Approximate Local Search in Combinatorial Optimization
SIAM Journal on Computing, Vol. 33, No. 5, pp. 1201-1214, 2004
Posted: 29 Jul 2007
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Approximate Local Search in Combinatorial Optimization
Abstract
Local search algorithms for combinatorial optimization problems are generally of pseudopolynomial running time, and polynomial-time algorithms are not often known for finding locally optimal solutions for NP-hard optimization problems. We introduce the concept of ε-local optimality and show that, for every ε > 0, an ε-local optimum can be identified in time polynomial in the problem size and 1/ε whenever the corresponding neighborhood can be searched in polynomial time. If the neighborhood can be searched in polynomial time for a δ-local optimum, a variation of our main algorithm produces a (δ + ε)-local optimum in time polynomial in the problem size and 1/ε. As a consequence, a combinatorial optimization problem has a fully polynomial-time approximation scheme if and only if the problem of determining a better neighbor in an exact neighborhood has a fully polynomial-time approximation scheme.
Keywords: Local Search, Neighborhood Search, Approximation Algorithms, Computational Complexity, Combinatorial Optimization, 0/1-Integer Programming
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