A new algorithm for optimal 2-constraint satisfaction and its implications

• Published in: Theoretical computer science 2005-12, Vol.348 (2), p.357-365
• Main Author: Williams, Ryan
• Format: Article
• Language: English
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Calculus of variations and optimal control; Computer science; control theory; systems; Constraint satisfaction; Exact algorithms; Exact sciences and technology; Mathematical analysis; Mathematics; MAX-2-SAT; MAX-CUT; Sciences and techniques of general use; Theoretical computing
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2005.09.023

We present a novel method for exactly solving (in fact, counting solutions to) general constraint satisfaction optimization with at most two variables per constraint (e.g. MAX-2-CSP and MIN-2-CSP), which gives the first exponential improvement over the trivial algorithm. More precisely, the runtime bound is a constant factor improvement in the base of the exponent: the algorithm can count the number of optima in MAX-2-SAT and MAX-CUT instances in O ( m 3 2 ω n / 3 ) time, where ω < 2.376 is the matrix product exponent over a ring. When the constraints have arbitrary weights, there is a ( 1 + ε ) -approximation with roughly the same runtime, modulo polynomial factors. Our construction shows that improvement in the runtime exponent of either k - clique solution (even when k = 3 ) or matrix multiplication over GF ( 2 ) would improve the runtime exponent for solving 2-CSP optimization. Our approach also yields connections between the complexity of some (polynomial time) high-dimensional search problems and some NP-hard problems. For example, if there are sufficiently faster algorithms for computing the diameter of n points in ℓ 1 , then there is an ( 2 - ε ) n algorithm for MAX-LIN. These results may be construed as either lower bounds on the high-dimensional problems, or hope that better algorithms exist for the corresponding hard problems.