Path Relinking Scheme for the Max-Cut Problem within Global Equilibrium Search

Path Relinking Scheme for the Max-Cut Problem within Global Equilibrium Search

Volodymyr P. Shylo (Institute of Cybernetics NAS of Ukraine, Ukraine) and Oleg V. Shylo (University of Pittsburgh, USA)
Copyright: © 2013 |Pages: 10
DOI: 10.4018/978-1-4666-2479-5.ch003
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In this paper, the potential of the path relinking method for the maximum cut problem is investigated. This method is embedded within global equilibrium search to utilize the set of high quality solutions provided by the latter. The computational experiment on a set of standard benchmark problems is provided to study the proposed approach. The empirical experiments reveal that the large sizes of the elite set lead to restart distribution of the running times, i.e., the algorithm can be accelerated by simply removing all of the accumulated data (set P) and re-initiating its execution after a certain number of elite solutions is obtained.
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Assuming that the weights are non-negative, the maximum cut problem can be formulated by the following mixed-integer program (Kahruman, Kolotoglu, Butenko, & Hicks, 2007):

The optimal solution vector x defines a graph partition 978-1-4666-2479-5.ch003.m09 (if 978-1-4666-2479-5.ch003.m10 then 978-1-4666-2479-5.ch003.m11 otherwise 978-1-4666-2479-5.ch003.m12) that has the maximum cut value. Let 978-1-4666-2479-5.ch003.m13 denote a cost of a cut corresponding to the solution vector x.

Local search based methods require an initial solution 978-1-4666-2479-5.ch003.m14 to start the chain of local improvements until the local optimum is obtained. GES provides an intelligent mechanism of generating initial solutions for local search based methods. Its metaheuristic framework proved to be extremely efficient for a variety of combinatorial problems (Pardalos, Prokopyev, Shylo, & Shylo, 2008; Shylo, Prokopyev, & Shylo, 2008).

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