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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.

68T20 Problem solving in the context of artificial intelligence (heuristics, search strategies, etc.)
68Q05 Models of computation (Turing machines, etc.) (MSC2010)
Full Text: DOI
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