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A new algorithm for optimal 2-constraint satisfaction and its implications. (English) Zbl 1081.68095
Summary: 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 \(O(m^{3}2^{\omega n/3})\) time, where \(\omega < 2.376\) is the matrix product exponent over a ring. When the constraints have arbitrary weights, there is a (\(1+\varepsilon\))-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 \(\ell _{1}\), then there is an (\(2-\varepsilon)^{n}\) 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.

MSC:
68T20 Problem solving in the context of artificial intelligence (heuristics, search strategies, etc.)
68Q05 Models of computation (Turing machines, etc.) (MSC2010)
Software:
MAX-2-SAT
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