Using critical sets to solve the maximum independent set problem

Butenko, Sergiy; Trukhanov, Svyatoslav

doi:10.1016/j.orl.2006.07.004

Cited by 33 publications

(49 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A cycle in a directed graph is a directed sequence of linked agents, in which all arcs are traversed in their prescribed directions and every agent appears at most once, except for the first and the last agents, who coincide. 9 We show that the existence of equilibria where some agents contribute and other agents free ride is closely related to the non-existence of a cycle in directed graphs.…”

mentioning

confidence: 89%

Governing a Common-Pool Resource in a Directed Network

Richefort

Point

2010

SSRN Journal

View full text Add to dashboard Cite

A local public-good game played on directed networks is analyzed. The model is motivated by one-way flows of hydrological influence between cities of a river basin that may shape the level of their contribution to the conservation of wetlands. It is shown that in many (but not all) directed networks, there exists an equilibrium, sometimes socially desirable, in which some stakeholders exert maximal effort and the others free ride. It is also shown that more directed links are not always better. Finally, the model is applied to the conservation of wetlands in the Gironde estuary (France). Keywords

show abstract

mentioning

confidence: 89%

Governing a Common-Pool Resource in a Directed Network

Richefort

Point

2010

SSRN Journal

View full text Add to dashboard Cite

show abstract

“…Their algorithm works by first applying isolated clique removal reductions, then solving the remaining graph with a branch-and-bound algorithm. Later, Butenko and Trukhanov [6] introduced the critical independent set reduction, which was able to solve graphs produced by the Sanchis graph generator. Larson [26] later proposed an algorithm to find a maximum critical independent set, but in experiments it proved to be slow in practice [36].…”

Section: Related Workmentioning

confidence: 99%

WeGotYouCovered: The Winning Solver from the PACE 2019 Challenge, Vertex Cover Track

Hespe

Lamm

Schulz

et al. 2020

2020 Proceedings of the SIAM Workshop on Combinatorial Scientific Computing

View full text Add to dashboard Cite

We present the winning solver of the PACE 2019 Implementation Challenge, Vertex Cover Track. The minimum vertex cover problem is one of a handful of problems for which kernelization-the repeated reducing of the input size via data reduction rules-is known to be highly effective in practice. Our algorithm uses a portfolio of techniques, including an aggressive kernelization strategy, local search, branch-and-reduce, and a stateof-the-art branch-and-bound solver. Of particular interest is that several of our techniques were not from the literature on the vertex over problem: they were originally published to solve the (complementary) maximum independent set and maximum clique problems.Aside from illustrating our solver's performance in the PACE 2019 Implementation Challenge, our experiments provide several key insights not yet seen before in the literature. First, kernelization can boost the performance of branch-and-bound clique solvers enough to outperform branch-and-reduce solvers. Second, local search can significantly boost the performance of branch-and-reduce solvers. And finally, somewhat surprisingly, kernelization can sometimes make branch-and-bound algorithms perform worse than running branch-and-bound alone.

show abstract

“…Butenko et al [10] and Butenko and Trukhanov [11] were able to find exact maximum independent sets in graphs with thousands of vertices by first applying reductions. Further works have introduced reductions to more quickly solve the maximum clique problem [35,43] and enumerate kplexes [14].…”

Section: Related Workmentioning

confidence: 99%

Scalable Kernelization for Maximum Independent Sets

Hespe¹,

Schulz²,

Strash³

2018

2018 Proceedings of the Twentieth Workshop on Algorithm Engineering and Experiments (ALENEX)

View full text Add to dashboard Cite

The most efficient algorithms for finding maximum independent sets in both theory and practice use reduction rules to obtain a much smaller problem instance called a kernel. The kernel can then be solved quickly using exact or heuristic algorithms-or by repeatedly kernelizing recursively in the branch-and-reduce paradigm. It is of critical importance for these algorithms that kernelization is fast and returns a small kernel. Current algorithms are either slow but produce a small kernel, or fast and give a large kernel. We attempt to accomplish both of these goals simultaneously, by giving an efficient parallel kernelization algorithm based on graph partitioning and parallel bipartite maximum matching.We combine our parallelization techniques with two techniques to accelerate kernelization further: dependency checking that prunes reductions that cannot be applied, and reduction tracking that allows us to stop kernelization when reductions become less fruitful. Our algorithm produces kernels that are orders of magnitude smaller than the fastest kernelization methods, while having a similar execution time. Furthermore, our algorithm is able to compute kernels with size comparable to the smallest known kernels, but up to two orders of magnitude faster than previously possible. Finally, we show that our kernelization algorithm can be used to accelerate existing state-of-the-art heuristic algorithms, allowing us to find larger independent sets faster on large real-world networks and synthetic instances.

show abstract

Using critical sets to solve the maximum independent set problem

Cited by 33 publications

References 8 publications

Governing a Common-Pool Resource in a Directed Network

Governing a Common-Pool Resource in a Directed Network

WeGotYouCovered: The Winning Solver from the PACE 2019 Challenge, Vertex Cover Track

Scalable Kernelization for Maximum Independent Sets

Contact Info

Product

Resources

About