Freuder, E.C.: A Sufficient Condition for Backtrack-Free Search. JACM 29 (1) (1982) Freuder, E.C.: A Sufficient Condition for Backtrack-Bounded Search. JACM 32 (4) (1985) Gottlob, G., Leone, N., Scarcello, F.: Hypertree Decompoition and Tractable Querie. Journal of Computer and Sytem Science 64 (2002) Harvey, P., Ghoe, A.: Reducing Redundancy in the Hypertree Decompoition Scheme. In: The 15 th IEEE International Conference on Tool with Artificial Intelligence (ICTAI 03). (2003) Dechter, R.: Contraint Proceing. Morgan Kaufmann (2003) 10. Gottlob, G., Leone, N., Scarcello, F.: On Tractable Querie and Contraint. In: 10 th International Conference and Workhop on Databae and Expert Sytem Application (DEXA 1999). (1999) Gottlob, G., Hutle, M., Wotawa, F.: Combining Hypertree, Bicomp, And Hinge Decompoition. In: Proc. of the 15 th ECAI, Lyon, France (2002)