On the Structure and the Number of Prime Implicants of 2-CNFs

Let $m(n, k)$ be the maximum number of prime implicants that any $k$-CNF on n variables can have. We show that $3^{n/3} \le m(n,2) \le (1+o(1))3^{n/3}$

Breaking the PPSZ Barrier for Unique 3-SAT

The PPSZ algorithm by Paturi, Pudl\'ak, Saks, and Zane (FOCS 1998) is the fastest known algorithm for (Promise) Unique k-SAT. We give an improved algorithm with exponentially faster bounds for Unique 3-SAT. For uniquely satisfiable 3-CNF formulas, we do the following case distinction: We call a clause critical if exactly one literal is satisfied by the unique satisfying assignment. If a formula has many critical clauses, we observe that PPSZ by itself is already faster. If there are only few clauses allover, we use an algorithm by Wahlstr\"om (ESA 2005) that is faster than PPSZ in this case. Otherwise we have a formula with few critical and many non-critical clauses. Non-critical clauses have at least two literals satisfied; we show how to exploit this to improve PPSZ.Comment: 13 pages; major revision with simplified algorithm but slightly worse constant

A Randomized Algorithm for 3-SAT

In this work we propose and analyze a simple randomized algorithm to find a satisfiable assignment for a Boolean formula in conjunctive normal form (CNF) having at most 3 literals in every clause. Given a k-CNF formula phi on n variables, and alpha in{0,1}^n that satisfies phi, a clause of phi is critical if exactly one literal of that clause is satisfied under assignment alpha. Paturi et. al. (Chicago Journal of Theoretical Computer Science 1999) proposed a simple randomized algorithm (PPZ) for k-SAT for which success probability increases with the number of critical clauses (with respect to a fixed satisfiable solution of the input formula). Here, we first describe another simple randomized algorithm DEL which performs better if the number of critical clauses are less (with respect to a fixed satisfiable solution of the input formula). Subsequently, we combine these two simple algorithms such that the success probability of the combined algorithm is maximum of the success probabilities of PPZ and DEL on every input instance. We show that when the average number of clauses per variable that appear as unique true literal in one or more critical clauses in phi is between 1 and 1.9317, combined algorithm performs better than the PPZ algorithm