The Ordered Covering Problem

We study the Ordered Covering (OC) problem. The input is a finite set of n elements X, a color function and a collection of subsets of X. A solution consists of an ordered tuple of sets from which covers X, and a coloring such that , the first set covering x in the tuple, namely with , has color . The minimization version is to find a solution using the minimum number of sets. Variants of OC include OC in which each element of color appears in at most sets of , and k-OC in which the first set of the solution is required to have color 0, and there are at most alternations of colors in the solution. Among other results we showThere is a polynomial time approximation algorithm for Min-OC(2, 2) with approximation ratio 2. (This is best possible unless Vertex Cover can be approximated within a ratio better than 2.) Moreover, Min-OC(2, 2) can be solved optimally in polynomial time if the underlying instance is bipartite. For every , there is a polynomial time approximation algorithm for Min-3-OC with approximation . Unless the unique games conjecture is false, this is best possible.