PTAS for Densest k-Subgraph in Interval Graphs

Abstract

Given an interval graph and integer k, we consider the problem of finding a subgraph of size k with a maximum number of induced edges, called densest k-subgraph problem in interval graphs. It has been shown that this problem is NP-hard even for chordal graphs [17], and there is probably no PTAS for general graphs [12]. However, the exact complexity status for interval graphs is a long-standing open problem [17], and the best known approximation result is a 3-approximation algorithm [16]. We shed light on the approximation complexity of finding a densest k-subgraph in interval graphs by presenting a polynomialtime approximation scheme (PTAS), that is, we show that there is an (1+e)-approximation algorithm for any e > 0, which is the first such approximation scheme for the densest k-subgraph problem in an important graph class without any further restrictions.