Proceedings of the forty-third annual ACM symposium on Theory of computing
Testing the (s,t)-disconnectivity of graphs and digraphs
Theoretical Computer Science
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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:\bit^{k}\to \bit 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-\epsilon)-far from satisfiability requires \Omega(n^{1/2+\delta}) queries where n is the number of variables and \delta0 is a constant that depends on P and \epsilon. This breaks a natural lower bound \Omega(n^{1/2}), which is obtained by the birthday paradox. We also show that every one-sided error tester requires \Omega(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 \txor (or, equivalently \textsf{E2LIN2}), we show an \Omega(n^{1/2+\delta}) lower bound for distinguishing instances between \epsilon-close to and (1/2-\epsilon)-far from satisfiability. Next, for the general \kcsp over the binary domain, we show that every algorithm that distinguishes satisfiable instances from instances (1-2k/2^k-\epsilon)-far from satisfiability requires \Omega(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 \mislong 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 \epsilon n requires \Omega(n) queries. Previously, only super-constant lower bounds were known.