Fixed-parameter complexity of λ-labelings

Fiala, Jiřı́; Kloks, Ton; Kratochvı́l, Jan

doi:10.1016/s0166-218x(00)00387-5

Cited by 117 publications

(60 citation statements)

References 10 publications

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“…The L(2, 1)-labeling problem has been shown to be NP-complete by Griggs and Yeh [12] by a reduction from Hamiltonian cycle (with λ = |V G |). Fiala, Kratochvíl and Kloks [7] showed that L(2, 1)-labeling remains NP-complete also for all fixed λ ≥ 4, while for λ ≤ 3 it is solvable in linear time.…”

Section: Parametersmentioning

confidence: 99%

Fixed Parameter Complexity of Distance Constrained Labeling and Uniform Channel Assignment Problems

Fiala

Gavenčiak

Knop

et al. 2016

Lecture Notes in Computer Science

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We study computational complexity of the class of distance-constrained graph labeling problems from the fixed parameter tractability point of view. The parameters studied are neighborhood diversity and clique width.We rephrase the distance constrained graph labeling problem as a specific uniform variant of the Channel Assignment problem and show that this problem is fixed parameter tractable when parameterized by the neighborhood diversity together with the largest weight. Consequently, every L(p 1 , p 2 , . . . , p k )-labeling problem is FPT when parameterized by the neighborhood diversity, the maximum p i and k.Our results yield also FPT algorithms for all L(p 1 , p 2 , . . . , p k )-labeling problems when parameterized by the size of a minimum vertex cover, answering an open question of Fiala et al.:Parameterized complexity of coloring problems: Treewidth versus vertex cover. The same consequence applies on Channel Assignment when the maximum weight is additionally included among the parameters.Finally, we show that the uniform variant of the Channel Assignment problem becomes NP-complete when generalized to graphs of bounded clique width. ACM Subject Classification G.2.2 Graph Theory

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Section: Parametersmentioning

confidence: 99%

Fixed Parameter Complexity of Distance Constrained Labeling and Uniform Channel Assignment Problems

Fiala

Gavenčiak

Knop

et al. 2016

Lecture Notes in Computer Science

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“…Related work. Several authors [5,10,19] have considered problems parameterized by cyclomatic number; this is also known as parameterizing by feedback edge set. In parameterized complexity, Hitting Set is often studied when the sets to be hit have constant size.…”

Section: Introductionmentioning

confidence: 99%

On Structural Parameterizations of Hitting Set: Hitting Paths in Graphs Using 2-SAT

Jansen

2016

Graph-Theoretic Concepts in Computer Science

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Abstract. Hitting Set is a classic problem in combinatorial optimization. Its input consists of a set system F over a finite universe U and an integer t; the question is whether there is a set of t elements that intersects every set in F. The Hitting Set problem parameterized by the size of the solution is a well-known W[2]-complete problem in parameterized complexity theory. In this paper we investigate the complexity of Hitting Set under various structural parameterizations of the input. Our starting point is the folklore result that Hitting Set is polynomialtime solvable if there is a tree T on vertex set U such that the sets in F induce connected subtrees of T . We consider the case that there is a treelike graph with vertex set U such that the sets in F induce connected subgraphs; the parameter of the problem is a measure of how treelike the graph is. Our main positive result is an algorithm that, given a graph G with cyclomatic number k, a collection P of simple paths in G, and an integer t, determines in time 2 5k (|G| + |P|) O(1) whether there is a vertex set of size t that hits all paths in P. It is based on a connection to the 2-SAT problem in multiple valued logic. For other parameterizations we derive W[1]-hardness and para-NP-completeness results.

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“…5,7,8,10,28 The L 2 1 -labeling problem was shown to be NP-hard for general graphs, 5 and even restricted to many special graphs such as planar graphs, bipartite graphs, chordal graphs 11 and graphs of treewidth two. 14 In order to solve the L 2 1 -labeling problem, many exact algorithms were proposed and they appear to be exponential 12,13 for general graphs. Until now, only a few graph classes such as paths, cycles, wheels are known to have polynomial time algorithms for this problem.…”

Section: Introductionmentioning

confidence: 99%

A Multistart Iterated Reactive Tabu Search for the <I>L</I>(2, 1)-Labeling Problem

Luo¹

2015

Jnl of Comp & Theo Nano

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The problem of assigning frequencies to transmitters in a radio network can be modeled through vertex labelings of a graph, wherein each vertex represents a transmitter and edges connect vertices whose corresponding transmitters are operating in close proximity. In one such model, a k-L 2 1 -labeling of a graph G is an assignment of labels from 0 1 k to the vertices of G such that vertices at distance two get different labels and adjacent vertices get labels that are at least two apart. The -number G of G is the minimum value k such that G admits a k-L 2 1 -labeling. The L 2 1 -labeling problem consists of finding the minimum -number of a graph. This paper presents a multistart iterated reactive tabu search heuristic (MIRTS) for the L 2 1 -labeling problem. MIRTS incorporates the multistart feature and iterated reactive tabu scheme. The algorithm has been successfully applied to the L 2 1 -labeling problem of interesting hypercubes graphs and find improved upper bounds of the -numbers for some of these graphs.

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Fixed-parameter complexity of λ-labelings

Cited by 117 publications

References 10 publications

Fixed Parameter Complexity of Distance Constrained Labeling and Uniform Channel Assignment Problems

Fixed Parameter Complexity of Distance Constrained Labeling and Uniform Channel Assignment Problems

On Structural Parameterizations of Hitting Set: Hitting Paths in Graphs Using 2-SAT

A Multistart Iterated Reactive Tabu Search for the <I>L</I>(2, 1)-Labeling Problem

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