Parameterized Complexity of Maximum Edge Colorable Subgraph

Abstract

A graph H is 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 $\psi (uv) \ne \psi (vw)$. The 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 $|E(H)| \ge l$. 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 Integer Linear Programming, and (2) l, a randomized algorithm via a reduction to 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) $l - {\texttt {mm}}(G)$, where ${\texttt {mm}}(G)$ 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 ${\textsf {NP}}\subseteq \textsf {coNP/poly}$.