Almost Optimal Canonical Property Testers for Satisfiability

In the (k, d)-Function-SAT problem we are given a set of n variables {X1, ... , Xn} that can take values from the set {1, . .. , d} and a set of Boolean constraints on these variables, where each constraint is of the form f : {1, ... , d}k → {0, 1}, i.e. the constraint depends on exactly k of these variables. We will treat k and d as constants. The goal is to determine whether the set of constraints has a satisfying assignment, i.e. an assignment to the variables such that all constraints simultanuously map to 1. In this paper, we study (k, d)-Function-SAT in the property testing model for dense instances. We call an instance ε-far from satisfiable, if every assignment violates more than εnk constraints. A property testing algorithm is a randomized algorithm that, given oracle access to the set of constraints, must accept with probability at least 3/4 all satisfiable inputs and rejects with probability at least 3/4 all inputs, which are ε-far from satisfiable. We analyze the canonical non-adaptive property testing algorithm with one-sided error: Sample r variables and accept, if and only if the induced set of constraints has a satisfying assignment. The value of r will be called the sample commlexity of the algorithm. We show that there is an r0 = O(1/ε) such that for any instance that is ε-far from satisfiable, the probability, that a random sample on r ≥ r0 variables is satisfiable, is at most 1/4. This implies that the above algorithm is a property tester. The obtained sample complexity is nearly optimal for canonical testers as a lower bound of Ω(1/ε) on the sample complexity is known. Previously, a tester with sample complexity o(1/ε2) was only known for the very special case of testing bipartiteness in the dense graph model [3]. Our new general result improves the best previous result for testing satisfiability (and even for the special case of 3-colorability in graphs) from sample complexity Õ(1/ε2) to Õ(1/ε). It also slightly improves the sample complexity for the special case of bipartiteness. Improving the sample complexity for (k, d)-Function-SAT (or special cases of it) had been posed in several papers as an open problem [3], [4], [17]. This paper solves this problem nearly optimally for canonical testers and, in the case of k = 2, also for nonadaptive testers as there is a lower bound of Ω(1/ε2) on the query complexity of non-adaptive testers for bipartiteness in the dense graph model [6], where the query complexity denotes the number of queries asked about the graph (for a canonical tester in graphs, the query complexity is the square of its sample complexity). As a byproduct, we obtain an algorithm, which, given a satisfiable set of constraints, computes in time O(n/εO(1) + 2Õ(1/ε)) a solution, which violates at most εnk constraints.