Approximation algorithm for maximum edge coloring

Feng, Wangsen; Zhang, Liang; Wang, Hanpin

doi:10.1016/j.tcs.2008.10.035

Cited by 14 publications

(13 citation statements)

References 11 publications

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“…Moreover, they showed that it is hard to approximate within a factor of (1+1/q) for every q ≥ 2, assuming the unique games conjecture. A simple 2-factor algorithm for maximum 2-colouring problem was reported in [7]. A description of the algorithm is provided in Algorithm 1.…”

Section: Introductionmentioning

confidence: 99%

Improved approximation for maximum edge colouring problem

Chandran

Lahiri

Singh

2022

Discrete Applied Mathematics

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The anti-Ramsey number, ar (G, H) is the minimum integer k such that in any edge colouring of G with k colours there is a rainbow subgraph isomorphic to H, namely, a copy of H with each of its edges assigned a different colour. The notion was introduced by Erdös and Simonovits in 1973. Since then the parameter has been studied extensively. The case when H is a star graph was considered by several graph theorists from the combinatorial point of view. Recently this case received the attention of researchers from the algorithm community because of its applications in interface modelling of wireless networks. To the algorithm community, the problem is known as maximum edge q-colouring problem: Find a colouring of the edges of G, maximizing the number of colours satisfying the constraint that each vertex spans at most q colours on its incident edges. It is easy to see that the maximum value of the above optimization problem equals ar(G, K 1,q+1 ) − 1.In this paper, we study the maximum edge 2-colouring problem from the approximation algorithm point of view. The case q = 2 is particularly interesting due to its application in real-life problems. Algorithmically, this problem is known to be NP-hard for q ≥ 2. For the case of q = 2, it is also known that no polynomial-time algorithm can approximate to a factor less than 3/2 assuming the unique games conjecture. Feng et al. showed a 2-approximation algorithm for this problem. Later Adamaszek and Popa presented a 5/3-approximation algorithm with the additional assumption that the input graph has a perfect matching. Note that the obvious but the only known algorithm issues different colours to the edges of a maximum matching (say M) and different colours to the connected components of G \ M. In this article, we give a new analysis of the aforementioned algorithm to show that for triangle-free graphs with perfect matching the approximation ratio is 8/5. We also show that this algorithm cannot achieve a factor better than 58/37 on triangle free graphs that has a perfect matching. The contribution of the paper is a completely new, deeper and closer analysis of how the optimum achieves a higher number of colours than the matching based algorithm, mentioned above.

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Section: Introductionmentioning

confidence: 99%

Improved approximation for maximum edge colouring problem

Chandran

Lahiri

Singh

2022

Discrete Applied Mathematics

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“…Related work. The problem of finding a maximum edge q-coloring of a given graph has been first studied by Feng et al [5,4,6]. They provide a 2-approximation algorithm for q = 2 and a (1 + 4q−2 3q 2 −5q+2 )approximation for q > 2.…”

Section: Introductionmentioning

confidence: 99%

The Min-Max Edge q-Coloring Problem

Larjomaa¹,

Popa²

2015

JGAA

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In this paper we introduce and study a new problem named min-max edge q-coloring which is motivated by applications in wireless mesh networks. The input of the problem consists of an undirected graph and an integer q. The goal is to color the edges of the graph with as many colors as possible such that: (a) any vertex is incident to at most q different colors, and (b) the maximum size of a color group (i.e. set of edges identically colored) is minimized. We show the following results:1. Min-max edge q-coloring is NP-hard, for any q ≥ 2. 2. A polynomial time exact algorithm for min-max edge q-coloring on trees.3. Exact formulas of the optimal solution for cliques and almost tight bounds for bicliques and hypergraphs. 4. A non-trivial lower bound of the optimal solution with respect to the average degree of the graph. 5. An approximation algorithm for planar graphs.

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“…We note that the flavor of this question is quite different from the classical edge coloring question, which is a minimization problem, and the constraints require a vertex to be incident to completely distinct colors. This problem definition is motivated by the problem of channel assignment in wireless networks (as pointed out in [1,10], see also [17]). The interference between the frequency channels is understood to be a bottleneck for bandwidth in wireless networks.…”

Section: Introductionmentioning

confidence: 99%

“…It is known to be NP-complete and APX-hard [1] and also admits a 2-approximation algorithm [10]. In their work on this problem, Feng et al [10] show the problem to be polynomial time for trees and complete graphs for q = 2, and Adamaszek and Popa [1] demonstrate a 5/3-approximation algorithm for graphs which have a perfect matching. Given these developments, it is natural to pursue the parameterized complexity of the problem.…”

Section: Introductionmentioning

confidence: 99%

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On the Parameterized Complexity of the Maximum Edge 2-Coloring Problem

Goyal

Kamat

Misra

2013

Mathematical Foundations of Computer Science 2013

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We investigate the parameterized complexity of the following edge coloring problem motivated by the problem of channel assignment in wireless networks. For an integer q ≥ 2 and a graph G, the goal is to find a coloring of the edges of G with the maximum number of colors such that every vertex of the graph sees at most q colors. This problem is NP-hard for q ≥ 2, and has been well-studied from the point of view of approximation. Our main focus is the case when q = 2, which is already theoretically intricate and practically relevant. We show fixed-parameter tractable algorithms for both the standard and the dual parameter, and for the latter problem, the result is based on a linear vertex kernel. I The O * notation is used to suppress polynomial factors in the running time (c.f. Section 2).

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Approximation algorithm for maximum edge coloring

Cited by 14 publications

References 11 publications

Improved approximation for maximum edge colouring problem

Improved approximation for maximum edge colouring problem

The Min-Max Edge q-Coloring Problem

On the Parameterized Complexity of the Maximum Edge 2-Coloring Problem

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