Algorithmic approaches for the minimum rainbow subgraph problem

We consider the Minimum Rainbow Subgraph problem (MRS): Given a graph G, whose edges are coloured with p colours. Find a subgraph F ⊆ G of minimum order and with p edges such that each colour occurs exactly once. This problem is APX-hard. In this paper we will show that the Greedy algorithm for the MRS problem has an approximation ratio of Δ 2 + ln Δ + 1 2 for graphs with maximum degree Δ. If the average degree d of a minimum rainbow subgraph is known, then the approximation ratio is d 2 + ln ⌈ d ⌉ + 1 2 .