Heuristic initialization for bipartite matching problems

Abstract

It is a well-established result that improved pivoting in linear solvers can be achieved by computing a bipartite matching between matrix entries and positions on the main diagonal. With the availability of increasingly faster linear solvers, the speed of bipartite matching computations must keep up to avoid slowing down the main computation. Fast algorithms for bipartite matching, which are usually initialized with simple heuristics, have been known for a long time. However, the performance of these algorithms is largely dependent on the quality of the heuristic. We compare combinations of several known heuristics and exact algorithms to find fast combined methods, using real-world matrices as well as randomly generated instances. In addition, we present a new heuristic aimed at obtaining high-quality matchings and compare its impact on bipartite matching algorithms with that of other heuristics. The experiments suggest that its performance compares favorably to the best-known heuristics, and that it is especially suited for application in linear solvers.

Push-relabel based algorithms for the maximum transversal problem

“…Thus, these algorithms can be initiated with a non-empty matching. In order to exploit this, several efficient and effective heuristics, which find initial matchings of considerable size, have been proposed in the literature [22,23,24,25,26].…”

“…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].…”

“…There are extended versions of these heuristics [24,25]. However, none of the extended heuristics could be shown to consistently provide performance superior to KSM or SGM.…”

Two approximation algorithms for bipartite matching on multicore architectures

“…There is considerable interest in simpler and faster algorithms that have some approximation guarantee [2]. Such cheap algorithms are used as a jump-start routine by the current state of the art matching 10 algorithms [2,3,4]. Furthermore, there are applications [5] where approximate cardinality matchings are used.…”

“…KS obtains very good results in practice. Currently, it is the suggested one to be used as a jump-start routine [2,4] for exact algorithms, especially for augmenting-path based ones [7]. Algorithms that achieve an approximation ratio of 1−1/e, where 20 e is the base of the natural logarithm are designed for the online case [8].…”