Determining the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.gif" display="inline" overflow="scroll"><mml:mi>L</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math>-span in polynomial space

Junosza-Szaniawski, Konstanty; Kratochvíl, Jan; Liedloff, Mathieu; Rzewski, Paweł

doi:10.1016/j.dam.2013.03.027

Cited by 4 publications

References 26 publications

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The L(h, k)-Labelling Problem: An Updated Survey and Annotated Bibliography

Calamoneri

2011

The Computer Journal

160

104

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Given any fixed nonnegative integer values h and k, the L(h, k)labelling problem consists in an assignment of nonnegative integers to the nodes of a graph such that adjacent nodes receive values which differ by at least h, and nodes connected by a 2 length path receive values which differ by at least k. The span of an L(h, k)-labelling is the difference between the largest and the smallest assigned frequency. The goal of the problem is to find out an L(h, k)-labelling with minimum span. The L(h, k)-labelling problem has been intensively studied following many approaches and restricted to many special cases, concerning both the values of h and k and the considered classes of graphs. This paper reviews the results from previous by published literature, looking at the problem with a graph algorithmic approach. It is an update of a previous survey written by the same author.

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The L(h, k)-Labelling Problem: An Updated Survey and Annotated Bibliography

Calamoneri

2011

The Computer Journal

160

104

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Solving Distance-constrained Labeling Problems for Small Diameter Graphs via TSP^*

Hanaka

Ono

Sugiyama

2023

2023 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW)

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Solving Distance-constrained Labeling Problems for Small Diameter Graphs via TSP

Hanaka,

Ono,

Sugiyama

2024

IJNC

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For an undirected graph G = (V, E) and a k-non-negative integer vector p = (p1, . . . , p k ), a mapping l : V → N ∪ {0} is called an L(p)-labeling of G if |l(u) − l(v)| ≥ p d for any two distinct vertices u, v ∈ V with distance d, and the maximum value of {l(v) | v ∈ V } is called the span of l. Originally, L(p)-labeling of G for p = (2, 1) is introduced in the context of frequency assignment in radio networks, where 'close' transmitters must receive different frequencies and 'very close' transmitters must receive frequencies that are at least two frequencies apart so that they can avoid interference. L(p)-Labeling is the problem of finding the minimum span λp among L(p)labelings of G, which is NP-hard for every non-zero p. L(p)-Labeling is well studied for specific p's; in particular, many (exact or approximation) algorithms for general graphs or restricted classes of graphs are proposed for p = (2, 1) or more generally p = (p, q). Unfortunately, most algorithms strongly depend on the values of p, and it is not apparent to extend algorithms for p to ones for another p ′ in general. In this paper, we give a simple polynomial-time reduction of L(p)-Labeling on graphs with a small diameter to Metric (Path) TSP, which enables us to use numerous results on (Metric) TSP. On the practical side, we can utilize various high-performance heuristics for TSP, such as Concordo and LKH, to solve our problem. On the theoretical side, we can see that the problem for any p under this framework is 1.5-approximable, and it can be solved by the Held-Karp algorithm in O(2 n n 2 ) time, where n is the number of vertices, and so on.

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Determining the L(2,1)-span in polynomial space

Cited by 4 publications

References 26 publications

The L(h, k)-Labelling Problem: An Updated Survey and Annotated Bibliography

The L(h, k)-Labelling Problem: An Updated Survey and Annotated Bibliography

Solving Distance-constrained Labeling Problems for Small Diameter Graphs via TSP^*

Solving Distance-constrained Labeling Problems for Small Diameter Graphs via TSP

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Determining the L(2,1)-span in polynomial space

Cited by 4 publications

References 26 publications

The L(h, k)-Labelling Problem: An Updated Survey and Annotated Bibliography

The L(h, k)-Labelling Problem: An Updated Survey and Annotated Bibliography

Solving Distance-constrained Labeling Problems for Small Diameter Graphs via TSP*

Solving Distance-constrained Labeling Problems for Small Diameter Graphs via TSP

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Product

Resources

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Solving Distance-constrained Labeling Problems for Small Diameter Graphs via TSP^*