Layered Graphs: Applications and Algorithms

The computation of distances between strings has applications in molecular biology, music theory and pattern recognition. One such measure, called short reversal distance, has applications in evolutionary distance computation. It has been shown that this problem can be reduced to the computation of a maximum independent set on the corresponding graph that is constructed from the given input strings. The constructed graphs primarily fall into a class that we call layered graphs. In a layered graph, each layer refers to a subgraph containing, at most, some k vertices. The inter-layer edges are restricted to the vertices in adjacent layers. We study the MIS, MVC, MDS, MCV and MCD problems on layered graphs where MIS computes the maximum independent set; MVC computes the minimum vertex cover; MDS computes the minimum dominating set; MCV computes the minimum connected vertex cover; and MCD computes the minimum connected dominating set. The MIS, MVC and MDS are computed in polynomial time if k = Θ(log |V|). MCV and MCD run in polynomial time if k=O((log |V|) ^α) where a < 1. If k = Θ((log |V|)^{1+e}), for e > 0, then MIS, MVC and MDS run in quasi-polynomial time. If k = Θ(log |V|), then MCV and MCD run in quasi-polynomial time.