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Register a new userImproved Approximation for Maximum Edge Colouring Problem
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Abhiruk Lahiri
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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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 $delta ge 3$.
We introduce and investigate the approximability of the maximum binary tree problem (MBT) in directed and undirected graphs. The goal in MBT is to find a maximum-sized binary tree in a given graph. MBT is a natural variant of the well-studied longest path problem, since both can be viewed as finding a maximum-sized tree of bounded degree in a given graph. The connection to longest path motivates the study of MBT in directed acyclic graphs (DAGs), since the longest path problem is solvable efficiently in DAGs. In contrast, we show that MBT in DAGs is in fact hard: it has no efficient $exp(-O(log n/ log log n))$-approximation algorithm under the exponential time hypothesis, where $n$ is the number of vertices in the input graph. In undirected graphs, we show that MBT has no efficient $exp(-O(log^{0.63}{n}))$-approximation under the exponential time hypothesis. Our inapproximability results rely on self-improving reductions and structural properties of binary trees. We also show constant-factor inapproximability assuming $text{P} eq text{NP}$. In addition to inapproximability results, we present algorithmic results along two different flavors: (1) We design a randomized algorithm to verify if a given directed graph on $n$ vertices contains a binary tree of size $k$ in $2^k text{poly}(n)$ time. (2) Motivated by the longest heapable subsequence problem, introduced by Byers, Heeringa, Mitzenmacher, and Zervas (ANALCO 2011), which is equivalent to MBT in permutation DAGs, we design efficient algorithms for MBT in bipartite permutation graphs.
In the W-streaming model, an algorithm is given $O(n mathrm{polylog} n)$ space and must process a large graph of up to $O(n^2)$ edges. In this short note we give two algorithms for edge colouring under the W-streaming model. For edge colouring in W-streaming, a colour for every edge must be determined by the time all the edges are streamed. Our first algorithm uses $Delta + o(Delta)$ colours in $O(n log^2 n)$ space when the edges arrive according to a uniformly random permutation. The second algorithm uses $(1 + o(1))Delta^2 / s$ colours in $tilde{O}(n s)$ space when edges arrival adversarially.
Threshold graphs are a class of graphs that have many equivalent definitions and have applications in integer programming and set packing problems. A graph is said to have a threshold cover of size $k$ if its edges can be covered using $k$ threshold graphs. Chvatal and Hammer, in 1977, defined the threshold dimension $mathrm{th}(G)$ of a graph $G$ to be the least integer $k$ such that $G$ has a threshold cover of size $k$ and observed that $mathrm{th}(G)geqchi(G^*)$, where $G^*$ is a suitably constructed auxiliary graph. Raschle and Simon~[Proceedings of the Twenty-seventh Annual ACM Symposium on Theory of Computing, STOC 95, pages 650--661, 1995] proved that $mathrm{th}(G)=chi(G^*)$ whenever $G^*$ is bipartite. We show how the lexicographic method of Hell and Huang can be used to obtain a completely new and, we believe, simpler proof for this result. For the case when $G$ is a split graph, our method yields a proof that is much shorter than the ones known in the literature.
We present improved results for approximating maximum-weight independent set ($MaxIS$) in the CONGEST and LOCAL models of distributed computing. Given an input graph, let $n$ and $Delta$ be the number of nodes and maximum degree, respectively, and let $MIS(n,Delta)$ be the the running time of finding a emph{maximal} independent set ($MIS$) in the CONGEST model. Bar-Yehuda et al. [PODC 2017] showed that there is an algorithm in the CONGEST model that finds a $Delta$-approximation for $MaxIS$ in $O(MIS(n,Delta)log W)$ rounds, where $W$ is the maximum weight of a node in the graph, which can be as high as $poly (n)$. Whether their algorithm is deterministic or randomized depends on the $MIS$ algorithm that is used as a black-box. Our main result in this work is a randomized $(poly(loglog n)/epsilon)$-round algorithm that finds, with high probability, a $(1+epsilon)Delta$-approximation for $MaxIS$ in the CONGEST model. That is, by sacrificing only a tiny fraction of the approximation guarantee, we achieve an emph{exponential} speed-up in the running time over the previous best known result. Due to a lower bound of $Omega(sqrt{log n/log log n})$ that was given by Kuhn, Moscibroda and Wattenhofer [JACM, 2016] on the number of rounds for any (possibly randomized) algorithm that finds a maximal independent set (even in the LOCAL model) this result implies that finding a $(1+epsilon)Delta$-approximation for $MaxIS$ is exponentially easier than $MIS$.
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