Heuristic Algorithm and Results of Computational Experiments of Solution of Graph Placement Problem

Abstract

Most of the problems on graphs are hard problems. Therefore, it is obvious that an exhaustive approach to solving problems will rarely succeed. Approaches considered by other authors are related to evolutionary modeling, genetic algorithms and other stochastic algorithms, and they have some success. However, some shortcomings are seen in these approaches. The authors propose an approach that is heuristic, but not stochastic. The paper presents a generalized mathematical model of the problem that enables considering the problem of placement in n-dimensional space as a problem of searching for permutations of n elements. This eliminates such shortcomings of algorithms for placing graphs, such as the possibility of control over the process of operation of the algorithm and the strong dependence of the search capabilities on the time complexity of the algorithm. The presented heuristic algorithm Hebene was built based on the corresponding mathematical description. Computational experiments were undertaken for all pairwise nonisomorphic connected graphs up to order 9 inclusive. The algorithm found the optimal solution in more than 50% of cases, the algorithm also yielded acceptable solutions in other situations.