On the Minimum Load Coloring Problem

Ahuja, Nitin K.; Baltz, Andreas; Doerr, Benjamin; Přívětivý, Aleš; Srivastav, Anand

doi:10.1007/11671411_2

Cited by 4 publications

(18 citation statements)

References 11 publications

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“…We will denote this minimum by λ(G) and call this problem Load Coloring Problem (LCP). The LCP arises in Wavelength Division Multiplexing, the technology used for constructing optical communication networks [1,9]. Ahuja et al [1] proved that the problem is NP-hard and gave a polynomial time algorithm for optimal colorings of trees.…”

Section: Introductionmentioning

confidence: 99%

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Parameterized algorithms for load coloring problem

Gutin

Jones

2014

Information Processing Letters

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One way to state the Load Coloring Problem (LCP) is as follows. Let G = (V, E) be graph and let f : V → {red, blue} be a 2-coloring. An edge e ∈ E is called red (blue) if both end-vertices of e are red (blue). For a 2-coloring f , let r ′ f and b ′ f be the number of red and blue edges and let µ f (G) = min{rWe introduce the parameterized problem k-LCP of deciding whether µ(G) ≥ k, where k is the parameter. We prove that this problem admits a kernel with at most 7k. Ahuja et al. (2007) proved that one can find an optimal 2-coloring on trees in polynomial time. We generalize this by showing that an optimal 2-coloring on graphs with tree decomposition of width t can be found in time O * (2 t ). We also show that either G is a Yes-instance of k-LCP or the treewidth of G is at most 2k. Thus, k-LCP can be solved in time O * (4 k ).

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Section: Introductionmentioning

confidence: 99%

“…We prove that this problem admits a kernel with at most 7k. Ahuja et al (2007) proved that one can find an optimal 2-coloring on trees in polynomial time. We generalize this by showing that an optimal 2-coloring on graphs with tree decomposition of width t can be found in time O * (2 t ).…”

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confidence: 99%

Parameterized algorithms for load coloring problem

Gutin

Jones

2014

Information Processing Letters

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“…This problem is NP-complete [1], and Gutin and Jones studied its parameterization by k [9]. They proved that 2-Load Coloring is fixed-parameter tractable (FPT) 1 by obtaining a kernel with at most 7k vertices.…”

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confidence: 99%

“…6. Graphs Following [1,9], we consider graphs without loops or multiple edges. (Actually, our results generalize to graphs with loops and multiple edges, see Sect.…”

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Parameterized and Approximation Algorithms for the Load Coloring Problem

et al. 2016

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Let c, k be two positive integers. Given a graph G = (V, E), the c-Load Coloring problem asks whether there is a c-coloring ϕ : V → [c] such that for every i ∈ [c], there are at least k edges with both endvertices colored i. Gutin and Jones (Inf Process Lett 114:446-449, 2014) studied this problem with c = 2. They showed 2-Load Coloring to be fixed-parameter tractable (FPT) with parameter k by obtaining a kernel with at most 7k vertices. In this paper, we extend the study to any fixed c by giving both a linear-vertex and a linear-edge kernel. In the particular case of c = 2, we obtain a kernel with less than 4k vertices and less than 6k + (3 + √ 2) √ k + 4 edges. These results imply that for any fixed c ≥ 2, c-Load Coloring is FPT and the optimization version of c-Load Coloring (where k is to be maximized) has an approximation algorithm with a constant ratio.

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