What’s Next? Future Directions in Parameterized Complexity

Marx, Dániel

doi:10.1007/978-3-642-30891-8_20

Cited by 21 publications

(19 citation statements)

References 114 publications

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“…They mention that the problem may be solved in linear-time on graphs of treewidth w by applying Courcelle's Theorem [2,6,8,9], or by a dynamic programming step with running time 2 w Ω(w) · n. The intuition behind this dynamic programming algorithm is to keep track of possible Kuratowski minors in the graph after deleting at most k vertices. We give a 2 O(w log w) · n time algorithm for (the weighted version of) Vertex Planarization on bounded treewidth graphs, thereby answering an open problem posed by Marx [28]. The states of the dynamic program are based on possible embeddings of the planar subgraph.…”

Section: K-vertex Planarizationmentioning

confidence: 99%

A Near-Optimal Planarization Algorithm

Jansen¹,

Lokshtanov²,

Saurabh³

2013

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

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The problem of testing whether a graph is planar has been studied for over half a century, and is known to be solvable in O(n) time using a myriad of different approaches and techniques. Robertson and Seymour established the existence of a cubic algorithm for the more general problem of deciding whether an n-vertex graph can be made planar by at most k vertex deletions, for every fixed k. Of the known algorithms for k-Vertex Planarization, the algorithm of Marx and Schlotter (WG 2007, Algorithmica 2012) running in time 22 achieves the best running time dependence on k. The algorithm ofthat is not stated explicitly, achieves the best dependence on n.In this paper we present an algorithm for k-Vertex Planarization with running time 2 O(k log k) · n, significantly improving the running time dependence on k without compromising the linear dependence on n. Our main technical contribution is a novel scheme to reduce the treewidth of the input graph to O(k) in time 2 O(k log k) · n. It combines new insights into the structure of graphs that become planar after contracting a matching, with a Baker-type subroutine that reduces the number of disjoint paths through planar parts of the graph that are not affected by the sought solution. To solve the reduced instances we formulate a dynamic programming algorithm for Weighted Vertex Planarization on graphs of treewidth w with running time 2 O(w log w) · n, thereby improving over previous double-exponential algorithms.While Kawarabayashi's planarization algorithm relies heavily on deep results from the graph minors project, our techniques are elementary and practically self-contained. We expect them to be applicable to related edge-deletion and contraction variants of planarization problems.

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Section: K-vertex Planarizationmentioning

confidence: 99%

A Near-Optimal Planarization Algorithm

Jansen¹,

Lokshtanov²,

Saurabh³

2013

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

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“…The most widely used complexity assumption for such tight lower bounds is the Exponential Time Hypothesis (ETH), which posits that no subexponential-time algorithms for k-CNF-SAT or CNF-SAT exist [34]. For more information about this "optimality program", we refer to a survey of Lokshtanov et al [36] and to an appropriate section of the recent survey of Marx [37].…”

Section: Cluster Editingmentioning

confidence: 99%

Tight bounds for parameterized complexity of Cluster Editing with a small number of clusters

Fomin

Kratsch

Pilipczuk

et al. 2014

Journal of Computer and System Sciences

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In the Cluster Editing problem, also known as Correlation Clustering, we are given an undirected n-vertex graph G and a positive integer k. The task is to decide if G can be transformed into a cluster graph, i.e., a disjoint union of cliques, by changing at most k adjacencies, i.e. by adding/deleting at most k edges. We give a subexponential-time parameterized algorithm that in time 2 O( √ pk) + n O(1) decides whether G can be transformed into a cluster graph with exactly p cliques by changing at most k adjacencies. Our algorithmic findings are complemented by the following tight lower bound on the asymptotic behavior of our algorithm. We show that unless ETH fails, for any constant 0 < σ ≤ 1, there is p = Θ(k σ ) such that there is no algorithm deciding in time 2 o( √ pk) · n O(1) whether G can be transformed into a cluster graph with at most p cliques by changing at most k adjacencies.

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“…A similar situation holds for Feedback Arc Set in Tournaments, for which the fastest known algorithms work in time 2 O( √ k) · n O(1) [23,33]. For this reason, the question of establishing tight upper and lower bounds for Minimum Fill-In was already asked explicitly by Fomin and Villanger [25], repeated by Marx in his survey on the optimality programme [38], and then reiterated for respective subclasses in all the works [6,7,19]. The goal of this paper is to remedy this situation by providing complexity foundations for proving that the square root in the exponent of the running time is hard to improve.…”

Section: Introductionmentioning

confidence: 96%

“…Hence, the study of this phenomenon is an important direction that naturally belongs to the optimality programme (cf. [35,38]): a trend in parameterized complexity that focuses on systematic study of parameterized problems by providing possibly tight upper and lower bounds on their complexity.…”

Section: Introductionmentioning

confidence: 99%

Lower bounds for the parameterized complexity of Minimum Fill-In and other completion problems

Bliznets

Cygan

Komosa

et al. 2015

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

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In this work, we focus on several completion problems for subclasses of chordal graphs: Minimum Fill-In, Interval Completion, Proper Interval Completion, Threshold Completion, and Trivially Perfect Completion. In these problems, the task is to add at most k edges to a given graph in order to obtain a chordal, interval, proper interval, threshold, or trivially perfect graph, respectively. We prove the following lower bounds for all these problems, as well as for the related Chain Completion problem:• Assuming the Exponential Time Hypothesis, none of these problems can be solved in time

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What’s Next? Future Directions in Parameterized Complexity

Cited by 21 publications

References 114 publications

A Near-Optimal Planarization Algorithm

A Near-Optimal Planarization Algorithm

Tight bounds for parameterized complexity of Cluster Editing with a small number of clusters

Lower bounds for the parameterized complexity of Minimum Fill-In and other completion problems

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