Improving the Performance of Consistency Algorithms by Localizing and Bolstering Propagation in a Tree Decomposition Shant Karakashian, Robert Woodward & Berthe Y. Choueiry Constraint Systems Laboratory University of Nebraska-Lincoln Acknowledgments: • Experiments conducted at UNL’s Holland Computing Center • NSF Award RI-111795 1
Outline • Introduction • Background – Tree decomposition – Relational consistency property R( ∗ ,m)C • Key ideas – Localize consistency to clusters of a tree decomposition – Bolstering propagation at separators • Evaluation – Theoretical: Comparing resulting consistency properties – Empirical: Solving CSPs in a backtrack-free manner • Conclusions & Future Work 2
Introduction • Constraint Satisfaction Problems (CSPs) – NP-complete in general – Islands of tractability are classes of CSPs solvable in polynomial time • One tractability condition links [Freuder 82] – Consistency level to – Width of the constraint network, a structural parameter • Our approach: exploit a tree decomposition – Localize application of the consistency algorithm – Add redundant constraints at separators to enhance propagation – Practical tractability aims to solve CSP instances in a backtrack-free manner 3
Tree Decomposition A tree decomposition: 〈 T , , 〉 • • Conditions – T : a tree of clusters – Each constraint appears in at least one cluster with all the – : maps variables to clusters variables in its scope – : maps constraints to clusters – For every variable, the clusters where the variable appears induce a connected subtree D R 4 R 5 C 1 (C1) (C1) G { A,B,C,E } , { R 2 ,R 3 } R 3 C 2 C 3 A C B { A,E,F } , { R 1 } { A,B,D } , { R 3 ,R 5 } F R 1 R 2 C 4 E { A,D,G } , { R 4 } Hypergraph Tree decomposition 4
Tree Decomposition: Separators • A separator of two adjacent clusters is the set of variables associated to both clusters C 1 C C 1 C 2 E { A,B,C,E } , { R 2 ,R 3 } B C 3 C 3 C 2 A F { A,E,F } , { R 1 } { A,B,D } , { R 3 ,R 5 } D C 4 C 4 G { A,D,G } , { R 4 } • Width of a decomposition/network – Treewidth = maximum number of variables in clusters - 1 5
Relational Consistency Property R( ∗ , m )C [Karakashian+ AAAI 10] • A CSP is R( ∗ , m )C iff – Every tuple in a relation can be extended – to the variables in the scope of any ( m -1) other relations – in an assignment satisfying all m relations simultaneously • R( ∗ , m )C ≡ Every set of m relations is minimal ∀ tuple ..… ∀ relation ∀ m -1 relations 6
Localize Consistency • Restricting R( ∗ , m )C to clusters: cl-R( ∗ , m )C • Two clusters communicate via their separator – Constraints common to the two clusters – Domains of variables common to the two clusters E R 2 R 1 R 3 R 4 R 4 A D B B A A D D C C R 6 R 5 R 7 F 7
Bolstering Propagation at Separators • Localization cl-R(*, m )C – Fewer combinations of m relations – Reduces the enforced consistency level E • Ideally: add unique constraint R 2 R 1 R 3 – Space overhead, major bottleneck R 4 • Enhance propagation by bolstering B A D C R sep – Projection of existing constraints R 6 R 5 R 7 – Adding binary constraints F – Adding clique constraints 8
Bolstering Schemas: Approximate Unique Separator Constraint R y A D A A D D B C B C B C R a R x E E E E E E R 1 R 2 R 2 R 1 R 3 R 3 R y R a R 4 B C B B A A D D C C B A D C B A D C R x ’ R 6 R 6 R 5 R 5 R 7 R 3 R’ 3 R’ 3 F F F F F Projection Binary constraints Clique constraints cl+proj-R(*, m )C cl+bin-R(*, m )C cl+clq-R(*, m )C 9
Resulting Consistency Properties cl+clq-R( ∗ ,2)C cl+clq-R( ∗ ,3)C cl+clq-R( ∗ ,4)C cl+clq-R( ∗ ,| ψ ( cl i ) |)C cl+bin-R( ∗ ,3)C cl+bin-R( ∗ ,4)C cl+bin-R( ∗ ,| ψ ( cl i ) |)C cl+bin-R( ∗ ,2)C R( ∗ ,4)C cl+proj-R( ∗ ,2)C R( ∗ ,2)C cl+proj-R( ∗ ,3)C cl+proj-R( ∗ ,4)C cl+proj-R( ∗ ,| ψ ( cl i ) |)C R( ∗ ,3)C [Bessiere+ 2008] maxRPWC cl-R( ∗ ,2)C cl-R( ∗ ,3)C cl-R( ∗ ,4)C cl-R( ∗ ,| ψ ( cl i ) |)C GAC 10
Empirical Evaluations wR( ∗ ,2)C R( ∗ ,| (cl i )|)C + maxRPWC, m =3,4 binary binary global clique clique #inst. local Proj. local Proj. GAC Completed UNSAT 167 170 167 172 169 162 285 286 282 271 479 34.9% 35.5% 34.9% 35.9% 35.3% 33.8% 59.5% 59.7% 58.9% 56.6% SAT 174 179 178 176 169 104 152 138 124 113 200 87.0% 89.5% 89.0% 88.0% 84.5% 52.0% 76.0% 69.0% 62.0% 56.5% UNSAT 0 70 39 70 70 74 187 223 223 213 BT-Free 479 0.0% 14.6% 8.1% 14.6% 14.6% 15.4% 39.0% 46.6% 46.6% 44.5% SAT 44 55 37 53 52 38 39 77 71 58 200 22.0% 27.5% 18.5% 26.5% 26.0% 19.0% 19.5% 38.5% 35.5% 29.0% UNSAT 17 73 43 72 72 77 220 249 248 239 Min(#NV) 479 3.5% 15.2% 9.0% 15.0% 15.0% 16.1% 45.9% 52.0% 51.8% 49.9% SAT 47 64 37 62 61 39 83 111 100 79 200 23.5% 32.0% 18.5% 31.0% 30.5% 19.5% 41.5% 55.5% 50.0% 39.5% UNSAT 72 13 35 5 1 1 176 108 42 37 Fastest 479 15.0% 2.7% 7.3% 1.0% 0.2% 0.2% 36.7% 22.5% 8.8% 7.7% SAT 121 45 47 23 14 12 34 18 13 12 200 60.5% 22.5% 23.5% 11.5% 7.0% 6.0% 17.0% 9.0% 6.5% 6.0% 11
Cumulative Count of Instances Solved w/o Backtracking 80 250 UNSAT SAT cl+proj-R( ∗ ,| ψ ( cl i )|)C 70 cl+proj-R( ∗ ,| ψ ( cl i )|)C 200 60 cl-proj-wR( ∗ ,3)C cl-R( ∗ ,| ψ ( cl i )|)C 50 cl-proj-wR( ∗ ,2)C 150 cl-proj-wR( ∗ ,3)C GAC 40 cl-R( ∗ ,| ψ ( cl i )|)C 100 30 20 cl-proj-wR( ∗ ,2)C 50 10 GAC 0 0 0 100 200 300 0 50 100 150 200 Treewidth Treewidth Acknowledgment: Charts suggested by Rina Dechter 12
Conclusions & Future Work • Adapted R( ∗ , m )C to a tree decomposition of the CSP – Localizing R( ∗ , m )C to the clusters – Bolstering separators to strengthen the enforced consistency • Directions for future work – R( ∗ , m )C on non-table constraints via domain filtering – Automating the selection of a consistency property • Inside clusters • During search – Modify the structure of a tree decomposition to improve performance (e.g., merging clusters [Fattah & Dechter 1996]) 13
Thank You for Your Attention 14
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