On the Approximability of Presidential Type Predicates

Given a predicate P: {-1, 1}^k ? {-1, 1}, let CSP(P) be the set of constraint satisfaction problems whose constraints are of the form P. We say that P is approximable if given a nearly satisfiable instance of CSP(P), there exists a probabilistic polynomial time algorithm that does better than a random assignment. Otherwise, we say that P is approximation resistant. In this paper, we analyze presidential type predicates, which are balanced linear threshold functions where all of the variables except the first variable (the president) have the same weight. We show that almost all presidential type predicates P are approximable. More precisely, we prove the following result: for any ?? > 0, there exists a k? such that if k ? k?, ? ? (??,1 - 2/k], and {?}k + k - 1 is an odd integer then the presidential type predicate P(x) = sign({?}k{x?} + ?_{i = 2}^{k} {x_i}) is approximable. To prove this, we construct a rounding scheme that makes use of biases and pairwise biases. We also give evidence that using pairwise biases is necessary for such rounding schemes

On the Decision Tree Complexity of String Matching

String matching is one of the most fundamental problems in computer science. A natural problem is to determine the number of characters that need to be queried (i.e. the decision tree complexity) in a string in order to decide whether this string contains a certain pattern. Rivest showed that for every pattern p, in the worst case any deterministic algorithm needs to query at least n-|p|+1 characters, where n is the length of the string and |p| is the length of the pattern. He further conjectured that this bound is tight. By using the adversary method, Tuza disproved this conjecture and showed that more than one half of binary patterns are evasive, i.e. any algorithm needs to query all the characters (see Section 1.1 for more details). In this paper, we give a query algorithm which settles the decision tree complexity of string matching except for a negligible fraction of patterns. Our algorithm shows that Tuza\u27s criteria of evasive patterns are almost complete. Using the algebraic approach of Rivest and Vuillemin, we also give a new sufficient condition for the evasiveness of patterns, which is beyond Tuza\u27s criteria. In addition, our result reveals an interesting connection to Skolem\u27s Problem in mathematics