Optimal Approximation Algorithms for Maximum Distance-Bounded Subgraph Problems

In this paper we study the (in)approximability of two distance-based relaxed variants of the maximum clique problem (Max Clique), named Maxd-Clique and Maxd-Club: A d-clique in a graph $$G = (V, E)$$G=(V,E) is a subset $$S\subseteq V$$S⊆V of vertices such that for every pair of vertices $$u, v\in S$$u,v∈S, the distance between u and v is at most d in G. A d-club in a graph $$G = (V, E)$$G=(V,E) is a subset $$S'\subseteq V$$S′⊆V of vertices that induces a subgraph of G of diameter at most d. Given a graph G with n vertices, the goal of Maxd-Clique (Maxd-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 $$n^{1-\varepsilon }$$n1-ε for any $$\varepsilon > 0$$ε>0 unless $$\mathcal{P} = \mathcal{NP}$$P=NP. Also, in 2002, Marin$$\breve{\mathrm{c}}$$c˘ek and Mohar showed that it is $$\mathcal{NP}$$NP-hard to approximate Maxd-Club to within a factor of $$n^{1/3-\varepsilon }$$n1/3-ε for any $$\varepsilon >0$$ε>0 and any fixed $$d\ge 2$$d≥2. In this paper, we strengthen the hardness result; we prove that, for any $$\varepsilon > 0$$ε>0 and any fixed $$d\ge 2$$d≥2, it is $$\mathcal{NP}$$NP-hard to approximate Maxd-Club to within a factor of $$n^{1/2-\varepsilon }$$n1/2-ε. Then, we design a polynomial-time algorithm which achieves an optimal approximation ratio of $$O(n^{1/2})$$O(n1/2) for any integer $$d\ge 2$$d≥2. By using the similar ideas, we show the $$O(n^{1/2})$$O(n1/2)-approximation algorithm for Maxd-Clique for any $$d\ge 2$$d≥2. This is the best possible in polynomial time unless $$\mathcal{P} = \mathcal{NP}$$P=NP, as we can prove the $$\varOmega (n^{1/2-\varepsilon })$$Ω(n1/2-ε)-inapproximability.