Random k-SAT and the Power of Two Choices

We study an Achlioptas-process version of the random k-SAT process: a bounded number of k-clauses are drawn uniformly at random at each step, and exactly one added to the growing formula according to a particular rule. We prove the existence of a rule that shifts the satisfiability threshold. This extends a well-studied area of probabilistic combinatorics (Achlioptas processes) to random CSP's. In particular, while a rule to delay the 2-SAT threshold was known previously, this is the first proof of a rule to shift the threshold of k-SAT for k >= 3. We then propose a gap decision problem based upon this semi-random model. The aim of the problem is to investigate the hardness of the random k-SAT decision problem, as opposed to the problem of finding an assignment or certificate of unsatisfiability. Finally, we discuss connections to the study of Achlioptas random graph processes.Comment: 13 page

Finding heavy hitters from lossy or noisy data

Abstract. Motivated by Dvir et al. and Wigderson and Yehudayoff [3

The Power of Choice for Random Satisfiability

We consider Achlioptas processes for k-SAT formulas. We create a semi-random formula with n variables and m clauses, where each clause is a choice, made on-line, between two or more uniformly random clauses. Our goal is to delay the satisfiability/unsatisfiability transition, keeping the formula satisfiable up to densities m/n beyond the satisfiability threshold αk for random k-SAT. We show that three choices suffice to delay the transition for any k ≥ 3, and that two choices suffice for all 3 ≤ k ≤ 25. Wealso showthat two choices suffice tolower the threshold for all k ≥ 3, makingthe formula unsatisfiable at a density below αk.