Arist Kojevnikov

A new approach to proving upper bounds for MAX-2-SAT

In this paper we present a new approach to proving upper bounds for the maximum 2-satisfiability problem (MAX-2-SAT). We present a new 2K/5.5-time algorithm for MAX-2-SAT, where K is the number of clauses in an input formula. We also obtain a 2N/6 bound, where N is the number of variables in an input formula, for a particular case of MAX-2-SAT, where each variable appears in at most three 2-clauses. This immediately implies a 2N/6 bound, where N is the number of vertices in an input graph, for the independent set problem on 3-regular graphs. The key point of our improvement is a combined complexity measure for estimating the running time of an algorithm. By using a new complexity measure we are able to provide a much simpler proof of new upper bounds for MAX-2-SAT than proofs of previously known bounds.

Finding Efficient Circuits Using SAT-Solvers

In this paper we report preliminary results of experiments with finding efficient circuits (over binary bases) using SAT-solvers. We present upper bounds for functions with constant number of inputs as well as general upper bounds that were found automatically. We focus mainly on MOD-functions. Besides theoretical interest, these functions are also interesting from a practical point of view as they are the core of the residue number system. In particular, we present a circuit of size 3n + c over the full binary basis computing

Solving Boolean Satisfiability Using Local Search Guided by Unit Clause Elimination

{\rm MOD}_3^n

Circuit complexity and multiplicative complexity of Boolean functions

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Lower bounds of static lovász-schrijver calculus proofs for tseitin tautologies

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.

Several notes on the power of Gomory–Chvátal cuts

In this note, we use lower bounds on Boolean multiplicative complexity to prove lower bounds on Boolean circuit complexity. We give a very simple proof of a 7n/3 - c lower bound on the circuit complexity of a large class of functions representable by high degree polynomials over GF(2). The key idea of the proof is a circuit complexity measure assigning different weights to XOR and AND gates.

NEW COMBINATORIAL COMPLETE ONE-WAY FUNCTIONS

We prove an exponential lower bound on the size of static Lovasz-Schrijver calculus refutations of Tseitin tautologies. We use several techniques, namely, translating static LS+ proof into Positivstellensatz proof of Grigoriev et al., extracting a “good” expander out of a given graph by removing edges and vertices of Alekhnovich et al., and proving linear lower bound on the degree of Positivstellensatz proofs for Tseitin tautologies.

Improved lower bounds for tree-like resolution over linear inequalities

Abstract We prove that the Cutting Plane proof system based on Gomory–Chvatal cuts polynomially simulates the lift-and-project system with integer coefficients written in unary. The restriction on the coefficients can be omitted when using Krajicek’s cut-free Gentzen-style extension of both systems. We also prove that Tseitin tautologies have short proofs in this extension (of any of these systems and with any coefficients).

Complexity of semialgebraic proofs with restricted degree of falsity

In 2003, Leonid A. Levin presented the idea of a combinatorial complete one-way function and a sketch of the proof that Tiling represents such a function. In this paper, we present two new one-way functions based on semi-Thue string rewriting systems and a version of the Post Correspondence Problem and prove their completeness. Besides, we present an alternative proof of Levins result. We also discuss the properties a combinatorial problem should have in order to hold a complete one-way function.

UnitWalk: A New SAT Solver that Uses Local Search Guided by Unit Clause Elimination

We continue a study initiated by Krajicek of a Resolution-like proof system working with clauses of linear inequalities, R(CP). For all proof systems of this kind Krajicek proved in [1] an exponential lower bound of the form: exp(nΩ(1))/MO(W log2 n), where M is the maximal absolute value of coefficients in a given proof and W is the maximal clause width. In this paper we improve this lower bound. For tree-like R(CP)-like proof systems we remove a dependence on the maximal absolute value of coefficients M, hence, we give the answer to an open question from [2]. Proof follows from an upper bound on the real communication complexity of a polyhedra.