Lower Bounds on Query Complexity for Testing Bounded-Degree CSPs

In this paper, we consider lower bounds on the query complexity for testing CSPs in the bounded-degree model. We mainly consider Boolean CSPs allowing literals. First, for any "symmetric" predicate P : {0, 1}^k → {0,1} except EQU where k ≥ 3, we show that every (randomized) algorithm that distinguishes satisfiable instances of CSP(P) from instances (|P^-1 (0)|/2^k - ϵ)-far from satisfiability requires Ω(n^1/2+δ) queries where n is the number of variables and δ >; 0 is a constant that depends on P and e. This breaks a natural lower bound Ω(n^1/2), which is obtained by the birthday paradox. We also show that every one-sided error tester requires Ω(n) queries for such P. These results are hereditary in the sense that the same results hold for any predicate Q such that P^-1 (1) ⊆ Q^-1(1). For EQU, we give a one-sided error tester whose query complexity is O̅(n^1/2). Also, for 2-XOR (or, equivalently E2LIN2), we show an Ω(n^1/2+δ) lower bound for distinguishing instances between e-close to and (1/2 -ϵ)-far from satisfiability. Next, for the general k-CSP over the binary domain, we show that every algorithm that distinguishes satisfiable instances from instances (1 - 2k/2^k - ϵ)-far from satisfiability requires Ω(n) queries. The matching NP-hardness is not known, even assuming the Unique Games Conjecture or the d-to-1 Conjecture. As a corollary, for Maximum Independent Set on graphs with n vertices and a degree bound d, we show that every approximation algorithm within a factor d/poly log d and an additive error of ϵn requires Ω(n) queries. Previously, only super-constant lower bounds were known.