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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. First, for any ``symmetric'' predicate $P:{0,1}^{k} \to {0,1}$ except \equ where $k\geq 3$, 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 $ε$. 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 $P^{-1}(1) \subseteq Q^{-1}(1)$. For EQU, we give a one-sided error tester whose query complexity is $\tilde{O}(n^{1/2})$. Also, for 2-XOR (or, equivalently E2LIN2), we show an $Ω(n^{1/2+δ})$ lower bound for distinguishing instances between $ε$-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.