Approximating Maximum Diameter-Bounded Subgraphs

The paper studies the maximum diameter-bounded subgraph problem (MaxDBS for short) which is defined as follows: Given an n-vertex graph G and a fixed integer d≥1, the goal is to find its largest subgraph of the diameter d. If d=1, the problem is identical to the maximum clique problem and thus it is ${\cal NP}$-hard to approximate MaxDBS to within a factor of n1−e for any e>0. Also, it is known to be ${\cal NP}$-hard to approximate MaxDBS to within a factor of n1/3−e for any e>0 and a fixed d≥2. In this paper, we first strengthen the hardness result; we prove that, for any e>0 and a fixed d≥2, it is ${\cal NP}$-hard to approximate MaxDBS to within a factor of n1/2−e. Then, we show that a simple polynomial-time algorithm achieves an approximation ratio of n1/2 for any even d≥2, and an approximation ratio of n2/3 for any odd d≥3. Furthermore, we investigate the (in)tractability and the (in)approximability of MaxDBS on subclasses of graphs, including chordal graphs, split graphs, interval graphs, and k-partite graphs.