The Ordered Covering Problem

We study the Ordered Covering (OC) problem. The input is a finite set of n elements X , a color function c:X →{0,1} and a collection 𝒮 of subsets of X . A solution consists of an ordered tuple T=(S_1,… ,S_ℓ) of sets from 𝒮 which covers X , and a coloring g:{S_i}_i=1^ℓ→{0,1} such that ∀ x ∈ X , the first set covering x in the tuple, namely S_j with j=min{i : x ∈ S_i} , has color g(S_j)=c(x) . The minimization version is to find a solution using the minimum number of sets. Variants of OC include OC (α _0,α _1) in which each element of color i ∈{0,1} appears in at most α _i sets of 𝒮 , and k –OC in which the first set of the solution S_1 is required to have color 0, and there are at most k-1 alternations of colors in the solution. Among other results we show: There 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 α _0, α _1 ≥ 2 , there is a polynomial time approximation algorithm for Min–3–OC (α _0,α _1) with approximation α _1(α _0 - 1) . Unless the unique games conjecture is false, this is best possible.