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

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...

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Veröffentlicht in:Theoretical computer science Jg. 348; H. 2; S. 357 - 365
1. Verfasser: Williams, Ryan
Format: Journal Article Tagungsbericht
Sprache:Englisch
Veröffentlicht: Amsterdam Elsevier B.V 08.12.2005
Elsevier
Schlagworte:
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
Constraint satisfaction
MAX-CUT
MAX-2-SAT
Exact algorithms
Search problem
Lower bound
Multiplication
Computer theory
Optimal algorithm
Optimization method
Matrix product
Computing
MAX 2 SAT
Complexity
Polynomial time
MAX CUT
Counting
ISSN:0304-3975, 1879-2294
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Zusammenfassung: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.
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ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2005.09.023