The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e216" altimg="si6.svg"><mml:mi>b</mml:mi></mml:math>-Matching problem in distance-hereditary graphs and beyond

Ducoffe, Guillaume; Popa, Alexandru

doi:10.1016/j.dam.2021.09.012

Cited by 4 publications

(4 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…About Theorem 2, we note that different reductions from Maximum Matching to Maximum b-Matching have already been considered for graphs of bounded modularwidth [51] or bounded split-width [28], that are subclasses of bounded clique-width graphs. However, from the algorithmic point of view, the instances of Maximum b-Matching outputted by these former reductions are of bounded size, a much more restricted case than bounded tree-width.…”

Section: Related Workmentioning

confidence: 99%

“…In its simplest form, the former is about the existence of O(f (k) • (n + m) 1+o (1) )-time algorithms for various graph problems when some fixed parameter is at most k 1 . As far as we are concerned here, such running times were obtained in [13,29,28,36,45,49,50,51,59] for Maximum Matching, for different parameterizations. For instance, for the graphs of tree-width at most k, Fomin et.…”

Section: Introductionmentioning

confidence: 99%

“…However, the study of its applications to polynomial-time solvable problems is comparatively much more G. Ducoffe 15:3 recent and, so far, limited to cycle problems [5,13] and distance problems [13,16,27,52]. Parameterized almost linear-time algorithms for Maximum Matching are known for the important subclasses of bounded tree-width graphs [36,50], graphs of bounded modularwidth [13,51], and some others [13,28,29]. However, as far as we know, the complexity of this problem on bounded clique-width graphs has been open until this article.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Maximum Matching in Almost Linear Time on Graphs of Bounded Clique-Width

Ducoffe

2022

Algorithmica

Self Cite

View full text Add to dashboard Cite

Recently, independent groups of researchers have presented algorithms to compute a maximum matching in Õ(f (k) • (n + m)) time, for some computable function f , within the graphs where some clique-width upper bound is at most k (e.g., tree-width, modular-width and P4-sparseness). However, to the best of our knowledge, the existence of such algorithm within the graphs of bounded clique-width has remained open until this paper. Indeed, we cannot even apply Courcelle's theorem to this problem directly, because a matching cannot be expressed in M SO1 logic.Our first contribution is an almost linear-time algorithm to compute a maximum matching in any bounded clique-width graph, being given a corresponding clique-width expression. It can be used to also compute the Edmonds-Gallai decomposition. For that, we do apply Courcelle's theorem, but in order to compute the cardinality of a maximum matching rather than the matching itself, via the classic Tutte-Berge formula. To obtain with this approach a maximum matching, we need to combine it with a recursive dissection scheme for bounded clique-width graphs based on the existence of balanced edge-cuts with bounded neighbourhood diversity, and with a distributed version of Courcelle's theorem (Courcelle and Vanicat, DAM 2016) -of which we present here a slightly stronger version than the standard one in the literature -in order to evaluate the Tutte-Berge formula on various subgraphs of the input.Finally, for the bipartite graphs of clique-width at most k, we present an alternative Õ(k 2 •(n+m))time algorithm for the problem. The algorithm is randomized and it is based on a completely different approach than above: combining various reductions to matching and flow problems on bounded tree-width graphs with a very recent result on the parameterized complexity of linear programming (Dong et. al., STOC'21). Our results for bounded clique-width graphs extend many prior works on the complexity of Maximum Matching within cographs, distance-hereditary graphs, series-parallel graphs and other subclasses. ACM Subject ClassificationTheory of computation → Design and analysis of algorithms; Theory of computation → Graph algorithms analysis

show abstract

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Maximum Matching in Almost Linear Time on Graphs of Bounded Clique-Width

Ducoffe

2022

Algorithmica

Self Cite

View full text Add to dashboard Cite

show abstract

“…Unlike for treewidth, it was open until this paper whether there does exist a parameterized quasi-linear-time algorithm for this problem on bounded clique-width graphs that matches their conditional lower bound. Their work has been continued in [28,27,46] and especially in [47], where the authors obtained an O((kn) 2 )-time algorithm for All-Pairs Shortest Paths (APSP) on n-vertex graphs of clique-width at most k.…”

Section: Introductionmentioning

confidence: 99%

Fast Diameter Computation Within Split Graphs

Ducoffe

Habib

Viennot

2019

Combinatorial Optimization and Applications

Self Cite

View full text Add to dashboard Cite

When can we compute the diameter of a graph in quasi linear time? We address this question for the class of split graphs, that we observe to be the hardest instances for deciding whether the diameter is at most two. We stress that although the diameter of a non-complete split graph can only be either 2 or 3, under the Strong Exponential-Time Hypothesis (SETH) we cannot compute the diameter of a split graph in less than quadratic time. Therefore it is worth to study the complexity of diameter computation on subclasses of split graphs, in order to better understand the complexity border. Specifically, we consider the split graphs with bounded clique-interval number and their complements, with the former being a natural variation of the concept of interval number for split graphs that we introduce in this paper. We first discuss the relations between the clique-interval number and other graph invariants such as the classic interval number of graphs, the treewidth, the VC-dimension and the stabbing number of a related hypergraph. Then, in part based on these above relations, we almost completely settle the complexity of diameter computation on these subclasses of split graphs:• For the k-clique-interval split graphs, we can compute their diameter in truly subquadratic time if k = O(1), and even in quasi linear time if k = o(log n) and in addition a corresponding ordering is given. However, under SETH this cannot be done in truly subquadratic time for any k = ω(log n).• For the complements of k-clique-interval split graphs, we can compute their diameter in truly subquadratic time if k = O(1), and even in time O(km) if a corresponding ordering is given. Again this latter result is optimal under SETH up to polylogarithmic factors.Our findings raise the question whether a k-clique interval ordering can always be computed in quasi linear time. We prove that it is the case for k = 1 and for some subclasses such as boundedtreewidth split graphs, threshold graphs and comparability split graphs. Finally, we prove that some important subclasses of split graphs -including the ones mentioned above -have a bounded clique-interval number.

show abstract

Getting linear time in graphs of bounded neighborhood diversity

Cordasco,

Gargano,

Rescigno

2024

Networks

View full text Add to dashboard Cite

Parameterized complexity, introduced to efficiently solve NP‐hard problems for small values of a fixed parameter, has been recently used as a tool to speed up algorithms for tractable problems. Following this line of research, we design algorithms parameterized by neighborhood diversity () for several graph theoretic problems in : Maximum ‐Matching, Triangle Counting and Listing, Girth, Global Minimum Vertex Cut, and Perfect Graphs Recognition. Such problems are known to admit algorithms parameterized by modular‐width () and consequently—as is a special case of —by . However, the proposed novel algorithms allow for improving the computational complexity from time —where and denote, respectively, the number of vertices and edges in the input graph—to time which is only additive in the size of the input. Then we consider some classical NP‐hard problems (Maximum independent set, Maximum clique, and Minimum dominating set) and show that for several classes of hereditary graphs, they admit linear time algorithms for sufficiently small—nonnecessarily constant—values of the neighborhood diversity parameter.

show abstract

The b-Matching problem in distance-hereditary graphs and beyond

Cited by 4 publications

References 31 publications

Maximum Matching in Almost Linear Time on Graphs of Bounded Clique-Width

Maximum Matching in Almost Linear Time on Graphs of Bounded Clique-Width

Fast Diameter Computation Within Split Graphs

Getting linear time in graphs of bounded neighborhood diversity

Contact Info

Product

Resources

About