Fully Polynomial-Time Parameterized Computations for Graphs and Matrices of Low Treewidth

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

doi:10.1145/3186898

Cited by 37 publications

(6 citation statements)

References 83 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Albeit it must be under some conditions called fairness, the fact that a linear time algorithm exists for this kind of popular matching helps in better understanding the underlying discrete structure of matchings. Surprisingly, while being a well-known and fundamentally polynomial algorithmic problem, Matching has lately attracted research interest in the parameterized areas of algorithmic as well [9,12]. Here, the overall effort has been put in reducing the polynomial time complexity to linear time, by means of factorising bits of the time complexity to depend on another parameter of the input instance rather than its size.…”

Section: Introductionmentioning

confidence: 99%

Temporal matching

Baste

Bui-Xuan

Roux

2020

Theoretical Computer Science

View full text Add to dashboard Cite

A link stream is a sequence of pairs of the form (t, {u, v}), where t ∈ N represents a time instant and u = v. Given an integer γ, the γ-edge between vertices u and v, starting at time t, is the set of temporally consecutive edges defined by {(t , {u, v}) | t ∈ t, t + γ − 1 }. We introduce the notion of temporal matching of a link stream to be an independent γ-edge set belonging to the link stream. We show that the problem of computing a temporal matching of maximum size is NP-hard as soon as γ > 1. We depict a kernelization algorithm parameterized by the solution size for the problem. As a byproduct we also give a 2-approximation algorithm.Both our 2-approximation and kernelization algorithms are implemented and confronted to link streams collected from real world graph data. We observe that finding temporal matchings is a sensitive question when mining our data from such a perspective as: managing peer-working when any pair of peers X and Y are to collaborate over a period of one month, at an average rate of at least two email exchanges every week. We furthermore design a link stream generating process by mimicking the behaviour of a random moving group of particles under natural simulation, and confront our algorithms to these generated instances of link streams. All the implementations are open source.

show abstract

Section: Introductionmentioning

confidence: 99%

Temporal matching

Baste

Bui-Xuan

Roux

2020

Theoretical Computer Science

View full text Add to dashboard Cite

show abstract

“…Husfeldt [62] shows that the eccentricity of every vertex in an undirected graph on n vertices can be computed in time n·exp [O(k log d)], where k and d are the treewidth and the diameter of the graph, respectively. More recently, a tour de force was achieved by Fomin et al [44] who were the first to design parameterized algorithms with polynomial dependency on the treewidth, for Maximum Matching and Maximum Flow. Furthermore they proved that for graphs with treewidth at most k, a tree decomposition of width O(k 2 ) can be computed in O(k 7 · n log n)-time.…”

Section: Introductionmentioning

confidence: 99%

Fully polynomial FPT algorithms for some classes of bounded clique-width graphs

Coudert¹,

Ducoffe²,

Popa³

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

Recently, hardness results for problems in P were achieved using reasonable complexity theoretic assumptions such as the Strong Exponential Time Hypothesis. According to these assumptions, many graph theoretic problems do not admit truly subquadratic algorithms. A central technique used to tackle the difficulty of the above mentioned problems is fixed-parameter algorithms with polynomial dependency in the fixed parameter (P-FPT). Applying this technique to clique-width, an important graph parameter, remained to be done. In this paper we study several graph theoretic problems for which hardness results exist such as cycle problems, distance problems and maximum matching. We give hardness results and P-FPT algorithms, using cliquewidth and some of its upper-bounds as parameters. We believe that our most important result is an O(k 4 · n + m)-time algorithm for computing a maximum matching where k is either the modular-width or the P 4 -sparseness. The latter generalizes many algorithms that have been introduced so far for specific subclasses such as cographs. Our algorithms are based on preprocessing methods using modular decomposition and split decomposition. Thus they can also be generalized to some graph classes with unbounded clique-width.

show abstract

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

Section: Introductionmentioning

confidence: 99%

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

Giannopoulou

Mertzios

Niedermeier

2017

Theoretical Computer Science

View full text Add to dashboard Cite

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.

show abstract

Fully Polynomial-Time Parameterized Computations for Graphs and Matrices of Low Treewidth

Cited by 37 publications

References 83 publications

Temporal matching

Temporal matching

Fully polynomial FPT algorithms for some classes of bounded clique-width graphs

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

Contact Info

Product

Resources

About