Optimal Approximation Algorithms for Maximum Distance-Bounded Subgraph Problems

• Published in: Algorithmica Vol. 80; no. 6; pp. 1834 - 1856
• Main Authors: Asahiro, Yuichi, Doi, Yuya, Miyano, Eiji, Samizo, Kazuaki, Shimizu, Hirotaka
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2018; Springer Nature B.V
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Approximation; Computer Science; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Decision trees; Graph theory; Graphs; Mathematical analysis; Mathematics of Computing; Polynomials; Set theory; Theory of Computation; club; clique; Maximum distance-bounded subgraph problems; (In)approximability
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-017-0344-y

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 ⊆ V of vertices such that for every pair of vertices u , v ∈ 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 ′ ⊆ 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 - ε for any ε > 0 unless P = NP . Also, in 2002, Marin c ˘ ek and Mohar showed that it is NP -hard to approximate Max d - Club to within a factor of n 1 / 3 - ε for any ε > 0 and any fixed d ≥ 2 . In this paper, we strengthen the hardness result; we prove that, for any ε > 0 and any fixed d ≥ 2 , it is NP -hard to approximate Max d - Club to within a factor of n 1 / 2 - ε . Then, we design a polynomial-time algorithm which achieves an optimal approximation ratio of O ( n 1 / 2 ) for any integer d ≥ 2 . By using the similar ideas, we show the O ( n 1 / 2 ) -approximation algorithm for Max d - Clique for any d ≥ 2 . This is the best possible in polynomial time unless P = NP , as we can prove the Ω ( n 1 / 2 - ε ) -inapproximability.