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In Search of a Phase Transition in the AC-Matching Problem

In Search of a Phase Transition in the AC-Matching Problem. Phokion G. Kolaitis Thomas Raffill Computer Science Department UC Santa Cruz. Phase Transitions. A phase transition is an abrupt change in the behavior of a property of a “system”.

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In Search of a Phase Transition in the AC-Matching Problem

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  1. In Search of a Phase Transitionin the AC-Matching Problem Phokion G. Kolaitis Thomas Raffill Computer Science Department UC Santa Cruz

  2. Phase Transitions • A phase transition is an abrupt change in the behavior of a property of a “system”. • Extensive study of phase transitions in physics (statistical mechanics). • Extensive study of phase transitions in NP-complete problems during the past decade.

  3. Motivation and Goals • Understand the “structure” of NP-complete problems. • Relate phase transitions to the average-case performance of particular algorithms for NP-complete problems.

  4. NP-Complete Problems • Introduce a “constrainedness” parameter to partition the space of instances. • Generate random instances at fixed parameter values. • For some problems, probability of a “yes” instance abruptly changes from 1 to 0 at some critical value. • For some problems and some solvers, average difficulty peaks sharply at the same critical value.

  5. Main Example: 3-SAT • Parameter: Ratio of number of clauses to number of variables. • Intuition: Low ratios are underconstrained, high ratios are overconstrained. • Critical Value: Experimental results suggest that it is about 4.3 clauses to variables. • Average Performance:DPLL procedure peaks around 4.3

  6. AC-Matching • Term matching under an operation that is associative & commutative (no unit). a1X1+ … + anXn = AC b1C1+ …+ bmCm • Example: 2X1+X2 = AC 4C1+ 5C2 • Solution 1:X1  2C1 , X2  5C2 • Solution 2:X1  C1 , X2  2 C1+ 5C2 • Solution 3: X1  2C1+C2 , X2  3C2 • Solution 4: …

  7. AC-Matching • AC-matching plays an important role in automated deduction. • AC-matching solvers are key components of many theorem-provers (eg., EQP). • AC-matching is strong NP-complete (it is NP-complete even if the coefficients are given in unary).

  8. Parametrization of AC-Matching • Several different parameters come into play: number of variables, number of constants, maximum coefficients, … a1X1+ … + anXn = AC b1C1+ …+ bmCm • Our chosen parameter: r = (  ai ) / (  bj) • Some intuition: • more variables  more constrained instance • more constants  less constrained instance • reflects both # of symbols and multiplicities.

  9. NP-Completeness for Fixed Ratios • Definition: AC(r)-Matching is the restriction of AC-Matching to instances of ratio r. • Fact: If r > 1, then every instance of AC(r)-Matching is negative. • Theorem: If r is such that 0 < r  1, then AC(r)-Matching is NP-complete. -- r = 1: 3-Partition is reducible to AC(1)-Matching (Eker – 1993). -- 0 < r < 1: By careful padding, can reduce AC(1)-Matching to AC(r)-Matching. 

  10. Phase Transition Conjecture • Pr(r,s) = probability that a random instance of AC(r)-Matching of size s is positive, where s =  ai +  bj . • Conjecture: There is critical ratio r* s.t. • If r < r*, then Pr(r,s)  1 , as s  ; • If r > r*, then Pr(r,s)  0 , as s  .

  11. Generating Random Instances • Fix size s. • Step through ratios u/v  1, where u+v = s. • Generate random partitions of u and v. • Use the partition of u for LHS coefficients; Use the partition of v for RHS coefficients. • 1200 samples give < 4% margin of error with 95% confidence. • 30000 samples give < 1% margin of error.

  12. Solvers Used in Experiments • Direct AC-Matching Solver developed by S. Eker at SRI as part of Maude, a high-performance system for equational logic and rewriting. • Reduction to Integer Linear Programming (ILP) and CPLEX, a commercial optimization package with a powerful ILP solver. • Reduction to SAT and Grasp, one of the main SAT solvers developed by J. Silva.

  13. Reductions to ILP and SAT • Given an instance of AC-Matching a1X1+ … + anXn = AC b1C1+ …+ bmCm express each Xi as a non-empty linear combination of the Cjs: Xi  S gijCj • Resulting instance of ILP is: iaiij = bj , 1  j  m jij  1 , 1  i  n. • Standard reduction of ILP to SAT.

  14. Prob. of solvability as function of r based on 1200 samples

  15. Large-Scale Experiments • Initial experiments based on instances of size up to 400 and on samples of size 1200 suggest a possible crossover near ratio 42:58 • Large-scale experiments were carried out on the interval of ratios [30:70, 50:50] • Instance sizes: 100, 200, 400, 800, 1600 • Sample size: 30000 random instances for each data point.

  16. Large-Scale Experiments: Close-up on Critical Region

  17. Finite-Size Scaling • Given a family of curves f(r,s) for various instance sizes s, rescale x-axis according to a power law r = [(r – r*)/r*]  s • Superimpose curves f(r,s) by replacing each data point (r,p) by the point ( [(r – r*)/r*]  s , p). • Check whether the curves f(r,s) collapse to a universal function f(r) which is monotone and takes values between 1 and 0 as r varies from - to . • The existence of a universal function supports phase transition conjecture: in the vicinity of r*, the values of f(r,s) jump from 1 to 0 as s  .

  18. Results of Finite-Size Scaling:Probability Curves Collapse

  19. Validation of Finite-Size Scaling

  20. Slowly Emerging Phase Transition? • Curve-fitting gives the power law r' = [(r  0.73)/0.73]  s 0.171 critical ratio r* = 0.73  42:58 scaling exponent  = 0.171 • Scaling exponent is rather small (scaling exponent for 3-SAT is in [0.625, 0.714]) . • This suggests that any phase transition for AC-matching emerges very slowly.

  21. Extrapolation to Very Large Sizes

  22. Comparison of Solvers • The three solvers were run on the instance sets and CPU time was recorded. • Maude and Reduction to ILP + CPLEX are fast on almost all instances. • Reduction to SAT + Grasp is much slower than either Maude or Reduction to ILP + CPLEX. • Reduction to SAT + Grasp has sharp peak in solving time near the critical ratio 0.73

  23. Median Time of Reduction to SAT + Grasp

  24. 70th percentile ofReduction to SAT + Grasp

  25. Concluding Remarks • There is some evidence for a phase transition in AC-Matching based on experimental results and finite-size scaling. • However, in contrast to 3-SAT and several other NP-complete problems, the phase-transition in AC-Matching emerges very slowly. • Limitation of experimental methods: analytical results are needed to provide more convincing evidence or demonstrate its existence.

  26. Concluding Remarks • Maude and CPLEX-based solver show no change in performance near the critical ratio. Will this change with larger-size instances? • Grasp-based solver peaks near the critical ratio. Will this change with a better reduction of AC-matching to SAT and/or a different SAT solver?

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