Tighter Approximations for Maximum Induced Matchings in Regular Graphs

It is an easy observation that a natural greedy approach yields a (d − O(1))-factor approximation algorithm for the maximum induced matching problem in d-regular graphs. The only considerable and non-trivial improvement of this approximation ratio was obtained by Gotthilf and Lewenstein using a combination of the greedy approach and local search, where understanding the performance of the local search was the challenging part of the analysis. We study the performance of their local search when applied to general graphs, C 4 -free graphs, {C 3 , C 4 }-free graphs, C 5 -free graphs, and claw-free graphs. As immediate consequences, we obtain approximation algorithms for the maximum induced matching problem restricted to the d-regular graphs in these classes.

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“…we obtain, using (6) and d xy ≥ 0, that (2d 2 + (d − 1) 2 )|M | ≥ 2m(G), which completes the proof of (ii).…”

Section: Greedy(f )supporting

confidence: 63%

Locally searching for large induced matchings

Fürst

Leichter

Rautenbach

2018

Theoretical Computer Science

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“…A similar technique can also be used to show that the induced matching problem on d-regular bipartite graphs is hard to approximate to within a factor of d 1/2− , improving upon the APX-hardness of [16,50]. (This result is not tight as the best known upper bound is Θ(d) [24]. )…”

Section: Using Subadditivity (Theorem 11)mentioning

confidence: 99%

Graph Products Revisited: Tight Approximation Hardness of Induced Matching, Poset Dimension and More

Chalermsook¹,

Laekhanukit²,

Nanongkai³

2013

Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…In general graphs, the problem cannot be approximated to within a factor of n 1/2−ǫ for any ǫ > 0, where n is the number of vertices of the input graph [38]. There exists an approximation algorithm for the problem on r-regular graphs (r ≥ 3) with asymptotic performance ratio r − 1 [16], which has subsequently been improved to 0.75r + 0.15 [25]. Moreover, there exists a polynomial-time approximation scheme (PTAS) for planar graphs of maximum degree 3 [16].…”

Section: Questionmentioning

confidence: 99%

The Parameterized Complexity of the Induced Matching Problem in Planar Graphs

Moser

Sikdar

Frontiers in Algorithmics

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Given a graph G and an integer k ≥ 0, the NP-complete Induced Matching problem asks whether there exists an edge subset M of size at least k such that M is a matching and no two edges of M are joined by an edge of G. The complexity of this problem on general graphs as well as on many restricted graph classes has been studied intensively. However, other than the fact that the problem is W[1]-hard on general graphs little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we provide first-time fixed-parameter tractability results for planar graphs, bounded-degree graphs, graphs with girth at least six, bipartite graphs, line graphs, and graphs of bounded treewidth. In particular, we give a linear-size problem kernel for planar graphs.

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Tighter Approximations for Maximum Induced Matchings in Regular Graphs

Cited by 13 publications

References 11 publications

Locally searching for large induced matchings

Locally searching for large induced matchings

Graph Products Revisited: Tight Approximation Hardness of Induced Matching, Poset Dimension and More

The Parameterized Complexity of the Induced Matching Problem in Planar Graphs

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