Conditional hardness for satisfiable 3-CSPs

In this paper we study a fundamental open problem in the area of probabilistic checkable proofs: What is the smallest s such that NP naPCP1,s[O(log n), 3]? In the language of hardness of approximation, this problem is equivalent to determining the smallest s such that getting an s-approximation for satisfiable 3-bit constraint satisfaction problems ("3-CSPs") is NP-hard. The previous best upper bound and lower bound for s are 20/27 + by Khot and Saket [19] and 5/8 by Zwick [29]. In this paper we close the gap assuming Khot's d-to-1 Conjecture [15]. Formally, we prove that if Khot's d-to-1 Conjecture holds for any finite constant integer d, then NP naPCP1,5/8+[O(log n), 3] for any constant > 0. Our conditional result also solves H?astad's open question [12] on determining the inapproximability of satisfiable Max-NTW ("Not Two") instances and confirms Zwick's conjecture [29] that the 5/8-approximation algorithm for satisfiable 3-CSPs is opt...

• Type: Conference
• Year: 2009
• Where: STOC
• Authors: Ryan O'Donnell, Yi Wu
• Keywords: Algorithms, Finite Constant Integer, Fundamental Open Problem, Probabilistic Checkable Proofs, STOC 2009