Design, implementation, and analysis of maximum transversal algorithms

Abstract

We report on careful implementations of seven algorithms for solving the problem of finding a maximum transversal of a sparse matrix. We analyse the algorithms and discuss the design choices. To the best of our knowledge, this is the most comprehensive comparison of maximum transversal algorithms based on augmenting paths. Previous papers with the same objective either do not have all the algorithms discussed in this paper or they used non-uniform implementations from different researchers. We use a common base to implement all of the algorithms and compare their relative performance on a wide range of graphs and matrices. We systematize, develop and use several ideas for enhancing performance. One of these ideas improves the performance of one of the existing algorithms in most cases, sometimes significantly. So much so that we use this as the eighth algorithm in comparisons.

Push-relabel based algorithms for the maximum transversal problem

“…The way in which augmenting paths are found constitutes the main difference between the algorithms based on augmenting path search, both in theory and in practice. In this paper, we use PFP (described in Section 5.1), the fastest augmenting path-based matching algorithm identified by Duff et al [12].…”

“…These heuristics have seen extensive experimental investigations, among others by Duff et al [12], Langguth et al [24], and Magun [25]. There are extended versions of these heuristics [24,25].…”

“…This raises the question of whether the algorithm could be improved by encouraging fairness in neighbor selection. This was proposed by Duff et al [12] for improving the Pothen and Fan (PF) algorithm [22] and resulted in significant performance gain (discussed in Section 5.1). By varying the direction of search through the adjacency list of an active column, the likelihood of the algorithm repeatedly pursuing an unpromising direction of search is reduced.…”

“…One class of algorithms is based on augmenting paths. Duff et al [12] discuss the design, analysis and implementation of eight augmenting path-based algorithms. Push-relabel-based algorithms [13] form a second class.…”

“…We report thorough results on all large enough problems corresponding to matrices from the University of Florida Sparse Matrix Collection [16]. Our third contribution is the adaptation of a simple strategy proposed by Duff et al [12] to the push-relabel algorithm, as well as an additional modification, which speed up the algorithm noticeably.…”