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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Extensive study of phase transitions in
physics (statistical mechanics).
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Extensive study of phase transitions in NPcomplete problems during the past decade.
Motivation and Goals

Understand the “structure” of NP-complete
problems.
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Relate phase transitions to the average-case
performance of particular algorithms for
NP-complete problems.
NP-Complete Problems
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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.
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
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AC-Matching
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Term matching under an operation that is
associative & commutative (no unit).
a1X1+ … + anXn = AC b1C1+ …+ bmCm
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Example:
–
–
–
–
Solution 1:
Solution 2:
Solution 3:
Solution 4:
2X1+X2 = AC 4C1+ 5C2
X1  2C1 ,
X2  5C2
X1  C1 ,
X2  2 C1+ 5C2
X1  2C1+C2 , X2  3C2
…
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).

Parametrization of AC-Matching
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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.
NP-Completeness for Fixed Ratios
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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. 
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  .
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.
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Solvers Used in Experiments
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Direct AC-Matching Solver developed by S. Eker
at SRI as part of Maude, a high-performance
system for equational logic and rewriting.
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Reduction to Integer Linear Programming (ILP)
and CPLEX, a commercial optimization package
with a powerful ILP solver.
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Reduction to SAT and Grasp, one of the main SAT
solvers developed by J. Silva.
Reductions to ILP and SAT
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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 gij Cj
Resulting instance of ILP is:
iaigij = bj , 1  j  m
jgij  1 , 1  i  n.
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
 Large-scale experiments were carried out on
the interval of ratios [30:70, 50:50]
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– Instance sizes: 100, 200, 400, 800, 1600
– Sample size: 30000 random instances for each
data point.
Large-Scale Experiments:
Close-up on Critical Region
Finite-Size Scaling
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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  .
Results of Finite-Size Scaling:
Probability Curves Collapse
Validation of Finite-Size Scaling
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.
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Extrapolation to Very Large Sizes
Comparison of Solvers
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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
Median Time of
Reduction to SAT + Grasp
th
70
percentile of
Reduction to SAT + Grasp
Concluding Remarks
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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.
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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