Computing Maximum Independent Sets over Large Sparse Graphs

Alsahafy, Maram; Chang, Lijun

doi:10.1007/978-3-030-34223-4_45

Cited by 2 publications

(1 citation statement)

References 21 publications

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“…How can one efficiently apply reductions in distributed memory? Alsahafy and Chang [6] recently proposed an algorithm that combines the reducing-peeling framework with the exact clique solver MoMC by Li et al [121]. Their algorithms splits reduction rules in two sets: ones that can be updated and applied incrementally (similar to Hespe et al [94]), and ones that can not.…”

Section: Heuristic Approachesmentioning

confidence: 99%

Recent Advances in Practical Data Reduction

Abu-Khzam¹,

Lamm²,

Mnich³

et al. 2020

Preprint

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Over the last two decades, significant advances have been made in the design and analysis of fixedparameter algorithms for a wide variety of graph-theoretic problems. This has resulted in an algorithmic toolbox that is by now well-established. However, these theoretical algorithmic ideas have received very little attention from the practical perspective. We survey recent trends in data reduction engineering results for selected problems. Moreover, we describe concrete techniques that may be useful for future implementations in the area and give open problems and research questions.2 Recent Advances for NP-Hard Problems Maximum Independent Set and Minimum Vertex CoverGiven an undirected graph G = (V, E), the goal of the maximum independent set (MIS) problem is to compute a set of vertices I ⊆ V such that (1) no two vertices in I are adjacent to one another, (2) the set I has maximum cardinality among all such sets. The complement of an independent set I is called a vertex cover V \ I. The MIS problem and the complementary problem of finding a minimum vertex cover (MVC) are well-studied NP-hard optimization problems [75] that attract both researchers and practitioners alike. Furthermore, there is no polynomial time approximation algorithm for MIS that can provide an O(n 1−ε ) guarantee for any constant ε > 0, unless P=NP [174]. Finally, MIS is W [1]-hard [58] when parameterized by solution size k. This makes it unlikely that the problem is fixed-parameter tractable in k [58]. On the other hand, MVC is fixed-parameter tractable in solution size k [58]. Exact ApproachesIn recent years, the bridge between theoretically efficient algorithms and their practical applicability has been significantly reduced. In particular, the branch-and-reduce paradigm, i.e., branching algorithms that use a wide variety of reduction rules, have been (1) shown to achieve theoretical running times that are among the best for both MIS and MVC [71,170], and (2) are able to solve large real-world networks in practice [5]. However, most often the approaches used in practice only use a small subset of the reduction rules that have been proposed to achieve good theoretical running times.Abu-Khzam et al.[4] introduced and analyzed the crown reduction rule (and the usage of data reduction rules in this context in practice). Even though the crown rule is not as powerful as the linear programming (LP)-based rule [135] when considering the worst-case size of the resulting kernel, they experimentally verified that it often performs as well as the LP-based rule and is significantly faster in many cases. Furthermore, they show that the LP-based rule is most useful for fairly sparse graphs and should be avoided for dense graphs, as it yields little to no reduction in size.Later, Akiba and Iwata [5] were the first to show the practicality of the branch-and-reduce paradigm for MVC (and MIS) compared to other state-of-the-art approaches like branch-and-bound and branch-and-cut. Their algorithm uses a wide spectrum of reduction rules that form the foundation...

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Section: Heuristic Approachesmentioning

confidence: 99%

Recent Advances in Practical Data Reduction

Abu-Khzam¹,

Lamm²,

Mnich³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

Recent Advances in Practical Data Reduction

Abu-Khzam

Lamm

Mnich

et al. 2022

Lecture Notes in Computer Science

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Over the last two decades, significant advances have been made in the design and analysis of fixed-parameter algorithms for a wide variety of graph-theoretic problems. This has resulted in an algorithmic toolbox that is by now well-established. However, these theoretical algorithmic ideas have received very little attention from the practical perspective. We survey recent trends in data reduction engineering results for selected problems. Moreover, we describe concrete techniques that may be useful for future implementations in the area and give open problems and research questions.

show abstract

Computing Maximum Independent Sets over Large Sparse Graphs

Cited by 2 publications

References 21 publications

Recent Advances in Practical Data Reduction

Recent Advances in Practical Data Reduction

Recent Advances in Practical Data Reduction

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