Optimal Approximation Algorithms for Maximum Distance-Bounded Subgraph Problems

Abstract

A d-clique in a graph $G = (V, E)$ is a subset $S\subseteq V$ of vertices such that for pairs of vertices $u, v\in S$, the distance between u and v is at most d in G. A d-club in a graph $G = (V, E)$ is a subset $S'\subseteq V$ of vertices that induces a subgraph of G of diameter at most d. Given a graph G with n vertices, the goal of Max d-Clique (Max d-Club, resp.) is to find a d-clique (d-club, resp.) of maximum cardinality in G. Max 1-Clique and Max 1-Club cannot be efficiently approximated within a factor of $n^{1-\varepsilon }$ for any $\varepsilon > 0$ unless $\mathcal{P} = \mathcal{NP}$ since they are identical to Max Clique [14, 21]. Also, it is known [3] that it is $\mathcal{NP}$-hard to approximate Max d-Club to within a factor of $n^{1/2 - \varepsilon }$ for any fixed $d\ge 2$ and for any $\varepsilon > 0$. As for approximability of Max d-Club, there exists a polynomial-time algorithm which achieves an optimal approximation ratio of $O(n^{1/2})$ for any even $d\ge 2$ [3]. For any odd $d\ge 3$, however, there still remains a gap between the $O(n^{2/3})$-approximability and the $\varOmega (n^{1/2-\varepsilon })$-inapproximability for Max d-Club [3]. In this paper, we first strengthen the approximability result for Max d-Club; we design a polynomial-time algorithm which achieves an optimal approximation ratio of $O(n^{1/2})$ for Max d-Club for any odd $d\ge 3$. Then, by using the similar ideas, we show the $O(n^{1/2})$-approximation algorithm for Max d-Clique on general graphs for any $d\ge 2$. This is the best possible in polynomial time unless $\mathcal{P} = \mathcal{NP}$, as we can prove the $\varOmega (n^{1/2-\varepsilon })$-inapproximability. Furthermore, we study the tractability of Max d-Clique and Max d-Club on subclasses of graphs.