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 + ǫ)-approximation algorithm for any ǫ > 0, which is the first such approximation scheme for the densest k-subgraph problem in an important graph class without any further restrictions.

Exact and superpolynomial approximation algorithms for the densest k-subgraph problem

“…In [24], three procedures are used in order to obtain a O(n − 1 /3 )-approximation ratio, while the best known approximation algorithm achieves a ratio of O(n −(( 1 /4)+ ) ) within n O( 1 / ) time, for any > 0 [5]. A polynomial time approximation scheme (PTAS) has been presented in [2] for a class of dense graphs known as everywhere-dense graphs, while several approximation results are known for special graph classes like bipartite graphs [4], chordal graphs [36] and interval graphs [39]. Moreover, the case where k = n /2 has been extensively studied in the literature (see for example [25,31]).…”

Approximating the Sparsest k-Subgraph in Chordal Graphs

“…The main results of the paper are the N P-hardness of SPARS-EST k-SUBGRAPH in chordal graphs (Section 3), and a tight 2-approximation greedy algorithm (Section 2). Finally, we show in Section 4 how the arguments of [17] (which provides a PTAS for DkS in interval graphs) can be adapted to SkS in proper interval graphs. Notice that our N P-hardness result implies the N P-hardness of PVC in chordal graphs, which supplements the recent N P-hardness of [2,14] for PVC in bipartite graphs.…”

“…Finally, as in [17] where the author designs a P T AS for DENSEST k-SUBGRAPH on interval graph (despite the unknown complexity status), we design a PTAS for SPARSEST k-SUBGRAPH in proper interval graphs. We first assume that the instance has one connected component.…”

Approximating the Sparsest k-Subgraph in Chordal Graphs

“…[3] exact O * (1, 4 C ) [14] chordal N P-h [9] N P-h [this paper] N P-h (c.f. SkS) 3-approx [15] 2-approx [this paper] interval OPEN OPEN, OPEN PTAS [16] F P T (C) [17] F P T (n-k) (c.f. SkS) proper interval OPEN OPEN, OPEN PTAS [16] P T AS [this paper] bipartite N P-h…”