Scalable Kernelization for Maximum Independent Sets

Hespe, Demian; Schulz, Christian; Strash, Darren

doi:10.1145/3355502

Cited by 20 publications

(10 citation statements)

References 56 publications

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“…The algorithm may repeatedly attempt to apply the same reduction to a vertex even though the graph has not changed sufficiently to allow the reduction to succeed. Checking dependencies between reductions [94], allows to avoid applying certain local reductions when they will provably not succeed, e.g. if their relevant neighborhood did not change since the reduction was last checked.…”

Section: Engineering Techniquesmentioning

confidence: 99%

“…Reduction tracking. The algorithm by Hespe et al [94] stops local reductions when they are not effectively reducing the global graph sizes. It is not always ideal to apply reductions exhaustively-for example, if only few reductions will succeed and they are costly.…”

Section: Engineering Techniquesmentioning

confidence: 99%

“…In order to quickly achieve smaller kernels than is possible with simple reduction rules, Hespe et al [94] provided the first shared-memory kernelization based on the rules of Akiba and Iwata. For this purpose they make use of both graph partitioning and parallel bipartite maximum matchings.…”

Section: Heuristic Approachesmentioning

confidence: 99%

See 2 more Smart Citations

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: Engineering Techniquesmentioning

confidence: 99%

Section: Engineering Techniquesmentioning

confidence: 99%

See 1 more Smart Citation

Recent Advances in Practical Data Reduction

Abu-Khzam¹,

Lamm²,

Mnich³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Note that it is still infeasible to solve our MIS problem with state of the art algorithms [ 18 , 19 ] when

. Segundo et al proposed a variation of BBMC [ 20 ] and found the maximum independent set of the graph induced from two deletion (

) channel [ 14 ].…”

Section: Preliminariesmentioning

confidence: 99%

Nonasymptotic Upper Bounds on Binary Single Deletion Codes via Mixed Integer Linear Programming

2019

Entropy

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The size of the largest binary single deletion code has been unknown for more than 50 years. It is known that Varshamov–Tenengolts (VT) code is an optimum single deletion code for block length n ≤ 10 ; however, only a few upper bounds of the size of single deletion code are proposed for larger n. We provide improved upper bounds using Mixed Integer Linear Programming (MILP) relaxation technique. Especially, we show the size of single deletion code is smaller than or equal to 173 when the block length n is 11. In the second half of the paper, we propose a conjecture that is equivalent to the long-lasting conjecture that “VT code is optimum for all n”. This equivalent formulation of the conjecture contains small sub-problems that can be numerically verified. We provide numerical results that support the conjecture.

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“…A rich body of theoretical and applied algorithmic research has been devoted to the exact solution of the Vertex Cover problem [5,23,32,33]. A standard 2-way branching algorithm can test whether a graph G has a vertex cover of size k in time O(2 k (n + m)), which can be improved by more sophisticated techniques [14].…”

Section: Introductionmentioning

confidence: 99%

Preprocessing to Reduce the Search Space: Antler Structures for Feedback Vertex Set

Donkers¹,

Jansen²

2021

Preprint

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The goal of this paper is to open up a new research direction aimed at understanding the power of preprocessing in speeding up algorithms that solve NP-hard problems exactly. We explore this direction for the classic Feedback Vertex Set problem on undirected graphs, leading to a new type of graph structure called antler decomposition, which identifies vertices that belong to an optimal solution. It is an analogue of the celebrated crown decomposition which has been used for Vertex Cover. We develop the graph structure theory around such decompositions and develop fixed-parameter tractable algorithms to find them, parameterized by the number of vertices for which they witness presence in an optimal solution. This reduces the search space of fixed-parameter tractable algorithms parameterized by the solution size that solve Feedback Vertex Set.

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Scalable Kernelization for Maximum Independent Sets

Cited by 20 publications

References 56 publications

Recent Advances in Practical Data Reduction

Recent Advances in Practical Data Reduction

Nonasymptotic Upper Bounds on Binary Single Deletion Codes via Mixed Integer Linear Programming

Preprocessing to Reduce the Search Space: Antler Structures for Feedback Vertex Set

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