Optimal Approximation Algorithms for Maximum Distance-Bounded Subgraph Problems

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 pairs 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 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 $$n^{1-\varepsilon }$$n1-ε for any $$\varepsilon > 0$$ε>0 unless $$\mathcal{P} = \mathcal{NP}$$P=NP since they are identical to Max Cliqueï ź[14, 21]. Also, it is knownï ź[3] that it is $$\mathcal{NP}$$NP-hard to approximate Max d-Club to within a factor of $$n^{1/2 - \varepsilon }$$n1/2-ε for any fixed $$d\ge 2$$dï ź2 and for any $$\varepsilon > 0$$ε>0. As for approximability of Max d-Club, there exists a polynomial-time algorithm which achieves an optimal approximation ratio of $$On^{1/2}$$On1/2 for any even $$d\ge 2$$dï ź2ï ź[3]. For any odd $$d\ge 3$$dï ź3, however, there still remains a gap between the $$On^{2/3}$$On2/3-approximability and the $$\varOmega n^{1/2-\varepsilon }$$Ωn1/2-ε-inapproximability for Max d-Clubï ź[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 $$On^{1/2}$$On1/2 for Max d-Club for any odd $$d\ge 3$$dï ź3. Then, by using the similar ideas, we show the $$On^{1/2}$$On1/2-approximation algorithm for Max d-Clique on general graphs for any $$d\ge 2$$dï ź2. This is the best possible in polynomial time unless $$\mathcal{P} = \mathcal{NP}$$P=NP, as we can prove the $$\varOmega n^{1/2-\varepsilon }$$Ωn1/2-ε-inapproximability. Furthermore, we study the tractability of Max d-Clique and Max d-Club on subclasses of graphs.