On the parameterized complexity of some optimization problems related to multiple-interval graphs

Abstract

We show that for any constant t ≥ 2 , k - Independent Set and k - Dominating Set in t -track interval graphs are W[1]-hard. This settles an open question recently raised by Fellows, Hermelin, Rosamond, and Vialette. We also give an FPT algorithm for k - Clique in t -interval graphs, parameterized by both k and t , with running time max { t O ( k ) , 2 O ( k log k ) } ⋅ poly ( n ) , where n is the number of vertices in the graph. This slightly improves the previous FPT algorithm by Fellows, Hermelin, Rosamond, and Vialette. Finally, we use the W[1]-hardness of k - Independent Set in t -track interval graphs to obtain the first parameterized intractability result for a recent bioinformatics problem called Maximal Strip Recovery (MSR). We show that MSR- d is W[1]-hard for any constant d ≥ 4 when the parameter is either the total length of the strips, or the total number of adjacencies in the strips, or the number of strips in the solution.