Dmitry Itsykson

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.

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

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.

Lower bounds for myopic DPLL algorithms with a cut heuristic

The paper is devoted to lower bounds on the time complexity of DPLL algorithms that solve the satisfiability problem using a splitting strategy. Exponential lower bounds on the running time of DPLL algorithms on unsatisfiable formulas follow from the lower bounds for resolution proofs. Lower bounds on satisfiable instances are also known for some classes of DPLL algorithms; this lower bounds are usually based on reductions to unsatisfiable instances. In this paper we consider DPLL algorithms with a cut heuristic that may decide that some branch of the splitting tree will not be investigated. DPLL algorithms with a cut heuristic always return correct answer on unsatisfiable formulas while they may err on satisfiable instances. We prove the theorem about effectiveness vs. correctness trade-off for deterministic myopic DPLL algorithms with cut heuristic. Myopic algorithms can see formulas with erased signs of negations; they may also request a small number of clauses to read them precisely. We construct a family of unsatisfiable formulas Φ(n) and a polynomial time samplable ensemble of distributions Qn concentrated on satisfiable formulas such that every deterministic myopic algorithm that gives a correct answer with probability 1−o(1) on a random formula from the ensemble Qn runs exponential time on the formulas Φ(n).

The complexity of inversion of explicit goldreich's function by DPLL algorithms

The Goldreichs function has n binary inputs and n binary outputs. Every output depends on d inputs and is computed from them by the fixed predicate of arity d. Every Goldreichs function is defined by its dependency graph G and predicate P. In 2000 O. Goldreich formulated a conjecture that if G is an expander and P is a random predicate of arity d then the corresponding function is one way. In 2005 M. Alekhnovich, E. Hirsch and D. Itsykson proved the exponential lower bound on the complexity of inversion of Goldreichs function based on linear predicate and random graph by myopic DPLL agorithms. In 2009 J. Cook, O. Etesami, R. Miller, and L. Trevisan extended this result to nonliniar predicates (but for a slightly weaker definition of myopic algorithms). Recently D. Itsykson and independently R. Miller proved the lower bound for drunken DPLL algorithms that invert Goldreichs function with nonlinear P and random G. All above lower bounds are randomized. The main contribution of this paper is the simpler proof of the exponential lower bound of the Goldreichs function inversion by myopic DPLL algorithms. A dependency graph in our construction may be based on an arbitrary expander, particulary it is possible to use an explicit expander; the predicate may be linear or slightly nonlinear. Our definition of myopic algorithms is more general than one used by J. Cook et al. Our construction may be used in the proof of lower bound for drunken algorithms as well.

Lower Bounds for Splittings by Linear Combinations

A typical DPLL algorithm for the Boolean satisfiability problem splits the input problem into two by assigning the two possible values to a variable; then it simplifies the two resulting formulas. In this paper we consider an extension of the DPLL paradigm. Our algorithms can split by an arbitrary linear combination of variables modulo two. These algorithms quickly solve formulas that explicitly encode linear systems modulo two, which were used for proving exponential lower bounds for conventional DPLL algorithms.

An Infinitely-Often One-Way Function Based on an Average-Case Assumption

We assume the existence of a function fthat is computable in polynomial time but its inverse function is not computable in randomized average-case polynomial time. The cryptographic setting is, however, different: even for a weak one-way function, every possible adversary should fail on a polynomial fraction of inputs. Nevertheless, we show how to construct an infinitely-oftenone-way function based on f.

Heuristic Time Hierarchies via Hierarchies for Sampling Distributions

We introduce a new framework for proving the time hierarchy theorems for heuristic classes. The main ingredient of our proof is a hierarchy theorem for sampling distributions recently proved by Watson [11]. Class \(\mathrm {Heur}_{\epsilon }{\mathbf {FBPP}}\) consists of functions with distributions on their inputs that can be computed in randomized polynomial time with bounded error on all except \(\epsilon \) fraction of inputs. We prove that for every a, \(\delta \) and integer k there exists a function \({F: \{0, 1\}^* \rightarrow \{0, 1, \dots , k - 1\}}\) such that \((F, U) \in \mathrm {Heur}_{\epsilon }{\mathbf {FBPP}}\) for all \(\epsilon > 0\) and for every ensemble of distributions \(D_n\) samplable in \(n^a\) steps, \((F, D) \notin \mathrm {Heur}_{1 - \frac{1}{k} - \delta }{\mathbf {FBPTime}}[n^a]\). This extends a previously known result for languages with uniform distributions proved by Pervyshev [9] by handling the case \(k > 2\). We also prove that \({\mathbf {P}}\not \subseteq \mathrm {Heur}_{\frac{1}{2} - \epsilon }{\mathbf {BPTime}}[n^k]\) if one-way functions exist.

Resolution complexity of perfect matching principles for sparse graphs

The resolution complexity of the perfect matching principle was studied by Razborov [Raz04], who developed a technique for proving its lower bounds for dense graphs. We construct a constant degree bipartite graph \(G_n\) such that the resolution complexity of the perfect matching principle for \(G_n\) is \(2^{\varOmega (n)}\), where n is the number of vertices in \(G_n\). This lower bound is tight up to some polynomial. Our result implies the \(2^{\varOmega (n)}\) lower bounds for the complete graph \(K_{2n+1}\) and the complete bipartite graph \(K_{n, O(n)}\) that improves the lower bounds following from [Raz04]. Our results also imply the well-known exponential lower bounds on the resolution complexity of the pigeonhole principle, the functional pigeonhole principle and the pigeonhole principle over a graph.

Reordering rule makes OBDD proof systems stronger

Atserias, Kolaitis, and Vardi showed that the proof system of Ordered Binary Decision Diagrams with conjunction and weakening, OBDD(∧, weakening), simulates CP* (Cutting Planes with unary coefficients). We show that OBDD(∧, weakening) can give exponentially shorter proofs than dag-like cutting planes. This is proved by showing that the Clique-Coloring tautologies have polynomial size proofs in the OBDD(∧, weakening) system. The reordering rule allows changing the variable order for OBDDs. We show that OBDD(∧, weakening, reordering) is strictly stronger than OBDD(∧, weakening). This is proved using the Clique-Coloring tautologies, and by transforming tautologies using coded permutations and orification. We also give CNF formulas which have polynomial size OBDD(∧) proofs but require superpolynomial (actually, quasipolynomial size) resolution proofs, and thus we partially resolve an open question proposed by Groote and Zantema. Applying dag-like and tree-like lifting techniques to the mentioned results, we completely analyze which of the systems among CP*, OBDD(∧), OBDD(∧, reordering), OBDD(∧, weakening) and OBDD(∧, weakening, reordering) polynomially simulate each other. For dag-like proof systems, some of our separations are quasipolynomial and some are exponential; for tree-like systems, all of our separations are exponential.

Hard Satisfiable Formulas for Splittings by Linear Combinations

Itsykson and Sokolov in 2014 introduced the class of \(\mathrm {DPLL}(\oplus )\) algorithms that solve Boolean satisfiability problem using the splitting by linear combinations of variables modulo 2. This class extends the class of \(\mathrm {DPLL}\) algorithms that split by variables. \(\mathrm {DPLL}(\oplus )\) algorithms solve in polynomial time systems of linear equations modulo 2 that are hard for \(\mathrm {DPLL}\), \(\mathrm {PPSZ}\) and \(\mathrm {CDCL}\) algorithms. Itsykson and Sokolov have proved first exponential lower bounds for \(\mathrm {DPLL}(\oplus )\) algorithms on unsatisfiable formulas.