Approximate Local Search in Combinatorial Optimization

SIAM Journal on Computing, Vol. 33, No. 5, pp. 1201-1214, 2004

Posted: 29 Jul 2007

See all articles by James B. Orlin

James B. Orlin

Massachusetts Institute of Technology (MIT) - Sloan School of Management

Abraham P. Punnen

University of New Brunswick - Department of Mathematical Sciences

Andreas S. Schulz

Massachusetts Institute of Technology (MIT) - Sloan School of Management

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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

Suggested Citation

Orlin, James B. and Punnen, Abraham P. and Schulz, Andreas S., Approximate Local Search in Combinatorial Optimization. SIAM Journal on Computing, Vol. 33, No. 5, pp. 1201-1214, 2004, Available at SSRN: https://ssrn.com/abstract=1003443 or http://dx.doi.org/10.2139/ssrn.423560

James B. Orlin

Massachusetts Institute of Technology (MIT) - Sloan School of Management ( email )

E53-357
Cambridge, MA 02142
United States
617-253-6606 (Phone)
617-258-7579 (Fax)

Abraham P. Punnen

University of New Brunswick - Department of Mathematical Sciences ( email )

Irving Hall 312
Saint John, New Brunswick E2L 4L5
Canada

Andreas S. Schulz (Contact Author)

Massachusetts Institute of Technology (MIT) - Sloan School of Management ( email )

E53-361
77 Massachusetts Avenue
Cambridge, MA 02139-4307
United States
617-258-7340 (Phone)

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