We study weighted edge coloring of graphs, where we are given an undirected edge-weighted general multi-graph $G := (V, E)$ with weights $w : E \rightarrow [0, 1]$. The goal is to find a proper weighted coloring of the edges with as few colors as possible. An edge coloring is called a proper weighted coloring if the sum of the weights of the edges incident to a vertex of any color is at most one. In the online setting, the edges are revealed one by one and have to be colored irrevocably as soon as they are revealed. We show that $3.39m+o(m)$ colors are enough when the maximum number of neighbors of a vertex over all the vertices is $o(m)$ and where $m$ is the maximum over all vertices of the minimum number of unit-sized bins needed to pack the weights of the incident edges to that vertex. We also prove the tightness of our analysis. This improves upon the previous best upper bound of $5m$ by Correa and Goemans [STOC 2004]. For the offline case, we show that for a simple graph with edge disjoint cycles, $m+1$ colors are sufficient and for a multi-graph tree, we show that $1.693m+12$ colors are sufficient.
翻译:我们研究图表的加权边边色, 以未定向边缘加权一般多面G = (V, E) = (V, E) 美元, 加权值为 $w : E\ rightrow [0, 1] 美元。 目标是找到一个适当加权色色, 边色尽可能少。 如果边缘事件重量与任何颜色的顶点之和最大, 边色被称为适当加权色。 在网上设置中, 边点会被一个一个地显示出来, 并且一旦显示, 就会被不可调出色 : 美元 : E\ rightrowror [0, 1] 美元。 我们显示, 339 m+o(m)$的颜色足够。 当所有脊椎的顶端的最大边点为 $(m) 时, 边色色色被称为适当加权色 。 边色是最小的单位大小的垃圾桶重量到该顶点。 我们还证明了我们分析的紧度 。 $93