Fully polynomial-time parameterized computations for graphs and matrices of low treewidth

Fomin, Fedor V.; Lokshtanov, Daniel; Pilipczuk, Michał; Saurabh, Saket; Wrochna, Marcin

doi:10.1137/1.9781611974782.92

Cited by 22 publications

(21 citation statements)

References 69 publications

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“…In this line of research, Fomin et al studied graph and matrix problems on instances with small treewidth. In particular the authors presented, among other results, an O(k 3 n log n) randomized algorithm for computing the cardinality of a maximum matching and an O(k 4 n log 2 n) randomized algorithm for actually constructing a maximum matching, where k is an upper bound for the treewidth of the given graph [25].…”

Section: Outlook and Discussionmentioning

confidence: 99%

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Polynomial fixed-parameter algorithms: A case study for longest path on interval graphs

Giannopoulou

Mertzios

Niedermeier

2017

Theoretical Computer Science

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We study the design of fixed-parameter algorithms for problems already known to be solvable in polynomial time. The main motivation is to get more efficient algorithms for problems with unattractive polynomial running times. Here, we focus on a fundamental graph problem: Longest Path, that is, given an undirected graph, find a maximum-length path in G. Longest Path is NP-hard in general but known to be solvable in O(n 4 ) time on n-vertex interval graphs. We show how to solve Longest Path on Interval Graphs, parameterized by vertex deletion number k to proper interval graphs, in O(k 9 n) time. Notably, Longest Path is trivially solvable in linear time on proper interval graphs, and the parameter value k can be approximated up to a factor of 4 in linear time. From a more general perspective, we believe that using parameterized complexity analysis may enable a refined understanding of efficiency aspects for polynomial-time solvable problems similarly to what classical parameterized complexity analysis does for NP-hard problems.

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Section: Outlook and Discussionmentioning

confidence: 99%

“…We finally mention that, very recently, two further works delved deeper into "FPT inside P" algorithms for Maximum Matching [25,42].…”

Section: Related Workmentioning

confidence: 99%

Polynomial fixed-parameter algorithms: A case study for longest path on interval graphs

Giannopoulou

Mertzios

Niedermeier

2017

Theoretical Computer Science

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“…All this and the more general spirit of "parameterization for polynomial-time solvable problems" (also referred to as "FPT in P" or "FPTP" for short) [12] forms the starting point of our research. Remarkably, Fomin et al [9] recently developed an algorithm to compute a maximum matching in graphs of treewidth k in O(k 4 n log 2 n) randomized time. Afterwards, Iwata, Ogasawara, and Ohsaka [16] provided an elegant algorithm computing a maximum matching in graphs of treedepth in O( ⋅ m) time.…”

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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“…time algorithm for Maximum-Cardinality Matching on graphs of treewidth at most k, see [13,19]. Turning our attention on denser graph classes of bounded clique-width, we proved in [5]…”

Section: :3mentioning

confidence: 99%

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

Ducoffe

Popa

2021

Discrete Applied Mathematics

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We make progress on the fine-grained complexity of Maximum-Cardinality Matching on graphs of bounded clique-width. Quasi linear-time algorithms for this problem have been recentlyproposed for the important subclasses of bounded-treewidth graphs (Fomin et al., SODA'17) and graphs of bounded modular-width (Coudert et al., SODA'18). We present such algorithm for bounded split-width graphs -a broad generalization of graphs of bounded modular-width, of which an interesting subclass are the distance-hereditary graphs. Specifically, we solve Maximum-Cardinality Matching in O((k log 2 k)•(m+n)•log n)-time on graphs with split-width at most k. We stress that the existence of such algorithm was not even known for distance-hereditary graphs until our work. Doing so, we improve the state of the art (Dragan, WG'97) and we answer an open question of (Coudert et al., SODA'18). Our work brings more insights on the relationships between matchings and splits, a.k.a., join operations between two vertex-subsets in different connected components. Furthermore, our analysis can be extended to the more general (unit cost) b-Matching problem. On the way, we introduce new tools for b-Matching and dynamic programming over split decompositions, that can be of independent interest.

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Fully polynomial-time parameterized computations for graphs and matrices of low treewidth

Cited by 22 publications

References 69 publications

Polynomial fixed-parameter algorithms: A case study for longest path on interval graphs

Polynomial fixed-parameter algorithms: A case study for longest path on interval graphs

The Power of Linear-Time Data Reduction for Maximum Matching

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

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