Solving the maxcut problem by the global equilibrium search

Abstract

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...

Breakout Local Search for the Max-Cutproblem

“…Updates are performed to reflect the current state of the search (lines [13][14][15][16][17][18][19][20][21][22]. In particular, if the last discovered local optimum is better than the best solution found so far (recorded in C best ), C best is updated with the last local optimum (lines 17-18).…”

“…Nevertheless, our experimental study provides interesting indications about the performance of the proposed BLS algorithm relative to these state of the art approaches. Table 2 summarizes the computational results of BLS on the set of 71 MaxCut instances (G1-G81) in comparison with the current best-known results (column f prev ) which are from references [7,16,19,21,22]. For BLS, we report the best objective value f best , the average objective value f avg , the standard deviation σ, and the average CPU time in seconds required for reaching f best over 20 executions.…”

“…Well-known exact methods for MaxCut, such as the Branch and Price procedure [17], are capable of solving to optimality medium size instances (i.e., |V | = 500). For larger instances, different heuristic methods have been proposed including global equilibrium search [21], projected gradient approach [6], rank-2 relaxation heuristic [7], and greedy heuristics [11]. Other well-known algorithms are based on popular metaheuristics such as variable neighbourhood search [8], tabu search [1,16], scatter search [19], grasp [22], and different hybrid approaches [8,22].…”

Probabilistic GRASP-Tabu Search algorithms for the UBQP problem

“…These instances are created by using a machine-independent graph generator, comprising of toroidal, planar and random weighted graphs with weight values 1, 0 or -1. Many authors including [8,10,24,28,33] employ these instances to test their algorithms. Note that we use the UBQP model to solve the MaxCut problem through a simple transformation according to [19].…”

Hypervolume-Based Multi-level Algorithm for the Bi-criteria Max-Cut Problem

“…As one of Karps 21 NP-complete problems with numerous practical applications [7], a large number of metaheuristics have been proposed to tackle this problem, including tabu search [2], breakout local search [7], variable neighborhood search [9], scatter search [11], global equilibrium search [13], hybrid approaches [14], etc.…”