Parameterized Complexity of Maximum Edge Colorable Subgraph

A graph $H$ is {\em $p$-edge colorable} if there is a coloring $\psi: E(H) \rightarrow \{1,2,\dots,p\}$, such that for distinct $uv, vw \in E(H)$, we have $\psi(uv) \neq \psi(vw)$. 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 $\mathcal{O}(k^{1-\epsilon} \cdot f(p))$, for any $\epsilon > 0$ and computable function $f$, unless \NP \subseteq \CONPpoly

Maximum Edge-Colorable Subgraph and Strong Triadic Closure Parameterized by Distance to Low-Degree Graphs

Given an undirected graph $G$ and integers $c$ and $k$, the Maximum Edge-Colorable Subgraph problem asks whether we can delete at most $k$ edges in $G$ to obtain a graph that has a proper edge coloring with at most $c$ colors. We show that Maximum Edge-Colorable Subgraph admits, for every fixed $c$, a linear-size problem kernel when parameterized by the edge deletion distance of $G$ to a graph with maximum degree $c-1$. This parameterization measures the distance to instances that, due to Vizing's famous theorem, are trivial yes-instances. For $c\le 4$, we also provide a linear-size kernel for the same parameterization for Multi Strong Triadic Closure, a related edge coloring problem with applications in social network analysis. We provide further results for Maximum Edge-Colorable Subgraph parameterized by the vertex deletion distance to graphs where every component has order at most $c$ and for the list-colored versions of both problems.Comment: 32 Page

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum