Parameterized Reductions and Algorithms for Another Vertex Cover Generalization

We study a novel generalization of the VERTEX COVER problem which is motivated by, e.g., error correction in the inference of chemical mixtures by their observable reaction products. We focus on the important case of deciding on one of two candidate substances. This problem has nice graph-theoretic formulations situated between VERTEX COVER and 3-HITTING: SET. In order to characterize their parameterized complexity we devise parameter-preserving reductions, and we show that some minimum solution can be computed faster than by solving 3-HITTING SET in general. More explicitly, we introduce the UNION EDITING problem: In a hypergraph with red and blue vertices, edit the colors so that the red set becomes the union of some hyperedges. The case of degree 2 is equivalent to STAR EDITING: In a graph with red and blue edges, edit the colors so that the red set becomes the union of some stars, i.e., vertices with all their incident edges.