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تسجيل مستخدم جديدImproved Approximation for Maximum Edge Colouring Problem
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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 $H$, i.e., a copy of $H$ with each of its edges assigned a different colour. The notion was introduced by Erd{{o}}s and Simonovits in 1973. Since then the parameter has been studied extensively in combinatorics, also the particular case when $H$ is a star graph. 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. 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 $qge 2$. 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 $G setminus M$. In this article, we give a new analysis of the aforementioned algorithm leading to an improved approximation bound for triangle-free graphs with perfect matching. We also show a new lower bound when the input graph is triangle-free. 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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For a graph $G$ and integer $qgeq 2$, an edge $q$-coloring of $G$ is an assignment of colors to edges of $G$, such that edges incident on a vertex span at most $q$ distinct colors. The maximum edge $q$-coloring problem seeks to maximize the number of
colors in an edge $q$-coloring of a graph $G$. The problem has been studied in combinatorics in the context of {em anti-Ramsey} numbers. Algorithmically, the problem is NP-Hard for $qgeq 2$ and assuming the unique games conjecture, it cannot be approximated in polynomial time to a factor less than $1+1/q$. The case $q=2$, is particularly relevant in practice, and has been well studied from the view point of approximation algorithms. A $2$-factor algorithm is known for general graphs, and recently a $5/3$-factor approximation bound was shown for graphs with perfect matching. The algorithm (which we refer to as the matching based algorithm) is as follows: Find a maximum matching $M$ of $G$. Give distinct colors to the edges of $M$. Let $C_1,C_2,ldots, C_t$ be the connected components that results when M is removed from G. To all edges of $C_i$ give the $(|M|+i)$th color. In this paper, we first show that the approximation guarantee of the matching based algorithm is $(1 + frac {2} {delta})$ for graphs with perfect matching and minimum degree $delta$. For $delta ge 4$, this is better than the $frac {5} {3}$ approximation guarantee proved in {AAAP}. For triangle free graphs with perfect matching, we prove that the approximation factor is $(1 + frac {1}{delta - 1})$, which is better than $5/3$ for $delta ge 3$.
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