Maximum weight induced matching in some subclasses of bipartite graphs

Panda, B. S.; Pandey, Arti; Chaudhary, Juhi; Dane, Piyush; Kashyap, Manav

doi:10.1007/s10878-020-00611-2

Cited by 13 publications

(11 citation statements)

References 20 publications

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“…Motivated by previous works on weighted P-matchings, such as Weighted Induced Matching [20] [28] and Weighted Acyclic Matching [14], in this paper we introduced and studied the Weighted Connected Matching problem.…”

Section: Discussionmentioning

confidence: 99%

“…Recently, some P-matchings concepts were extended to edge-weighted problems, in which, in addition to the matching to have a certain property P, the sum of the weights of the matched edges is sufficiently large. It was shown that Maximum Weight Induced Matching can be solved in linear time for convex bipartite graphs [20] and in polynomial time for circular-convex and triad-convex bipartite graphs [28]. Also, Maximum Weight Acyclic Matching was approached in [14], showing that the problem is polynomially solvable for P 4 -free graphs and 2P 3 -free graphs.…”

Section: Introductionmentioning

confidence: 99%

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Weighted Connected Matchings

Gomes¹,

Masquio²,

Pinto³

et al. 2022

Preprint

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A matching M is a P-matching if the subgraph induced by the endpoints of the edges of M satisfies property P. As examples, for appropriate choices of P, the problems Induced Matching, Uniquely Restricted Matching, Connected Matching and Disconnected Matching arise. For many of these problems, finding a maximum P-matching is a knowingly NP-hard problem, with few exceptions, such as connected matchings, which has the same time complexity as the usual Maximum Matching problem. The weighted variant of Maximum Matching has been studied for decades, with many applications, including the well-known Assignment problem. Motivated by this fact, in addition to some recent researches in weighted versions of acyclic and induced matchings, we study the Maximum Weight Connected Matching. In this problem, we want to find a matching M such that the endpoint vertices of its edges induce a connected subgraph and the sum of the edge weights of M is maximum. Unlike the unweighted Connected Matching problem, which is in P for general graphs, we show that Maximum Weight Connected Matching is NP-hard even for bounded diameter bipartite graphs, starlike graphs, planar bipartite, and bounded degree planar graphs, while solvable in linear time for trees and subcubic graphs. When we restrict edge weights to be non negative only, we show that the problem turns to be polynomially solvable for chordal graphs, while it remains NP-hard for most of the cases when weights can be negative. Our final contributions are on parameterized complexity. On the positive side, we present a single exponential time algorithm when parameterized by treewidth. In terms of kernelization, we show that, even when restricted to binary weights, Weighted Connected Matching does not admit a polynomial kernel when parameterized by vertex cover under standard complexity-theoretical hypotheses.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Weighted Connected Matchings

Gomes¹,

Masquio²,

Pinto³

et al. 2022

Preprint

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“…Recently, Panda et al [20] have built on our results to obtain polynomial algorithms for finding maximum-weight induced matchings in circular-convex and triad-convex bipartite graphs. These graph classes generalize convex bipartite graphs.…”

Section: Related Work and Motivationmentioning

confidence: 99%

“…The bold entries collect the provisional maxima P r in each group. By way of example, the encircled entry P 27 [20] = 54 is the maximum among the shaded entries of the intervals that end at R i = 27, ignoring the yet unprocessed entries. As we proceed from j = 27 to j = 28 in row 30, the intervals with R i = 27 become relevant.…”

Section: Lemma 2 the Recursion (1) Is Correctmentioning

confidence: 99%

Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs

Klemz

Rote

2022

Algorithmica

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A bipartite graph $$G=(U,V,E)$$ G = ( U , V , E ) is convex if the vertices in V can be linearly ordered such that for each vertex $$u\in U$$ u ∈ U , the neighbors of u are consecutive in the ordering of V. An induced matchingH of G is a matching for which no edge of E connects endpoints of two different edges of H. We show that in a convex bipartite graph with n vertices and mweighted edges, an induced matching of maximum total weight can be computed in $$O(n+m)$$ O ( n + m ) time. An unweighted convex bipartite graph has a representation of size O(n) that records for each vertex $$u\in U$$ u ∈ U the first and last neighbor in the ordering of V. Given such a compact representation, we compute an induced matching of maximum cardinality in O(n) time. In convex bipartite graphs, maximum-cardinality induced matchings are dual to minimum chain covers. A chain cover is a covering of the edge set by chain subgraphs, that is, subgraphs that do not contain induced matchings of more than one edge. Given a compact representation, we compute a representation of a minimum chain cover in O(n) time. If no compact representation is given, the cover can be computed in $$O(n+m)$$ O ( n + m ) time. All of our algorithms achieve optimal linear running time for the respective problem and model, and they improve and generalize the previous results in several ways: The best algorithms for the unweighted problem versions had a running time of $$O(n^2)$$ O ( n 2 ) (Brandstädt et al. in Theor. Comput. Sci. 381(1–3):260–265, 2007. 10.1016/j.tcs.2007.04.006). The weighted case has not been considered before.

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“…• the Maximum Weight Induced Matching problem (see, e.g., [2,97]), which corresponds to the case H = {K 2 }, and…”

Section: Implications For the Maximum Weight Independent H-packing Pr...mentioning

confidence: 99%

Treewidth versus clique number. II. Tree-independence number

Dallard¹,

Milanič²,

Štorgel³

2021

Preprint

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In 2020, Dallard, Milanič, and Štorgel initiated a systematic study of graph classes in which the treewidth can only be large due the presence of a large clique, which they call (tw, ω)bounded. The family of (tw, ω)-bounded graph classes provides a unifying framework for a variety of very different families of graph classes, including graph classes of bounded treewidth, graph classes of bounded independence number, intersection graphs of connected subgraphs of graphs with bounded treewidth, and graphs in which all minimal separators are of bounded size. While Chaplick and Zeman showed in 2017 that (tw, ω)-bounded graph classes enjoy some good algorithmic properties related to clique and coloring problems, an interesting open problem is whether (tw, ω)-boundedness has useful algorithmic implications for problems related to independent sets. We provide a partial answer to this question by identifying a sufficient condition for (tw, ω)-bounded graph classes to admit a polynomial-time algorithm for the Maximum Weight Independent H-Packing problem, for any fixed finite set H of connected graphs. This family of problems generalizes several other problems studied in the literature, including the Maximum Weight Independent Set and Maximum Weight Induced Matching problems.Our approach is based on a new min-max graph parameter related to tree decompositions. We define the independence number of a tree decomposition T of a graph as the maximum independence number over all subgraphs of G induced by some bag of T . The tree-independence number of a graph G is then defined as the minimum independence number over all tree decompositions of G. Boundedness of the tree-independence number is a refinement of (tw, ω)-boundedness that is still general enough to hold for all the aforementioned families of graph classes. We show that if a graph is given together with a tree decomposition with bounded independence number, then for any fixed finite set H of connected graphs, the Maximum Weight Independent H-Packing problem can be solved in polynomial time. Motivated by this result, we consider six graph containment relations-the subgraph, topological minor, and minor relations, as well as their induced variants-and for each of them characterize the graphs H for which any graph excluding H with respect to the relation admits a tree decomposition with bounded independence number. These results build on and refine the analogous characterizations due to Dallard, Milanič, and Štorgel for (tw, ω)-boundedness, and shows that in all these cases, (tw, ω)boundedness is actually equivalent to bounded tree-independence number. We use a variety of tools including SPQR trees and potential maximal cliques, and show that in the bounded cases, one can also obtain tree decompositions with bounded independence number efficiently. This leads to polynomial-time algorithms for the Maximum Weight Independent Set problem in an infinite family of graph classes, each of which properly contains the class of chordal graphs. These results also apply to the class of...

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Maximum weight induced matching in some subclasses of bipartite graphs

Cited by 13 publications

References 20 publications

Weighted Connected Matchings

Weighted Connected Matchings

Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs

Treewidth versus clique number. II. Tree-independence number

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