PTAS for Densest -Subgraph in Interval Graphs

Given an interval graph and integer , we consider the problem of finding a subgraph of size with a maximum number of induced edges, called densest k -subgraph problem in interval graphs. This problem is NP-hard even for chordal graphs (Perl and Corneil in Discret Appl Math 9(1):27-39, 1984), and there is probably no PTAS for general graphs (Khot and Subhash in SIAM J Comput 36(4):1025-1071, 2006). However, the exact complexity status for interval graphs is a long-standing open problem (Perl and Corneil in Discret Appl Math 9(1):27-39, 1984), and the best known approximation result is a -approximation algorithm (Liazi et al. in Inf Process Lett 108(1):29-32, 2008). We shed light on the approximation complexity of finding a densest -subgraph in interval graphs by presenting a polynomial-time approximation scheme (PTAS), that is, we show that there is an -approximation algorithm for any , which is the first such approximation scheme for the densest -subgraph problem in an important graph class without any further restrictions.