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”. 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 physics (statistical mechanics). Extensive study of phase transitions in NP- complete problems during the past decade. Extensive study of phase transitions in NP- complete problems during the past decade.

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

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

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

AC-Matching Term matching under an operation that is associative & commutative (no unit). Term matching under an operation that is associative & commutative (no unit). a 1 X 1 + … + a n X n = AC b 1 C 1 + …+ b m C m a 1 X 1 + … + a n X n = AC b 1 C 1 + …+ b m C m Example: 2X 1 +X 2 = AC 4C 1 + 5C 2 Example: 2X 1 +X 2 = AC 4C 1 + 5C 2 –Solution 1: X 1  2C 1, X 2  5C 2 –Solution 2: X 1  C 1, X 2  2 C 1 + 5C 2 –Solution 3: X 1  2C 1 +C 2, X 2  3C 2 –Solution 4: …

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

Parametrization of AC-Matching Several different parameters come into play: number of variables, number of constants, maximum coefficients, … Several different parameters come into play: number of variables, number of constants, maximum coefficients, … a 1 X 1 + … + a n X n = AC b 1 C 1 + …+ b m C m a 1 X 1 + … + a n X n = AC b 1 C 1 + …+ b m C m Our chosen parameter: Our chosen parameter: r = (  a i ) / (  b j ) r = (  a i ) / (  b j ) Some intuition: Some intuition: –more variables  more constrained instance –more constants  less constrained instance –reflects both # of symbols and multiplicities.

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

Phase Transition Conjecture Pr(r,s) = probability that a random instance of AC(r)-Matching of size s is positive, where s =  a i +  b j. Pr(r,s) = probability that a random instance of AC(r)-Matching of size s is positive, where s =  a i +  b j. Conjecture: There is critical ratio r* s.t. 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  .

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

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. 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 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. Reduction to SAT and Grasp, one of the main SAT solvers developed by J. Silva.

Reductions to ILP and SAT Given an instance of AC-Matching Given an instance of AC-Matching a 1 X 1 + … + a n X n = AC b 1 C 1 + …+ b m C m a 1 X 1 + … + a n X n = AC b 1 C 1 + …+ b m C m express each X i as a non-empty linear combination of the C j s: express each X i as a non-empty linear combination of the C j s: X i   ij  C j X i   ij  C j Resulting instance of ILP is: Resulting instance of ILP is:  i a i  ij = b j, 1  j  m  i a i  ij = b j, 1  j  m  j  ij  1, 1  i  n.  j  ij  1, 1  i  n. Standard reduction of ILP to SAT. Standard reduction of ILP to SAT.

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

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 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] Large-scale experiments were carried out on the interval of ratios [30:70, 50:50] –Instance sizes: 100, 200, 400, 800, 1600 –Sample size: random instances for each data point.

Large-Scale Experiments: Close-up on Critical Region

Finite-Size Scaling Given a family of curves f(r,s) for various instance sizes s, rescale x-axis according to a power law 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 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). 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 . 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  . 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  .

Results of Finite-Size Scaling: Probability Curves Collapse

Validation of Finite-Size Scaling

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

Extrapolation to Very Large Sizes

Comparison of Solvers The three solvers were run on the instance sets and CPU time was recorded. 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. 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 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 Reduction to SAT + Grasp has sharp peak in solving time near the critical ratio 0.73

Median Time of Reduction to SAT + Grasp

70 th percentile of Reduction to SAT + Grasp 70 th percentile of Reduction to SAT + Grasp

Concluding Remarks There is some evidence for a phase transition in AC-Matching based on experimental results and finite-size scaling. 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. 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: Limitation of experimental methods: analytical results are needed to provide more convincing evidence or demonstrate its existence. analytical results are needed to provide more convincing evidence or demonstrate its existence.

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