Improved approximation bounds for the minimum rainbow subgraph problem

The Parameterized Complexity of the Rainbow Subgraph Problem

“…Our main results are as follows: RAINBOW SUBGRAPH is APX-hard even if the input graph is a properly edge-colored path with q = 2; this strengthens a previous hardness result [4]. RAINBOW SUBGRAPH is W [1]-hard on general graphs for each of the considered parameters; this rules out fixed-parameter algorithms for most natural parameters.…”

“…Here the optimization goal is to minimize the number of vertices in the solution; we refer to this problem as MINIMUM RAINBOW SUBGRAPH. MINIMUM RAINBOW SUBGRAPH is APX-hard even on graphs with maximum vertex degree ∆ ≥ 2 in which every color occurs at most twice [4]. Moreover, MINIMUM RAINBOW SUBGRAPH cannot be approximated within a factor of c ln ∆ for some constant c unless NP has slightly superpolynomial time algorithms [5].…”

“…MINIMUM RAINBOW SUBGRAPH can be approximated within a ratio of (δ + ln δ + 1)/2, where δ is the average vertex degree in the solution [6]. Katrenič and Schiermeyer [4] presented an exact algorithm for RAINBOW SUBGRAPH that has running time 2 p ·∆ 2p ·n O (1) , where n is the order of the input graph and ∆ is the maximum vertex degree of the input graph. This is the only previous fixed-parameter algorithm for MINIMUM RAINBOW SUBGRAPH that we are aware of.…”

“…Since RAINBOW SUBGRAPH is NP-hard even on collections of paths and cycles [4], we perform a broad parameterized complexity analysis. Table 1 gives an overview of the complexity of MINIMUM RAINBOW SUBGRAPH on paths, trees, and general graphs, when parameterized by…”

“…For the number of colors p, solution order k, and number of vertex deletions, the complexity seems to depend on the density of the graph as the problem is W[1]-hard for each of these parameters but it becomes tractable if any of these parameters is combined with the maximum degree ∆. Our algorithm for the parameter combination (∆, p) improves a previous algorithm for the same parameters [4]. For trees, we show a difference between the parameters p and : in this case, RAINBOW SUBGRAPH is fixed-parameter tractable for the parameters k or p, but W[1]-hard for the parameter .…”