New Structural Decomposition Techniques for Constraint Satisfaction Problems

Transcription

1 New Structural Decompoition Technique for Contraint Satifaction Problem Yaling Zheng and Berthe Y. Choueiry Contraint Sytem Laboratory Univerity of Nebraka-Lincoln yzheng Abtract. Technique for decompoing the contraint network of a Contraint Satifaction Problem (CSP) have yielded the identification of new tractable clae of CSP [1]. In thi paper, we propoe three new decompoition technique, namely HINGE +, CUT, and TRAVERSE, where HINGE + i a generalization of HINGE by Gyen et al. [2]. Further, we combine CUT and TRAVERSE into a new trategy, which we call Cutand-Travere (CaT). We introduce thee technique and compare them according to the criteria introduced by Gottlob et al. [1]. 1 Introduction Many important practical problem uch a cheduling, reource allocation, deign and product configuration can be modeled a a Contraint Satifaction Problem (CSP), where a et of deciion need to be made, each deciion ha a number of option, and the allowable combination of thee option are retricted by a et of contraint. Becaue a CSP i in general NP-complete, earch remain the ultimate mechanim for olving it. Decompoition i a common trategy for improving the performance of earch [3]. Recently, decompoition technique borrowed from the area of databae (i.e., acyclic conjunctive querie) have been ued to characterize tractable clae of CSP [4,2,5, 1]. 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 [6] to yield a olution to the initial CSP. We propoe new decompoition technique and we poition them in the context of the hierarchy pecified by Gottlob et al. [1], which unifie major decompoition trategie reported in the literature and compare them in term of generality. The main technique are the biconnected decompoition (BICOMP) [6], hinge decompoition (HINGE) [2, 5], tree clutering (TCLUSTER) [4], hinge decompoition combined with tree clutering (HINGE TCLUSTER ) [2], and hypertree decompoition (HYPERTREE) [7]. Thee technique can be further characterized by the width of the tree they generate and the their computational complexity. Among the above method, HYPERTREE i the mot general and yield tree with the mallet poible width. However, it cot i high. HINGE i a more efficient but le general trategy than HYPERTREE. In thi paper, we

2 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 new technique CaT. In ummary, HINGE + generalize HINGE, and CaT generalize CUT. 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. Finally, Section 8 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 C i a relation over a et S i of variable, and pecifie the et of allowed tuple a a ubet of the Carteian product of the domain of S i. We denote the et of 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. To olve a CSP we need to determine whether the CSP ha a olution and, if o, find one olution. Although CSP are NP-complete in general, ome of the CSP are tractable due to their retricted tructure. The CSP can be repreented with a contraint network, which i, in general, a hypergraph. The contraint hypergraph of a CSP P = (V, D, C) i given by H = (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 Fig.1. A contraint hypergraph H cg Fig.2. The primal graph of H cg. primal graph of a contraint hypergraph H = (V, S) i a graph G = (V, E) where E i a et of edge relating any two 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.

3 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., any cycle of length greater than 3 ha a chord) 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) [8]. H cg of Figure 1 i not acyclic. Following [9], 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 two 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= Fig.3. A join tree of H cg A tructural decompoition technique 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. 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. Figure 4 how the hierarchy, propoed by Gottlob et al. of known decompoition technique [1]. Thi hierarchy 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] D 1 D 2 indicate that CUTSET [Dechter, 1987] D 2 i trongly more general than D 1 Fig.4. The hierarchy of contraint tractability of [1]. characterized the relationhip between thee decompoition technique in term of the following criteria, where C(D i, k) i a cla of CSP for which there exit

4 a decompoition of width k by the decompoition method D i that can be olved in polynomial time [1] 1 : 1. Generalization. D 2 generalize D 1 if there exit a contant δ uch that, for each level k, C(D 1, k) C(D 2, k) 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) i not contained in cla 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. 3 Hinge + decompoition (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 two 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 one hyperedge (a uggeted by k-hinge of Jeavon et al. [5]), we can obtain better decompoition uch a the one we how in Figure 6. We introduce the concept of k-cut to achieve uch a reult, which yield HINGE +, our improvment on HINGE. Conider a contraint hypergraph H = (V, S) and a et of hyperedge F S. We define H r = (V r, S r ), denoted Remain-hg(F, H), a the remaining contraint hypergraph obtained after removing F from H. More formally: V r = V \ Var(F) S r = h S h \ Var(F). 1 In thi hierarchy method that are not related are guaranteed not comparable.

5 Fig.5. Applying HINGE to H cg. Fig. 6. A better decompoition than that of Figure 5. Definition 1 (i-cut). Given a connected contraint hypergraph H = (V, S) where S i+1, a i-cut of H i a et of hyperedge F that atifie the following condition: 1. F S and F = i; and 2. The remaining contraint hypergraph ha at leat 2 component. Definition 2 (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). Therefore, 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 a given i (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 3 (k-hinge + -tree). Given a contraint hypergraph H = (V, S), a k-hinge + -tree of H i a tree, T = (N, A), with node N and labeled arc A, uch that: 1. Each tree node p S; 2. For each hyperedge h S, there exit a tree node p uch that h p; 3. Two adjacent tree node p 1 and p 2, there exit an i-cut C (1 i k) that Var(p 1 ) Var(p 2 ) = Var(C); and 4. For each variable Y V, the et {p N Y Var(p)} induce a connected ubtree of T. Given a contraint hypergraph H and a contant number k, the hinge + decompoition algorithm hown in Figure 1 return a k-hinge + -tree by finding 1-cut through k-cut. In Algorithm 1, k i the maximum cut ize. The wort cae of the algorithm occur when there are no i-cut 1 i (k 1). In the wort cae,

6 Input: A hypergraph H = (V, S) and a maximum cut-ize k. Output: An k-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, 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: Algorithm of HINGE +.

7 line 11 loop at mot S k time, and each loop can be performed in O( V (S ) time. So the wort time complexity of HINGE + i O( V (S k+1 ). Since k i ued to limit the cut ize, Algorithm 1 remain polynomial. Figure 6 how a 2-hinge + -tree for H cg Fig.7. Applying HINGE + on H cg with k = 2. 4 Cut decompoition (CUT) Notice that, in general, the arc incident to a given node in a join tree may be labeled by two or more ditinct cut. For example, in the join tree of Figure 7, the arc incident to the node { 4, 5, 6,, 12 } are labeled with three different cut, namely { 4, 5 }, { 6, 12 }, and { }. In thi ection, we conider a variation of HINGE + called CUT, which retrict to at mot 2 the number of different cut labeling the arc incident to any given node in the join tree. Thi i achieved by replacing the condition in line 13 with the following one: 1. Γ > 1; 2. C q S cut, there exit Γ p Γ uch that C q Γ p ; and 3. For every two et of hyperedge C i and C j S cut, if C i C j, and C i Γ i, C j Γ j, then Γ i Γ j. The above condition guarantee that no more than two 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 +, and it thu i polynomial. Figure 8 how the reult of applying CUT to the contraint hypergraph H cg of Figure 1 with k =2. 5 Travere decompoition (TRAVERSE) Given a contraint hypergraph H = (V, S) and a et of hyperedge F S, TRAVERSE return a unique join tree obtained by Algorithm 2 via weeping through the contraint hypergraph tarting from the hyperedge in F. We denote by Travere-I(H, F) the reult obtained by applying Algorithm 2 with F on H.

8 Fig.8. Applying CUT on H cg. Definition 4 (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) 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, the et of all 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: Algorithm for TRAVERSE-I. The loop in line 7 of the Algorithm 2 execute at mot S time, and each execution can be performed in O( V (S ) time. Therefore, the complexity TRA- VERSE i V S 2 ) and i polynomial. Figure 9 how the join tree computed by travere decompoition from { 1 } in H cg. 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 yield a join tree of width d = 10. Starting from { 1 }, the width i d = 3 (ee Figure 9).

9 Fig.9. Applying TRAVERSE on H cg. We can combine CUT with TRAVERSE to improve the k-hinge + -tree computed by CUT. To thi, we need to modify the Algorithm 2 in order to allow it to weep the contraint hypergraph between two cut. In order to travere a contraint hypergraph from one et of hyperedge to another et of hyperedge, which will be ued in Section 6, we only need to revie Algorithm 2 to Algorithm 3. We denote by Travere-II(H, C 1, C 2 ) the reult of applying Algorithm 3 C 1 to C 2 on H The complexity of Algorithm 3 i alo O( V S 2 ). 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. N ; A ; Mark any hyperedge e S a unviited ; F d {e Var(e) Var(C 2)}; F v {e Var(e) Var(C 1)}; N N {F v}; Mark any hyperedge in F jv a viited ; while (F v F d ) and (not all hyperedge in S are viited ) do F Neighbor(F jv \ F d, the et of all unviited hyperedge F d ); F v {e Var(e) Var(F )}; N N {F v}; A A {(F jv, F v)}; F jv F v; Mark every hyperedge in F jv a viited ; T (N, A); Algorithm 3: Algorithm for TRAVERSE-II. 6 Cut-and-Travere decompoition (CaT) In thi ection, we introduce CaT, which combine CUT with TRAVERSE. The algorithm for CaT i given in Algorithm 4. Theorem 1. Given a contraint hypergraph H and a contant number k, the CaT algorithm compute an equivalent join tree for H.

10 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 (N i, A i) Travere-I(P i, C) where C i the only cut labeling the arc incident to P i ; cae 2 (N i, A i) Travere-II(P i, C 1, C 2) where C 1 and C 2 are the cut labeling the arc incident to P i ; N N {N i}; A A {A i}; T (N, A); Algorithm 4: The algorithm for CaT. Proof. Algorithm 4 firt applie CUT to compute T 1 a k-hinge + -tree equivalent to a contraint hypergraph H = (V, S). The arc incident to any tree node in T 1 are labeled with at mot two cut. The algorithm travere each tree node in T 1 by applying TRAVERSE. If there i only one tree node in T 1, then CaT become TRAVERSE, which exactly compute an equivalent join tree for H. If there are at mot 2 tree node in T 1, then each tree node contain at leat 1 cut and at mot 2 cut. Since the reult of TRAVERSE on each tree node containing 1 cut i a chain-like join tree that begin with one cut, the reult of TRAVERSE on each tree node containing 2 cut i alo a chain-like join tree that begin with one cut and with another cut. The ub join tree for all tree node are exactly connected through thee cut, which guarantee the connectedne property of the combined join tree of thee ub join tree. Thu, the CaT algorithm compute an equivalent join tree for H. HYPERTREE [9] compute an optimal hypertree of H with a width within a given bound k; the algorithm return failure if no uch decompoition exit. In our approach, the contant k only retrict the ize of cut; it doe not retrict the width of the join tree computed by CaT. Therefore, CaT i more flexible than HYPERTREE.

11 Given a contraint hypergraph H = (V, S) and a contant number k, CaT firt compute an k-hinge + -tree T 1 by CUT, which can be implemented in O( V (S k+1 ). The travere proce can be performed in O( V (S 2 ). 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 ). Figure 10 and Figure 11 how the equivalent tree of H cg computed by CaT and HYPERTREE. In thi cae, the width of the decompoition obtained by CaT and HYPERTREE are the ame and equal to {0, 1, 2, 3} {, } {1, 4, 3, 11} { 3, 4 } {4, 5, 6, 11, 12, 13} {, } {13, 14, 22} {, } {6, 7, 13, 14} {, 22 } {7, 8, 9, 14, 15, 16}{, 13 } {9 10, 16, 17} { 8, 14 } {10, 17, 18, 21} {, 16 } {10, 17, 18, 20}{, 15 } {10, 17, 18, 19} {, 10 } Fig.10. Applying CaT on H cg. Fig. 11. Applying HYPERTREE on H cg. 7 Characterization In thi ection, we compare HINGE, HINGE +, CUT, TRAVERSE, CaT and HYPERTREE in term of the criteria propoed by Gottlob et al. in [1]. Finally, we enrich the new contraint tractability hierarchy propoed by thee reearcher. Theorem 2. CaT trongly generalize CUT. Proof. The firt tep of CaT i CUT. The econd tep of CaT i TRAVERSE. TRAVERSE will improve or keep the ame decompoition reult by CUT, thu CaT generalize CUT. For H cg hown in Figure 1, CaT compute a join tree with width 2 when limiting cut ize to 2, a hown in Figure 10. CUT compute a join tree with width 4 when limiting cut ize to 2, a hown in Figure 8. Thu, CaT beat CUT. Therefore, CaT trongly generalize CUT. Theorem 3. HINGE + trongly generalize HINGE. Proof. When k = 1, HINGE + i exactly the ame a HINGE. When k > 1, HINGE + firt perform HINGE, then continuouly find 2-cut through k-cut, which will improve or keep the ame decompoition reult by HINGE. Thu, HINGE + generalize HINGE. For H cg hown in Figure 1, HINGE + compute a

12 join tree with width 5 when limiting cut ize to 2, a hown in Figure 7, while HINGE compute a join tree with width 12, a hown in Figure 5. Thu, CaT beat CUT. Therefore, HINGE + trongly generalize HINGE. Alo, Since HYPERTREE alway compute an optimal decompoition, while HINGE +, CUT and CaT are baed on heuritic function, HYPERTREE generalize thee method. For TRAVERSE, the decompoition reult dep on the et of hyperedge it begin with, thu it cannot compare with HINGE, HINGE +, CUT, and CaT. Gottlob et al. [1] introduce the following comparion criteria to compare 2 different decompoition method D 1 and D 2. To compare HINGE, HINGE +, CUT, TRAVERSE, CaT and HYPERTREE with each other, we give two additional contraint hypergraph borrowed from [1]. For any n > 0, let triangle(n) be the graph (V, E) define a follow. The et of vertice V contain 2n + 1 vertice p 1,...,p 2n+1. For each even index i, 2 i 2n, {p i, p i 1 }, {p i, p i+1 }, and {p i 1, p i+1 } are edge in E. No other edge belong to E. Figure 12 how the triangle(3). The hypertree width of triangle(n) i 2. In fact, a hypertree (T, χ, λ), where T i a imple chain of n vertice v 1,..., v n, and, for each v i (1 i n), χ(v i ) = {p 2i 1, p 2i, p 2i+1 } and λ(v i ) contain the two edge {p 2i 1, p 2i } and {p 2i, p 2i+1 }, i a width 2 hypertree decompoition of triangle(n). For any n > 0, let book(n) be a graph having 2n + 2 vertice and 3n + 1 edge that form n quare (page of the book) having exactly one common edge {X, Y }. It i eay to ee that the hypertree width of book(n) i 2. Figure 13 how the graph book(4). p 2 p 4 p 6 p 1 p 3 p 5 p 7 Fig. 12. Triangle(3). Fig. 13. Book(4). Theorem 4. HYPERTREE trongly generalize CaT. Proof. HYPERTREE compute a join tree with the mallet poible width for a contraint hypergraph. Becaue CaT a heuritic decompoition method, it only chooe one poible decompoition. Therefore, HYPERTREE alway produce a join tree whoe width i at mot a large a CaT. Thu HYPERTREE generalize CaT. For triangle(3) hown in Figure 12, HYPERTREE compute a hypertree with width 2 a hown in Figure 14, while CaT compute a join tree with width 3 when limiting cut ize to 2, a hown in Figure 15. Note that here we denote 1 a the hyperedge {P 1, P 2 }, 2 a the hyperedge {P 2, P 3 }, 3 a the hyperedge {P 1, P 3 }, 4 a the hyperedge {P 3, P 4 }, 5 a the hyperedge {P 4, P 5 }, 6 a the hyperedge {P 3, P 5 }, a the hyperedge {P 5, P 6 }, 8 a the hyperedge {P 6, P 7 },

13 { p 1, p2, p 3 } { 1, 2 } { p 4, p 5, p 6 }{ 4, 5 } p 7 p 8 p 9 {,, } {, 8 } Fig. 14. Applying HYPERTREE on triangle(3). Fig.15. Applying CaT on triangle(3). and a the hyperedge {P 5, P 7 }. Thu, HYPERTREE beat CaT. Therefore, HYPERTREE trongly generalize CaT. Theorem 5. HYPERTREE trongly generalize TRAVERSE. Proof. HYPERTREE compute a join tree for a contraint hypergraph with the mallet hypertree width. TRAVERSE compute one poibility of an equivalent join tree for a contraint hypergraph. Therefore, for a ame contraint hypergraph, HYPERTREE alway produce a join tree whoe width i maller than or equal to the width of the join tree that TRAVERSE compute. HYPERTREE thu generalize TRAVERSE. For H cg hown in Figure 1, TRAVERSE compute a join tree with width 3, a hown in Figure 9, while HYPERTREE compute a join tree with width 2, a hown in Figure 11. Thu, HYPERTREE beat TRAVERSE. Therefore, HYPERTREE trongly generalize TRAVERSE. Theorem 6. HYPERTREE trongly generalize HINGE +. Proof. HYPERTREE compute a join tree for a contraint hypergraph with the mallet hypertree width. HINGE + compute one poible equivalent join tree of a contraint hypergraph. Therefore, for a ame contraint hypergraph, HYPER- TREE alway produce a join tree whoe width i maller than or equal to the width of the join tree that HINGE + compute. Thu HYPERTREE generalize HINGE +. For H cg hown IN Figure 1, HINGE + compute a join tree with width 4 when limiting cut ize to 2, a hown in Figure 7, while HYPERTREE compute a hypertree tree with width 2, a hown in Figure 10. Thu, HYPERTREE beat HINGE +. Therefore, HYPERTREE trongly generalize HINGE +. Theorem 7. TRAVERSE and CUT are trongly incomparable. Proof. For H cg hown in Figure 1, TRAVERSE compute a join tree with width 3, a hown in Figure 9, while CUT compute a join tree with width 4 when limiting cut ize to 2, a hown in Figure 8. Thu, TRAVERSE beat CUT. For book(4) hown in Figure 13, a travere decompoition from hyperedge {X 1, X, Y, Y 1 } ha width 3, while a cut decompoition limiting cut ize to 2 ha width 2. Thu, CUT beat TRAVERSE. Therefore, TRAVERSE and CUT are trongly incomparable. Theorem 8. TRAVERSE and HINGE + are trongly incomparable.

14 Proof. For H cg a hown in Figure 1, TRAVERSE compute a join tree with width 3, a hown in Figure 9, while HINGE + compute a join tree with width 5 when limiting cut ize to 2, a hown in Figure 8. Thu, TRAVERSE beat HINGE +. For book(4) hown in Figure 13, a travere decompoition tarting from hyperedge {X 1, X, Y, Y 1 } ha width 3, while a hinge + decompoition limiting cut ize to 2 ha width 2. Thu, HINGE + beat TRAVERSE. Therefore, TRAVERSE and HINGE + are trongly incomparable. Figure 16 how a contraint tractability hierarchy baed on the above comparion reult and Gottlob et al. concluion [1]. Thi hierarchy i not yet complete. HYPERTREE [Gottlob et al., 2002] TRAVERSE CaT HINGE + CUT D 1 D 2 indicate that D 2 i trongly more general than D 1 HINGE [Gyen et al., 1994] Fig.16. Comparing our technique to previou one. Table 1 ummarize the complexity of the decompoition technique we dicu in thi ection. Technique Table 1. Complexity of decompoition method. Complexity HYPERTREE Normal form: opt-d-decomp [7] O( S 2d V 2 ) Reduced normal form [10] 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 dlog 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

15 8 Concluion In thi paper, we propoed new technique to further improve the decompoition reult of HINGE. We preented a different view of HINGE and generalize thi method a HINGE +. Then we introduced a variation of HINGE + called CUT. We alo propoed a new decompoition method called TRAVERSE, which can be further combined with CUT a CaT. We howed that HINGE +, CUT, TRA- VERSE, and CaT can be performed in polynomial time. We compared HINGE, HINGE +, CUT, TRAVERSE, CaT and HYPERTREE with each other. We enriched the contraint tractability hierarchy introduced by Gottlob et al. in [1] by adding the comparion reult. In the future, we plan to compare HINGE +, CUT, TRAVERSE, and CaT with HINGE TCLUSTER and HINGE+BICOMP+HYPERTREE. Acknowledgment: Thi work i upported by CAREER Award # from the National Science Foundation. Deb Derrick provided invaluable editorial help. Reference 1. Gottlob, G., Leone, N., Scarcello, F.: A Comparion of Structural CSP Decompoition Method. Artificial Intelligence 124 (2000) Gyen, M., Jeavon, P.G., Cohen, D.A.: Decompoing Contraint Satifaction Problem Uing Databae Technique. Artificial Intelligence 66 (1994) Freuder, E.C., Hubbe, P.D.: A Dijunctive Decompoition Control Schema for Contraint Satifaction. In Sarawat, V., Hentenryck, P.V., ed.: Principle and Practice of Contraint Programming. MIT Pre, Cambridge, MA (1995) Dechter, R., Pearl, J.: Tree Clutering for Contraint Network. Artificial Intelligence 38 (1989) Jeavon, P.G., Cohen, D.A., Gyen, M.: A Structural Decompoition for Hypergraph. Contemporary Mathematic 178 (1994) 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) Dechter, R.: Contraint Proceing. Morgan Kaufmann (2003) 9. 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) 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)