A Linear-Time Algorithm for Maximum-Cardinality Matching on Cocomparability Graphs

Mertzios, George B.; Nichterlein, André; Niedermeier, Rolf

doi:10.1137/17m1120920

Cited by 18 publications

(15 citation statements)

References 56 publications

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“…Yuster [26] developed an O(rn 2 log n)-time algorithm, where r is the difference between maximum and minimum degree of the input graph. Moreover, there are linear-time algorithms for computing maximum matchings in special graph classes, including convex bipartite [25], strongly chordal [7], chordal bipartite graphs [6], and cocomparability graphs [20].…”

Section: Introductionmentioning

confidence: 99%

The Power of Linear-Time Data Reduction for Maximum Matching

2020

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Finding maximum-cardinality matchings in undirected graphs is arguably one of the most central graph primitives. For m-edge and n-vertex graphs, it is well-known to be solvable in $$O(m\sqrt{n})$$ O ( m n ) time; however, for several applications this running time is still too slow. We investigate how linear-time (and almost linear-time) data reduction (used as preprocessing) can alleviate the situation. More specifically, we focus on linear-time kernelization. We start a deeper and systematic study both for general graphs and for bipartite graphs. Our data reduction algorithms easily comply (in form of preprocessing) with every solution strategy (exact, approximate, heuristic), thus making them attractive in various settings.

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Section: Introductionmentioning

confidence: 99%

The Power of Linear-Time Data Reduction for Maximum Matching

2020

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“…LexDFS and LexBFS are graph searching algorithms that have proven powerful on a number of graph families, cocomparability being one of them. We refer the reader to [31,11,9,29,10,28] for more on this topic. Unfortunately, one can show that LexDFS cocomparability orderings are not preserved under the ı and • rules, and thus computing such a solution would require computing a LexDFS ordering on ‡ ú , ‡ • .…”

Section: Discussionmentioning

confidence: 99%

“…p 3 : For this pattern, it su ces to notice that a ª e ª f always produces a p 3 in ‡. p 4 : This configuration in ‡ ú implies the following adjacencies in G: Since a ª c ª e ª f , it follows (using(29,30)) that af oe E would imply a, c, f © p 4 . Thus af / oe E. Suppose now that be oe E. Given that a ª b and d ª e, we try to place b with respect to e.…”

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confidence: 98%

Maximum Induced Matching Algorithms via Vertex Ordering Characterizations

Habib

Mouatadid

2019

Algorithmica

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We study the maximum induced matching problem on a graph G. Induced matchings correspond to independent sets in L 2 (G), the square of the line graph of G. The problem is NP-complete on bipartite graphs. In this work, we show that for a number of graph families with forbidden vertex orderings, almost all forbidden patterns on three vertices are preserved when taking the square of the line graph. These orderings can be computed in linear time in the size of the input graph. In particular, given a graph class G characterized by a vertex ordering, and a graph G = (V, E) oe G with a corresponding vertex ordering ‡ of V , one can produce (in linear time in the size of G) an ordering on the vertices of L 2 (G), that shows that L 2 (G) oe G -for a number of graph classes G -without computing the line graph or the square of the line graph of G. These results generalize and unify previous ones on showing closure under L 2 (·) for various graph families. Furthermore, these orderings on L 2 (G) can be exploited algorithmically to compute a maximum induced matching on G faster. We illustrate this latter fact in the second half of the paper where we focus on cocomparability graphs, a large graph class that includes interval, permutation, trapezoid graphs, and co-graphs, and we present the first O(mn) time algorithm to compute a maximum weighted induced matching on cocomparability graphs; an improvement from the best known O(n 4 ) time algorithm for the unweighted case.

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“…Initially, d = 0. The general approach is common to many results that upper-bound the size of the graph in terms of its feedback edge number [3,20,25,32]: To this end, it is sufficient to reduce the number of degree-one vertices and the lengths of paths of degree-two vertices. We will see that the second part-shrinking paths-is the challenging one.…”

Section: Parameterizing By the Feedback Edge Numbermentioning

confidence: 99%

Parameterized Algorithms for Power-Efficient Connected Symmetric Wireless Sensor Networks

Bentert

Bevern

Nichterlein

et al. 2017

Algorithms for Sensor Systems

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We study an NP-hard problem motivated by energy-efficiently maintaining the connectivity of a symmetric wireless sensor communication network. Given an edge-weighted n-vertex graph, find a connected spanning subgraph of minimum cost, where the cost is determined by letting each vertex pay the most expensive edge incident to it in the subgraph. We provide an algorithm that works in polynomial time if one can find a set of obligatory edges that yield a spanning subgraph with O(log n) connected components. We also provide a linear-time algorithm that reduces any input graph that consists of a tree together with g additional edges to an equivalent graph with O(g) vertices. Based on this, we obtain a polynomial-time algorithm for g ∈ O(log n). On the negative side, we show that o(log n)-approximating the difference d between the optimal solution cost and a natural lower bound is NP-hard and that there are presumably no exact algorithms running in 2 o(n) time or in f (d) · n O(1) time for any computable function f .

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A Linear-Time Algorithm for Maximum-Cardinality Matching on Cocomparability Graphs

Cited by 18 publications

References 56 publications

The Power of Linear-Time Data Reduction for Maximum Matching

The Power of Linear-Time Data Reduction for Maximum Matching

Maximum Induced Matching Algorithms via Vertex Ordering Characterizations

Parameterized Algorithms for Power-Efficient Connected Symmetric Wireless Sensor Networks

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