Edward A. Hirsch

New Worst-Case Upper Bounds for SAT

In 1980 Monien and Speckenmeyer proved that satisfiability of a propositional formula consisting of K clauses (of arbitrary length) can be checked in time of the order 2K / 3. Recently Kullmann and Luckhardt proved the worst-case upper bound 2L / 9, where L is the length of the input formula. The algorithms leading to these bounds are based on the splitting method, which goes back to the Davis–Putnam procedure. Transformation rules (pure literal elimination, unit propagation, etc.) constitute a substantial part of this method. In this paper we present a new transformation rule and two algorithms using this rule. We prove that these algorithms have the worst-case upper bounds 20. 30897 K and 20. 10299 L, respectively.

Exponential Lower Bounds for the Running Time of DPLL Algorithms on Satisfiable Formulas

DPLL (for Davis, Putnam, Logemann, and Loveland) algorithms form the largest family of contemporary algorithms for SAT (the propositional satisfiability problem) and are widely used in applications. The recursion trees of DPLL algorithm executions on unsatisfiable formulas are equivalent to treelike resolution proofs. Therefore, lower bounds for treelike resolution (known since the 1960s) apply to them. However, these lower bounds say nothing about the behavior of such algorithms on satisfiable formulas. Proving exponential lower bounds for them in the most general setting is impossible without proving P ≠ NP; therefore, to prove lower bounds, one has to restrict the power of branching heuristics. In this paper, we give exponential lower bounds for two families of DPLL algorithms: generalized myopic algorithms, which read up to n1−ε of clauses at each step and see the remaining part of the formula without negations, and drunk algorithms, which choose a variable using any complicated rule and then pick its value at random.

Complexity of Semi-algebraic Proofs

Proof systems for polynomial inequalities in 0-1 variables include the well-studied Cutting Planes proof system (CP) and the Lovasz-Schrijver calculi (LS) utilizing linear, respectively, quadratic, inequalities. We introduce generalizations LSd of LS involving polynomial inequalities of degree at most d.Surprisingly, the systems LSd turn out to be very strong. We construct polynomial-size bounded degree LSd proofs of the clique-coloring tautologies (which have no polynomial-size CP proofs), the symmetric knapsack problem (which has no bounded degree Positivstellensatz Calculus (PC) proofs), and Tseitins tautologies (hard for many known proof systems). Extending our systems with a division rule yields a polynomial simulation of CP with polynomially bounded coefficients, while other extra rules further reduce the proof degrees for the aforementioned examples.Finally, we prove lower bounds on Lovasz-Schrijver ranks, demonstrating, in particular, their rather limited applicability for proof complexity.

A New Algorithm for MAX-2-SAT

Recently there was a significant progress in proving (exponential-time) worst-case upper bounds for the propositional satisfiability problem (SAT) and related problems. In particular, for MAX-2-SAT Niedermeier and Rossmanith recently presented an algorithm with worstcase upper bound O(K ċ2K/2.88...), and the bound O(K ċ2K/3.44...) is implicit from the paper by Bansal and Raman (K is the number of clauses). In this paper we improve this bound to p(K)2K2/4, where K2 is the number of 2-clauses, and p is a polynomial. In addition, our algorithm and the proof are much simpler than the previous ones. The key ideas are to use the symmetric flow algorithm of Yannakakis and to count only 2-clauses (and not 1-clauses).

SAT Local Search Algorithms: Worst-Case Study

Recent experiments demonstrated that local search algorithms (e.g. GSAT) are able to find satisfying assignments for many “hard” Boolean formulas. A wide experimental study of these algorithms demonstrated their good performance on some inportant classes of formulas as well as poor performance on some other ones. In contrast, theoretical knowledge of their worst-case behavior is very limited. However, many worst-case upper and lower bounds of the form 2α n (α<1 is a constant) are known for other SAT algorithms, for example, resolution-like algorithms. In the present paper we prove both upper and lower bounds of this form for local search algorithms. The class of linear-size formulas we consider for the upper bound covers most of the DIMACS benchmarks; the satisfiability problem for this class of formulas is NP-complete.

Max sat approximation beyond the limits of polynomial-time approximation

We describe approximation algorithms for (unweighted) MAX SAT with performance ratios arbitrarily close to 1, in particular, when performance ratios exceed the limits of polynomial-time approximation. Namely, given a polynomial-time α-approximation algorithm A0, we construct an (α+e)-approximation algorithm A. The algorithm A runs in time of the order cek, where k is the number of clauses in the input formula and c is a constant depending on α. Thus we estimate the cost of improving a performance ratio. Similar constructions for MAX 2SAT and MAX 3SAT are also described. Taking known algorithms as A0 (for example, the Karloff–Zwick algorithm for MAX 3SAT), we obtain particular upper bounds on the running time of A.

Algebraic proof systems over formulas

We introduce two algebraic propositional proof systems F-NS and F-PC. The main difference of our systems from (customary) Nullstellensatz and polynomial calculus is that the polynomials are represented as arbitrary formulas (rather than sums of monomials). Short proofs of Tseitins tautologies in the constant-depth version of F-NS provide an exponential separation between this system and Polynomial Calculus.We prove that F-NS (and hence F-PC) polynomially simulates Frege systems, and that the constant-depth version of F-PC over finite field polynomially simulates constant-depth Frege systems with modular counting. We also present a short constant-depth F-PC (in fact, F-NS) proof of the propositional pigeon-hole principle. Finally, we introduce several extensions of our systems and pose numerous open questions.

Worst-case study of local search for MAX- k -SAT

During the past 3 years there was a considerable growth in the number of algorithms solving MAX-SAT and MAX-2-SAT in worst-case time of the order cK, where c 0.

A Complete Public-Key Cryptosystem

We present a cryptosystem which is complete for the class of probabilistic public-key cryptosystems with bounded error. Besides traditional encryption schemes such as RSA and El Gamal and probabilistic encryption of Goldwasser and Micali, this class contains also Ajtai-Dwork and NTRU cryptosystems. The latter two make errors with a small positive probability.

Solving Boolean Satisfiability Using Local Search Guided by Unit Clause Elimination

In this paper we present a new randomized algorithm for SAT combining unit clause elimination and local search. The algorithm is inspired by two randomized algorithms having the best current worstcase upper bounds ([9] and [11,12]). Despite its simplicity, our algorithm performs well on many common benchmarks (we present results of its empirical evaluation). It is also probabilistically approximately complete.