An improved approximation for Maximum
 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e183" altimg="si9.svg"><mml:mi>k</mml:mi></mml:math>-dependent Set on bipartite graphs

Hosseinian, Seyedmohammadhossein; Butenko, Sergiy

doi:10.1016/j.dam.2021.10.015

Cited by 9 publications

(13 citation statements)

References 12 publications

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“…By the structure of H, we have diss(H) = α(H) if and only if H has an independent set that contains, for every i in [m], exactly one of the vertices with exponent (i, 1). As noted above, this is equivalent to the satisfiability of f , which shows the NP-hardness of deciding equality in (5).…”

Section: Hardness Of Deciding Equality In (mentioning

confidence: 88%

“…Our initial motivation to consider the extremal graphs for (2) was the attempt to improve the approximation algorithm of Hosseinian and Butenko [5]. This remains to be done.…”

Section: Discussionmentioning

confidence: 99%

“…Recently, Hosseinian and Butenko [5] proposed a simple 4 3 -approximation algorithm for the maximum dissociation set problem restricted to bipartite graphs. Their result can be derived from the following two simple inequalities: Let G be a graph and let M be a maximum matching in G. Since every independent set in G − M is a dissociation set in G, we have…”

Section: Introductionmentioning

confidence: 99%

“…Since a maximum matching M in a given bipartite graph G as well as a maximum independent set I in the bipartite graph G − M can be determined efficiently, the combination of (1) and (2) implies that returning I yields a 4 3 -approximation for the maximum dissociation set problem in G. We will give the simple proof of (2) that is implicit in [5] below. As our first contribution we show that the extremal graphs for (2) have a very restricted structure, which yields the following.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 2. For each of the inequalities (3), ( 4), (5), and (6), it is NP-hard to decide whether a given graph satisfies it with equality.…”

Section: Introductionmentioning

confidence: 99%

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Relating dissociation, independence, and matchings

Bock¹,

Pardey²,

Penso³

et al. 2022

Preprint

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A dissociation set in a graph is a set of vertices inducing a subgraph of maximum degree at most 1. Computing the dissociation number diss(G) of a given graph G, defined as the order of a maximum dissociation set in G, is algorithmically hard even when G is restricted to be bipartite. Recently, Hosseinian and Butenko proposed a simple 4 3 -approximation algorithm for the dissociation number problem in bipartite graphs. Their result relies on the inequality diss(G) ≤ 4 3 α(G − M ) implicit in their work, where G is a bipartite graph, M is a maximum matching in G, and α(G − M ) denotes the independence number of G − M . We show that the pairs (G, M ) for which this inequality holds with equality can be recognized efficiently, and that a maximum dissociation set can be determined for them efficiently. The dissociation number of a graph G satisfies max{α(G), 2ν s (G)} ≤ diss(G) ≤ α(G) + ν s (G) ≤ 2α(G), where ν s (G) denotes the induced matching number of G. We show that deciding whether diss(G) equals any of the four terms lower and upper bounding diss(G) is NP-hard.

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Section: Hardness Of Deciding Equality In (mentioning

confidence: 88%

“…Our initial motivation to consider the extremal graphs for (2) was the attempt to improve the approximation algorithm of Hosseinian and Butenko [5]. This remains to be done.…”

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Theorem 2. For each of the inequalities (3), ( 4), (5), and (6), it is NP-hard to decide whether a given graph satisfies it with equality.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Relating dissociation, independence, and matchings

Bock¹,

Pardey²,

Penso³

et al. 2022

Preprint

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show abstract

A bound on the dissociation number

Bock

Pardey

Penso

et al. 2023

Journal of Graph Theory

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The dissociation number diss ( G ) $\text{diss}(G)$ of a graph G $G$ is the maximum order of a set of vertices of G $G$ inducing a subgraph that is of maximum degree at most 1. Computing the dissociation number of a given graph is algorithmically hard even when restricted to subcubic bipartite graphs. For a graph G $G$ with n $n$ vertices, m $m$ edges, k $k$ components, and c 1 ${c}_{1}$ induced cycles of length 1 modulo 3, we show diss ( G ) ≥ n − 1 3 ( m + k + c 1 ) $\text{diss}(G)\ge n-\frac{1}{3}(m+k+{c}_{1})$. Furthermore, we characterize the extremal graphs in which every two cycles are vertex‐disjoint.

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Relating the independence number and the dissociation number

Bock

Pardey

Penso

et al. 2023

Journal of Graph Theory

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The independence number and the dissociation number of a graph are the largest orders of induced subgraphs of of maximum degree at most 0 and at most 1, respectively. We consider possible improvements of the obvious inequality . For connected cubic graphs distinct from , we show , and describe the rich and interesting structure of the extremal graphs in detail. For bipartite graphs, and, more generally, triangle‐free graphs, we also obtain improvements. For subcubic graphs though, the inequality cannot be improved in general, and we characterize all extremal subcubic graphs.

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An improved approximation for Maximum k-dependent Set on bipartite graphs

Cited by 9 publications

References 12 publications

Relating dissociation, independence, and matchings

Relating dissociation, independence, and matchings

A bound on the dissociation number

Relating the independence number and the dissociation number

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