2012 IEEE 27th Conference on Computational Complexity (2011)

San Jose, California USA

June 8, 2011 to June 11, 2011

ISSN: 1093-0159

ISBN: 978-0-7695-4411-3

pp: 34-44

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/CCC.2011.10

ABSTRACT

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 \delta>0 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.

INDEX TERMS

Property testing, constraint satisfaction problems, bounded-degree model, lower bound

CITATION

Yuichi Yoshida,
"Lower Bounds on Query Complexity for Testing Bounded-Degree CSPs",

*2012 IEEE 27th Conference on Computational Complexity*, vol. 00, no. , pp. 34-44, 2011, doi:10.1109/CCC.2011.10