Fixed-Parameter Algorithms in Analysis of Heuristics for Extracting Networks in Linear Programs

Abstract: We consider the problem of extracting a maximum-size reflected network in a linear program. This problem has been studied before and a state-of-the-art SGA heuristic with two variations have been proposed. In this paper we apply a new approach to evaluate the quality of SGA\@. In particular, we solve majority of the instances in the testbed to optimality using a new fixed-parameter algorithm, i.e., an algorithm whose runtime is polynomial in the input size but exponential in terms of an additional parameter associated with the given problem. This analysis allows us to conclude that the the existing SGA heuristic, in fact, produces solutions of a very high quality and often reaches the optimal objective values. However, SGA contain two components which leave some space for improvement: building of a spanning tree and searching for an independent set in a graph. In the hope of obtaining even better heuristic, we tried to replace both of these components with some equivalent algorithms. We tried to use a fixed-parameter algorithm instead of a greedy one for searching of an independent set. But even the exact solution of this subproblem improved the whole heuristic insignificantly. Hence, the crucial part of SGA is building of a spanning tree. We tried three different algorithms, and it appears that the Depth-First search is clearly superior to the other ones in building of the spanning tree for SGA. Thereby, by application of fixed-parameter algorithms, we managed to check that the existing SGA heuristic is of a high quality and selected the component which required an improvement. This allowed us to intensify the research in a proper direction which yielded a superior variation of SGA.