Optimal Approximation Algorithms for Maximum Distance-Bounded Subgraph Problems

Abstract

In this paper we study the (in)approximability of two distance-based relaxed variants of the maximum clique problem (Max Clique), named Max d-Clique and Max d-Club: A d-clique in a graph $$G = (V, E)$$ is a subset $$S\subseteq V$$ of vertices such that for every pair 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. Since Max 1-Clique and Max 1-Club are identical to Max Clique, the inapproximabilty for Max Clique shown by Zuckerman in 2007 is transferred to them. Namely, Max 1-Clique and Max 1-Club cannot be efficiently approximated within a factor of $$n^{1-\varepsilon }$$ for any $$\varepsilon > 0$$ unless $$\mathcal{P} = \mathcal{NP}$$ . Also, in 2002, Marin $$\breve{\mathrm{c}}$$ ek and Mohar showed that it is $$\mathcal{NP}$$ -hard to approximate Max d-Club to within a factor of $$n^{1/3-\varepsilon }$$ for any $$\varepsilon >0$$ and any fixed $$d\ge 2$$ . In this paper, we strengthen the hardness result; we prove that, for any $$\varepsilon > 0$$ and any fixed $$d\ge 2$$ , it is $$\mathcal{NP}$$ -hard to approximate Max d-Club to within a factor of $$n^{1/2-\varepsilon }$$ . Then, we design a polynomial-time algorithm which achieves an optimal approximation ratio of $$O(n^{1/2})$$ for any integer $$d\ge 2$$ . By using the similar ideas, we show the $$O(n^{1/2})$$ -approximation algorithm for Max d-Clique 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.