On the Approximability of the Minimum Rainbow Subgraph Problem and Other Related Problems

Abstract

In this paper, we study the approximability of the minimum rainbow subgraph (MRS) problem and other related problems. The input to the problem is an n-vertex undirected graph, with each edge colored with one of p colors. The goal is to find a subgraph on a minimum number of vertices which has one induced edge of each color. The problem is known to be NP-hard, and has an upper bound of $O(\sqrt{n})$ and a lower bound of $\Omega (\log n)$ on its approximation ratio. We define a new problem called the densest colored k-subgraph problem, which has the same input as the MRS problem along with a parameter k. The goal is to output a subgraph on k vertices, which has the maximum number of edges of distinct colors. We give an $O(n^{1/3})$-approximation algorithm for it, and then, using that algorithm, give an $O(n^{1/3}\log n)$-approximation algorithm for the MRS problem. We observe that the Min-Rep problem (the minimization variant of the famous Label Cover problem) is indeed a special case of the MRS problem. This also implies a combinatorial $O(n^{1/3}\log n)$-approximation algorithm for the Min-Rep problem. Previously, Charikar et al. (Algorithmica 61(1):190–206, 2011) showed an ingenious LP-rounding based algorithm with an approximation ratio of $O(n^{1/3}\log ^{2/3}n)$ for Min-Rep. It is quasi-NP-hard to approximate the Min-Rep problem to within a factor of $2^{\log ^{1-\epsilon }n}$ (Kortsarz in Algorithmica 30(3): 432–450, 2001). The same hardness result now applies to the MRS problem. We also give approximation preserving reductions between various problems related to the MRS problem for which the best known approximation ratio is $O(n^c)$ where n is the size of the input and c is a fixed constant less than one.