Parameterized Complexity of Maximum Edge Colorable Subgraph

A graph H is p-edge colorable if there is a coloring ψ: E(H) → { 1 , 2 , ⋯ , p} , such that for distinct uv, vw∈ E(H) , we have ψ(uv) ≠ ψ(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) | ≥ 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- 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 O(k· p) vertices. Furthermore, we show that there is no kernel of size O(k^1-ϵ· f(p)) , for any ϵ> 0 and computable function f, unless NP⊆ coNP/ poly.