The authors propose an approach to the solution of the maxcut problem. It is based on the global equilibrium search method, which is currently one of the most efficient discrete programming methods. The efficiency of the proposed algorithm is analyzed.Keywords: cutset, approximate methods, global equilibrium search, computational experiment, algorithm efficiency.The maxcut problem, which has attracted considerable interest of researchers over the last two decades, is a classical discrete optimization problem with numerous practical applications [1][2][3]. To solve it, we propose an algorithm based on the global equilibrium search (GES) method [4], which has found applications in the solution of various classes of discrete optimization problems of complex nature [5][6][7][8]. The efficiency of the proposed algorithm was compared with that of currently the best approximate solution algorithms for the problem under study; the comparative analysis confirmed that the GES algorithm has advantages both in speed and the quality of solutions.Let there be given an undirected graph G G V E = ( , ) with the set of vertices V and the set of edges E, each edge ( , ) i j E Î being associated with a number w ij called the weight of the edge ( , ) i j . Assume that ( , ) V V 1 2 is the partition of the set V of vertices of the graph G into two disjoint subsets, V 1 and V 2 . Then the subset of edges ( , ) i j E Î that possess the property i V Î 1 , j V Î 2 is called the cut ( , ) V V 1 2 of the graph G. Obviously, each such partition generates a cut of the graph.The maxcut problem for the undirected graph G is to find a partition of the set V of vertices into disjoint subsets V 1 and V 2 so as to maximize the sum of the weights w V V w1 2 = Î Î Î å of edges of the respective cut.Let us briefly analyze the state of the art of solution methods for the problem under study, which is NP-hard even if all the edges have unit weight. Therefore, the amount of calculation in exact algorithms exponentially grows with problem dimension, which allows them solving only low-or medium-dimensional problems.For high-dimensional problems, aproximate methods considered in many publications, for example in [3,[9][10][11], are efficient. The paper [9] proposes a CirCut algorithm. Test calculations have shown that this algorithm is much better in terms of accuracy and run time than all the other then available approximate algorithms.Six algorithms proposed in [10] are based on the methodology of variable neighborhood search (VNS), greedy adaptive search procedure (GRASP), and path relinking. It is shown that VNSPR algorithm [10] uses much computer resources to find qualitative solutions. The paper [12] considers a randomized algorithm whose guaranteed accuracy for positive weights is 0.878 of the optimal value of the objective function. However, this algorithm is computationally intensive. For example, to solve the problem for n = 200 vertices, it needs about three hours of machine time. Two modifications (RRT and MST) of the multistart tabu search a...
New results are presented concerning binary correcting codes, such as deletion-correcting codes, transposition-correction codes, and codes for the g-eh-nnel. These codes are important due to the possibility of packet loss and corruption on internet transmissions. It is known that the problem of finding the largest correcting codes can be reduced to a well-known combinatorial optimization problem on graphs, the maximum independent set problem. A general scheme of organizing a local search for the maximum independent set problem is discussed. Based on the developed heuristics, an exact branch-and-bound algorithm is proposed, which is able to find exact solutions for graphs with over 500 vertices within a reasonable time.
We describe a heuristic method for solving the unconstrained binary quadratic optimization problem based on a global equilibrium search framework. We investigate performance of the proposed approach and compare it with the best available solver [G. Palubeckis, Multistart tabu search strategies for the unconstrained binary quadratic optimization problem, Ann. Oper. Res., 131 (2004), pp. 259-282; G. Palubeckis, Unconstrained binary quadratic optimization. Available at http://www.soften.ktu.lt/∼gintaras/binqopt.html (Last accessed December 2006).] on well-known benchmarks instances. The reported computational results indicate a high efficiency of the heuristic.
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