Michael Alekhnovich

More on average case vs approximation complexity

AbstractWe consider the problem to determine the maximal number of satisfiable equations in a linear system chosen at random. We make several plausible conjectures about the average case hardness of this problem for some natural distributions on the instances, and relate them to several interesting questions in the theory of approximation algorithms and in cryptography. Namely we show that our conjectures imply the following facts: ◦ Feige’s hypothesis about the hardness of refuting a random 3CNF is true, which in turn implies inapproximability within a constant for several combinatorial problems, for which no NP-hardness of approximation is known. ◦ It is hard to approximate the NEAREST CODEWORD within factor n1-ε. ◦ It is hard to estimate the rigidity of a matrix. More exactly, it is hard to distinguish between matrices of low rigidity and random ones. ◦ There exists a secure public-key (probabilistic) cryptosystem, based on the intractability of decoding of random binary codes.

An exponential separation between regular and general resolution

Two distinct proofs of an exponential separation between regular resolution and unrestricted resolution are given. The previous best known separation between these systems was quasi-polynomial.

Resolution Is Not Automatizable Unless W[P] Is Tractable

We show that neither resolution nor tree-like resolution is automatizable unless the class W[P] from the hierarchy of parameterized problems is fixed-parameter tractable by randomized algorithms with one-sided error.

Linear diophantine equations over polynomials and soft decoding of Reed-Solomon codes

This paper generalizes the classical Knuth-Schoumlnhage algorithm computing the greatest common divisor (gcd) of two polynomials for solving arbitrary linear Diophantine systems over polynomials in time, quasi-linear in the maximal degree. As an application, the following weighted curve fitting problem is considered: given a set of points in the plane, find an algebraic curve (satisfying certain degree conditions) that goes through each point the prescribed number of times. The main motivation for this problem comes from coding theory, namely, it is ultimately related to the list decoding of Reed-Solomon codes. This paper presents a new fast algorithm for the weighted curve fitting problem, based on the explicit construction of a Groebner basis. This gives another fast algorithm for the soft decoding of Reed-Solomon codes different from the procedure proposed by Feng, which works in time (w/r) O(1)nlog2n, where r is the rate of the code, and w is the maximal weight assigned to a vertical line

Lower bounds for polynomial calculus: non-binomial case

We generalize recent linear lower bounds for Polynomial Calculus based on binomial ideals. We produce a general hardness criterion (that we call immunity) which is satisfied by a random function and prove linear lower bounds on the degree of PC refutations for a wide class of tautologies based on immune functions. As some applications of our techniques, we introduce mod/sub p/ Tseitin tautologies in the Boolean case (e.g. in the presence of axioms x/sub i//sup 2/=x/sub i/), prove that they are hard for PC over fields with characteristic different from p, and generalize them to Flow tautologies which are based on the MAJORITY function and are proved to be hard over any field. We also show the /spl Omega/(n) lower bound for random k-CNFs over fields of characteristic 2.

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.

Satisfiability, branch-width and Tseitin tautologies

For a CNF τ, let wb(τ) be the branch-width of its underlying hypergraph, that is the smallest w for which the clauses of τ can be arranged in the form of leaves of a rooted binary tree in such a way that for every vertex its descendants and non-descendants have at most w variables in common. In this paper we design an algorithm for solving SAT in time

Learnability and automatizability

{n^{O(1)}2^{O(w_b(\tau))}}