Algorithmic Meta-theorems for Restrictions of Treewidth

Lampis, Michael

doi:10.1007/s00453-011-9554-x

Cited by 142 publications

(55 citation statements)

References 22 publications

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“…Mimicking the development for treewidth would point to extending this result to MSO. Sadly, a proven double exponential dependency on the run-time of model-checking MSO parameterized by the size of a vertex cover implies that no such result is possible [42]. Is there a characterization that better captures for which problems this is possible?…”

Section: Discussionmentioning

confidence: 99%

Width, Depth, and Space: Tradeoffs between Branching and Dynamic Programming

Chen¹,

Reidl

Rossmanith

et al. 2018

Algorithms

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Treedepth is a well-established width measure which has recently seen a resurgence of interest. Since graphs of bounded treedepth are more restricted than graphs of bounded treeor pathwidth, we are interested in the algorithmic utility of this additional structure. On the negative side, we show with a novel approach that the space consumption of any (single-pass) dynamic programming algorithm on treedepth decompositions of depth d cannot be bounded by (2 −) d • log O(1) n for VERTEX COVER, (3 −) d • log O(1) n for 3-COLORING and (3 −) d • log O(1) n for DOMINATING SET for any > 0. This formalizes the common intuition that dynamic programming algorithms on graph decompositions necessarily consume a lot of space and complements known results of the time-complexity of problems restricted to low-treewidth classes. We then show that treedepth lends itself to the design of branching algorithms. Specifically, we design two novel algorithms for DOMINATING SET on graphs of treedepth d: A pure branching algorithm that runs in time d O(d 2) • n and uses space O(d 3 log d + d log n) and a hybrid of branching and dynamic programming that achieves a running time of O(3 d log d • n) while using O(2 d d log d + d log n) space.

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Section: Discussionmentioning

confidence: 99%

Width, Depth, and Space: Tradeoffs between Branching and Dynamic Programming

Chen¹,

Reidl

Rossmanith

et al. 2018

Algorithms

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“…the tree-width, these parameters focus on dense graphs. First, up to our knowledge, of these parameters is the neighborhood diversity defined by Lampis [17]. We denote the neighborhood diversity of a graph G = (V, E) as nd(G).…”

Section: Preliminaries On Structural Graph Parametersmentioning

confidence: 99%

Target Set Selection in Dense Graph Classes

Dvořák¹,

Knop²,

Toufar³

2022

SIAM J. Discrete Math.

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In this paper we study the Target Set Selection problem from a parameterized complexity perspective. Here for a given graph and a threshold for each vertex the task is to find a set of vertices (called a target set) to activate at the beginning which activates the whole graph during the following iterative process. A vertex outside the active set becomes active if the number of so far activated vertices in its neighborhood is at least its threshold.We give two parameterized algorithms for a special case where each vertex has the threshold set to the half of its neighbors (the so called Majority Target Set Selection problem) for parameterizations by the neighborhood diversity and the twin cover number of the input graph.We complement these results from the negative side. We give a hardness proof for the Majority Target Set Selection problem when parameterized by (a restriction of) the modularwidth -a natural generalization of both previous structural parameters. We show that the Target Set Selection problem parameterized by the neighborhood diversity when there is no restriction on the thresholds is W[1]-hard.

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“…It is known that such a minimum partition can be found in linear time using fast modular decomposition algorithms [28,33]. It is also known that nd(G) 2 vc(G) + vc(G) for every graph G [24].…”

Section: Fpt Algorithm Parameterized By Neighborhood Diversitymentioning

confidence: 99%

Parameterized Complexity of Safe Set

Belmonte¹,

Hanaka²,

Katsikarelis³

et al. 2020

JGAA

Self Cite

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In this paper we study the problem of finding a small safe set S in a graph G, i.e. a non-empty set of vertices such that no connected component of G[S] is adjacent to a larger component in G − S. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that (1) the problem is W[2]-hard when parameterized by the pathwidth pw and cannot be solved in time n o(pw) unless the ETH is false, (2) it admits no polynomial kernel parameterized by the vertex cover number vc unless PH = Σ p 3 , but (3) it is fixed-parameter tractable (FPT) when parameterized by the neighborhood diversity nd, and (4) it can be solved in time n f (cw) for some double exponential function f where cw is the clique-width. We also present (5) a faster FPT algorithm when parameterized by solution size.

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Algorithmic Meta-theorems for Restrictions of Treewidth

Cited by 142 publications

References 22 publications

Width, Depth, and Space: Tradeoffs between Branching and Dynamic Programming

Width, Depth, and Space: Tradeoffs between Branching and Dynamic Programming

Target Set Selection in Dense Graph Classes

Parameterized Complexity of Safe Set

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