A Linear Lower Bound on the Query Complexity of Property Testing Algorithms for 3-Coloring in Bounded-degree Graphs

We consider the problem of testing 3-colorability in the bounded-degree model. A 3-colorability tester is an algorithm A that is given oracle access to the adjacency list representation of a graph G of maximum degree d with n vertices; A is required to, say, accept with probability at least 2/3 if G is 3-colorable, and to accept with probability at most 1/3 if G is -far from 3-colorable (meaning that at least an fraction of edges must be removed from G to make it 3-colorable); there is no requirement on A in the remaining cases. If A accepts 3-colorable graphs with probability one, then it is said to have one-sided error. For sufficiently small , the testing problem is NP-complete, so it is unlikely that polynomial-time, or even sub-exponential time testers exist. In this paper we are interested in unconditional lower bounds on query complexity. The strongest known lower bound is due to Goldreich and Ron, who show that, for small enough , every tester must have query complexity Ω( √ n). In this paper we show unconditionally that, for small enough , every tester for 3colorability must have query complexity Ω(n). This is the first linear lower bound for testing a natural graph property in the bounded-degree model. For one-sided error testers, we also show an Ω(n) lower bound for testers that distinguish 3-colorable graphs from graphs that are (1/3 − α)-far from 3-colorable, for arbitrarily small α. In contrast, a polynomial time algorithm by Frieze and Jerrum distinguishes 3-colorable graphs from graphs that are 1/5-far from 3-colorable. As a by-product of our techniques, we obtain tight unconditional lower bounds on the approximation ratios achievable by sub-linear time algorithms for Max E3SAT and Max E3LIN-2. ∗ [email protected]. Computer Science Division, University of California, Berkeley. † [email protected]. Computer Science Division, University of California, Berkeley. Work supported by an NSF graduate fellowship. ‡ [email protected]. Computer Science Division, University of California, Berkeley. Work supported by NSF grant CCR 9984703 and a Sloan Research Fellowship.