Using Maximal Independent Sets to Decompose a CSP

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Using Maximal Independent Sets to Decompose a CSP (Validation on Local Search) Joel Gompert

Using Maximal Independent Sets to Decompose a CSP (Validation on Local Search) Joel Gompert Constraint Systems Laboratory Department of Computer Science & Engineering University of Nebraska-Lincoln Constraint Systems Laboratory DP-CP 04

Overview 1. Ind. Set decomposes a CSP into I and Ī • • How

Overview 1. Ind. Set decomposes a CSP into I and Ī • • How to account for the constraints between I and Ī when solving Ī with LS+min-conflict: 3 heuristics Technique yields multiple, compact solutions 2. Identifying dangling trees • • Ind. Set becomes (improved) Cycle-Cutset [Dechter & Pearl 89] Allows parallelization of (dangling) trees 3. Applying Ind. Set recursively to Ī • Yields a new variable ordering heuristic Constraint Systems Laboratory DP-CP 04

1. Decompose CSP into Ī and I Constraint Systems Laboratory DP-CP 04

1. Decompose CSP into Ī and I Constraint Systems Laboratory DP-CP 04

Solve Ī Constraint Systems Laboratory DP-CP 04

Solve Ī Constraint Systems Laboratory DP-CP 04

REVISE(I, Ī) If no domain in I is empty, we have • solved the

REVISE(I, Ī) If no domain in I is empty, we have • solved the original CSP • found multiple solutions (cross product of domains in I) Constraint Systems Laboratory DP-CP 04

Accounting for the constraints between Ī and I When solving Ī with Local Search

Accounting for the constraints between Ī and I When solving Ī with Local Search and min-conflict a None: Ignore constraints between I and Ī Ī b b, c I c ü Some: Count the constraints incident to a variable in I as its domain is annihilated a Ī b b, c I c o All: Count all constraints incident to variable in I a Ī b c Constraint Systems Laboratory DP-CP 04 b, c I

Local search over Ī 80 variables, 8 values, constraint tightness: 0. 583 Constraint Systems

Local search over Ī 80 variables, 8 values, constraint tightness: 0. 583 Constraint Systems Laboratory DP-CP 04

Using Ind. Set information 80 variables, 8 values, constraint tightness: 0. 583 Constraint Systems

Using Ind. Set information 80 variables, 8 values, constraint tightness: 0. 583 Constraint Systems Laboratory DP-CP 04

Current investigations I 2. Remove dangles I • Ind. Set becomes Cycle Cutset •

Current investigations I 2. Remove dangles I • Ind. Set becomes Cycle Cutset • With parallel trees 3. Apply Ind. Set recursively • New variable ordering Constraint Systems Laboratory DP-CP 04

Conclusion • Contributions – Independent-set decomposition – Heuristics for local search – New technique

Conclusion • Contributions – Independent-set decomposition – Heuristics for local search – New technique for finding cycle-cutsets • Future Work – Generalize technique to cycle cutset decomposition – Decompose the graph recursively Constraint Systems Laboratory DP-CP 04