Parameterized Complexity of Maximum Edge Colorable Subgraph.

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 NP \(\subseteq \) coNP/poly.