arXivDaily arXiv每日学术速递 周一至周五更新
arXiv周末暂无论文更新，休息一下吧，周末愉快~~
arXiv 2008.07953cs.DM
收藏

极大边可着色子图的参数化复杂度

Parameterized Complexity of Maximum Edge Colorable Subgraph

Akanksha Agrawal, Madhumita Kundu, Abhishek Sahu, Saket Saurabh, Prafullkumar Tale

阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

一个图$H$是{ \em $p$ -edge colorable}，如果存在一个着色$ψ: E(H) \rightarrow \{1,2,\dots,p\}$，使得对于不同的$uv, vw \in E(H)$，我们有$ψ(uv) \neq ψ(vw)$。{ \sc最大边着色子图}问题以图$G$和整数$l$和$p$作为输入，并且目标是找到$G$的子图$H$和$H$的$p$ -边着色，使得$|E(H)| \geq l$。我们从参数化复杂性的角度研究了上述问题。我们获得\FPT \算法参数化时：$(1)$的顶点覆盖数的$G$，通过使用{ \sc线性规划}，和$(2)$ $l$，一个随机算法通过减少到\textsc{Rainbow Matching}，和一个确定性算法通过使用颜色编码，并划分和颜色。相对于参数$p+k$，其中$k$是以下之一：$(1)$的解决方案的大小，$l$，$(2)$的顶点覆盖数的$G$，和$(3)$ $l - {\mm}(G)$，其中${\mm}(G)$是一个最大的匹配在$G$的大小，我们表明，（决策版本的）问题承认一个内核与$\mathcal{O}(k \cdot p)$顶点。此外，我们表明，没有大小为$\mathcal{O}(k^{1-ε} \cdot f(p))$的内核，对于任何$ε> 0$和可计算函数$f$，除非$\NP \subseteq \CONPpoly$。

英文摘要：

A graph $H$ is {\em $p$-edge colorable} if there is a coloring $ψ: E(H) \rightarrow \{1,2,\dots,p\}$, such that for distinct $uv, vw \in E(H)$, we have $ψ(uv) \neq ψ(vw)$. The {\sc Maximum Edge-Colorable Subgraph} problem takes as input a graph $G$ and integers $l$ and $p$, and the objective is to find a subgraph $H$ of $G$ and a $p$-edge-coloring of $H$, such that $|E(H)| \geq l$. We study the above problem from the viewpoint of Parameterized Complexity. We obtain \FPT\ algorithms when parameterized by: $(1)$ the vertex cover number of $G$, by using {\sc Integer Linear Programming}, and $(2)$ $l$, a randomized algorithm via a reduction to \textsc{Rainbow Matching}, and a deterministic algorithm by using color coding, and divide and color. With respect to the parameters $p+k$, where $k$ is one of the following: $(1)$ the solution size, $l$, $(2)$ the vertex cover number of $G$, and $(3)$ $l - {\mm}(G)$, where ${\mm}(G)$ is the size of a maximum matching in $G$; we show that the (decision version of the) problem admits a kernel with $\mathcal{O}(k \cdot p)$ vertices. Furthermore, we show that there is no kernel of size $\mathcal{O}(k^{1-ε} \cdot f(p))$, for any $ε> 0$ and computable function $f$, unless $\NP \subseteq \CONPpoly$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑