Random 3-CNF Research Articles

Sufficient condition for polynomial solvability of random 3-CNF formulas

This paper is devoted to the localisation of random 3-CNF formulas that are polynomially solvable by the resolution algorithm. It is shown that random formulas with the number of clauses proportional to the square of the number of variables, are polynomially solvable with probability close to unity when the proportionality coefficient exceeds the found threshold.

Non‐orderability of random triangular groups by using random 3CNF formulas

We show that a random group Γ in the triangular binomial model Γ ( n , p ) is asymptotically almost surely (a.a.s.) not left-orderable for p ∈ ( c n − 2 , n − 3 / 2 − ε ) , where c , ε are any constants satisfying ε > 0 , c > ( 1 / 8 ) log 4 / 3 2 ≈ 0.3012 . We also prove that if p ⩾ ( 1 + ε ) ( log n ) n − 2 for any fixed ε > 0 , then a random Γ ∈ Γ ( n , p ) has a.a.s. no non-trivial left-orderable quotients. We proceed by constructing 3CNF formulas, which encode necessary conditions for left orderability and then proving their unsatisfiability a.a.s.

Counting Solutions to Random CNF Formulas

We give the first efficient algorithm to approximately count the number of solutions in the random $k$-SAT model when the density of the formula scales exponentially with $k$. The best previous cou...

Probabilities of first-order sentences on sparse random relational structures: An application to definability on random CNF formulas

Abstract We extend the convergence law for sparse random graphs proven by Lynch to arbitrary relational languages. We consider a finite relational vocabulary $\sigma $ and a first-order theory $T$ for $\sigma $ composed of symmetry and anti-reflexivity axioms. We define a binomial random model of finite $\sigma $-structures that satisfy $T$ and show that first-order properties have well defined asymptotic probabilities when the expected number of tuples satisfying each relation in $\sigma $ is linear. It is also shown that these limit probabilities are well behaved with respect to several parameters that represent the density of tuples in each relation $R$ in the vocabulary $\sigma $. An application of these results to the problem of random Boolean satisfiability is presented. We show that in a random $k$-CNF formula on $n$ variables, where each possible clause occurs with probability $\sim c/n^{k-1}$, independently any first-order property of $k$-CNF formulas that implies unsatisfiability does almost surely not hold as $n$ tends to infinity.

Properties of the satisfiability threshold of the strictly d-regular random (3,2s)-SAT problem

A k-CNF (conjunctive normal form) formula is a regular (k, s)-CNF one if every variable occurs s times in the formula, where k ⩾ 2 and s > 0 are integers. Regular (3, s)-CNF formulas have some good structural properties, so carrying out a probability analysis of the structure for random formulas of this type is easier than conducting such an analysis for random 3-CNF formulas. Some subclasses of the regular (3, s)-CNF formula have also characteristics of intractability that differ from random 3-CNF formulas. For this purpose, we propose strictly d-regular (k, 2s)-CNF formula, which is a regular (k, 2s)-CNF formula for which d ⩾ 0 is an even number and each literal occurs $$s - {d \over 2}$$ or $$s + {d \over 2}$$ times (the literals from a variable x are x and ¬x, where x is positive and ¬x is negative). In this paper, we present a new model to generate strictly d-regular random (k, 2s)-CNF formulas, and focus on the strictly d-regular random (3, 2s)-CNF formulas. Let F be a strictly d-regular random (3, 2s)-CNF formula such that 2s > d. We show that there exists a real number s0 such that the formula F is unsatisfiable with high probability when s > s0, and present a numerical solution for the real number s0. The result is supported by simulated experiments, and is consistent with the existing conclusion for the case of d = 0. Furthermore, we have a conjecture: for a given d, the strictly d-regular random (3, 2s)-SAT problem has an SAT-UNSAT (satisfiable-unsatisfiable) phase transition. Our experiments support this conjecture. Finally, our experiments also show that the parameter d is correlated with the intractability of the 3-SAT problem. Therefore, our research maybe helpful for generating random hard instances of the 3-CNF formula.

Space proof complexity for random 3-CNFs

We investigate the space complexity of refuting 3-CNFs in Resolution and algebraic systems. We prove that every Polynomial Calculus with Resolution refutation of a random 3-CNF φ in n variables requires, with high probability, Ω(n)distinct monomials to be kept simultaneously in memory. The same construction also proves that every Resolution refutation of φ requires, with high probability, Ω(n) clauses each of width Ω(n) to be kept at the same time in memory. This gives a Ω(n2) lower bound for the total space needed in Resolution to refute φ. These results are best possible (up to a constant factor) and answer questions about space complexity of 3-CNFs.The main technical innovation is a variant of Hall's Lemma. We show that in bipartite graphs with bipartition (L,R) and left-degree at most 3, L can be covered by certain families of disjoint paths, called VW-matchings, provided that L expands in R by a factor of (2−ϵ), for ϵ<15.

Short propositional refutations for dense random 3CNF formulas

Random 3CNF formulas constitute an important distribution for measuring the average-case behavior of propositional proof systems. Lower bounds for random 3CNF refutations in many propositional proof systems are known. Most notable are the exponential-size resolution refutation lower bounds for random 3CNF formulas with Ω(n1.5−ε) clauses (Chvátal and Szemerédi [14], Ben-Sasson and Wigderson [10]). On the other hand, the only known non-trivial upper bound on the size of random 3CNF refutations in a non-abstract propositional proof system is for resolution with Ω(n2/log⁡n) clauses, shown by Beame et al. [6]. In this paper we show that already standard propositional proof systems, within the hierarchy of Frege proofs, admit short refutations for random 3CNF formulas, for sufficiently large clause-to-variable ratio. Specifically, we demonstrate polynomial-size propositional refutations whose lines are TC0 formulas (i.e., TC0-Frege proofs) for random 3CNF formulas with n variables and Ω(n1.4) clauses.The idea is based on demonstrating efficient propositional correctness proofs of the random 3CNF unsatisfiability witnesses given by Feige, Kim and Ofek [23]. Since the soundness of these witnesses is verified using spectral techniques, we develop an appropriate way to reason about eigenvectors in propositional systems. To carry out the full argument we work inside weak formal systems of arithmetic and use a general translation scheme to propositional proofs.

Complete Convergence of Message Passing Algorithms for Some Satisfiability Problems.

Experimental results show that certain message passing algorithms, namely, survey propagation, are very effective in finding satisfying assignments in random satisfiable 3CNF formulas. In this paper we make a modest step towards providing rigorous analysis that proves the effectiveness of message passing algorithms for random 3SAT. We analyze the performance of Warning Propagation, a popular message passing algorithm that is simpler than survey propagation. We show that for 3CNF formulas generated under the planted assignment distribution, running warning propagation in the standard way works when the clause-to-variable ratio is a sufficiently large constant. We are not aware of previous rigorous analysis of message passing algorithms for satisfiability instances, though such analysis was performed for decoding of Low Density Parity Check (LDPC) Codes. We discuss some of the differences between results for the LDPC setting and our results.

Optimizing with minimum satisfiability

MinSAT is the problem of finding a truth assignment that minimizes the number of satisfied clauses in a CNF formula. When we distinguish between hard and soft clauses, and soft clauses have an associated weight, then the problem, called Weighted Partial MinSAT, consists in finding a truth assignment that satisfies all the hard clauses and minimizes the sum of weights of satisfied soft clauses. In this paper we describe a branch-and-bound solver for Weighted Partial MinSAT equipped with original upper bounds that exploit both clique partitioning algorithms and MaxSAT technology. Then, we report on an empirical investigation that shows that solving combinatorial optimization problems by reducing them to MinSAT is a competitive generic problem solving approach when solving MaxClique and combinatorial auction instances. Finally, we investigate an interesting correlation between the minimum number and the maximum number of satisfied clauses on random CNF formulae.

Lower Bounds for k-DNF Resolution on Random 3-CNFs

We prove exponential lower bounds on refutations of a random 3-CNF with linear number of clauses by k-DNF Resolution for $${k\leq \sqrt{\log n/\log\log n}}$$. For this we design a specially tailored random restrictions that preserve the structure of the input random 3-CNF while mapping every k-DNF with large covering number to one with high probability. Next we make use of the switching lemma for small restrictions by Segerlind, Buss and Impagliazzo to prove the lower bound. This work improves the previously known lower bound for Res(2) system on random 3-CNFs by Atserias, Bonet and Esteban and the result of Segerlind, Buss, Impagliazzo stating that random O(k 2)-CNF do not possess short Res(k) refutations.

More on Average Case vs Approximation Complexity

We 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-e. ◦ 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.

Buying Cheap Is Expensive: Approximability of Combinatorial Pricing Problems

We investigate nonparametric multiproduct pricing problems, in which we want to find revenue maximizing prices for products $\mathcal{P}$ based on a set of customer samples $\mathcal{C}$. We mostly focus on the unit-demand case, in which products constitute strict substitutes and each customer aims to purchase a single product. In this setting a customer sample consists of a number of nonzero values for different products and possibly an additional product ranking. Once prices are fixed, each customer chooses to buy one of the products she can afford based on some predefined selection rule. We distinguish between the min-buying, max-buying, and rank-buying models. Some of our results also extend to single-minded pricing, in which case products are strict complements and every customer seeks to buy a single set of products, which she purchases if the sum of prices is below her valuation for that set. For the min-buying model we show that the revenue maximization problem is not approximable within factor $\mathcal{O}(\log^{\varepsilon}|\mathcal{C}|)$ for some constant $\varepsilon>0$, unless $\mathrm{NP}\subseteq\mathrm{DTIME}(n^{\mathcal{O}(\log\log n)})$, thereby almost closing the gap between the known algorithmic results and previous lower bounds. We also prove inapproximability within $\mathcal{O}(\ell^{\varepsilon})$, $\ell$ being an upper bound on the number of nonzero values per customer, and $\mathcal{O}(|\mathcal{P}|^{\varepsilon})$ under slightly stronger assumptions and provide matching upper bounds. Surprisingly, these hardness results hold even if a price ladder constraint, i.e., a predefined order on the prices of all products, is given. Without the price ladder constraint we obtain similar hardness results for the special case of uniform valuations, i.e., the case that every customer has identical values for all the products she is interested in, assuming specific hardness of the balanced bipartite independent set problem in constant degree graphs or hardness of refuting random 3CNF formulas. Introducing a slightly more general problem definition in which customers are given as an explicit probability distribution, we obtain inapproximability within $\mathcal{O}(|\mathcal{P}|^{\varepsilon})$ assuming $\mathrm{NP}\nsubseteq\bigcap_{\delta>0}\mathrm{BPTIME}(2^{\mathcal{O}(n^{\delta})})$. These results apply to single-minded pricing as well. For the max-buying model a polynomial-time approximation scheme exists if a price ladder is given. We give a matching lower bound by proving strong NP-hardness. Assuming limited product supply, we analyze a generic local search algorithm and prove that it is 2-approximate. Finally, we discuss implications for the rank-buying model.

Random Cnf’s are Hard for the Polynomial Calculus

We prove linear lower bounds on the Polynomial Calculus (PC) refutation-degree of random CNF whenever the underlying field has characteristic greater than 2. Our proof follows by showing the PC refutation-degree of a unsatisfiable system of linear equations modulo 2 is equivalent to its Gaussian width, a concept defined by the late Mikhail Alekhnovich. The equivalence of refutation-degree and Gaussian width which is the main contribution of this paper, allows us to also simplify the refutation-degree lower bounds of Buss et al. (2001) and additionally prove non-trivial upper bounds on the resolution and PC complexity of refuting unsatisfiable systems of linear equations.

Complexity of propositional proofs under a promise

We study—within the framework of propositional proof complexity—the problem of certifying unsatisfiability of CNF formulas under the promise that any satisfiable formula has many satisfying assignments, where many stands for an explicitly specified function Λ in the number of variables n . To this end, we develop propositional proof systems under different measures of promises (i.e., different Λ) as extensions of resolution. This is done by augmenting resolution with axioms that, roughly, can eliminate sets of truth assignments defined by Boolean circuits. We then investigate the complexity of such systems, obtaining an exponential separation in the average case between resolution under different size promises: (1) Resolution has polynomial-size refutations for all unsatisfiable 3CNF formulas when the promise is ϵ…2 n , for any constant 0&lt;ϵ&lt;1. (2) There are no subexponential size resolution refutations for random 3CNF formulas, when the promise is 2 Δ n , for any constant 0&lt;δ&lt;1 (and the number of clauses is O ( n 3/2-ϵ ), for 0&lt;ϵ&lt;1/2). “ Goods Satisfactory or Money Refunded ” —The Eaton Promise

Information loss in knowledge compilation: A comparison of Boolean envelopes

Since Selman and Kautz's seminal work on the use of Horn approximation to speed up the querying of knowledge bases, there has been great interest in Boolean approximation for AI applications. There are several Boolean classes with desirable computational properties similar to those of the Horn class. The class of affine Boolean functions, for example, has been proposed as an interesting alternative to Horn for knowledge compilation. To investigate the trade-offs between precision and efficiency in knowledge compilation, we compare, analytically and empirically, four well-known Boolean classes, and their combinations, for ability to preserve information. We note that traditional evaluation which explores unit-clause consequences of random hard 3-CNF formulas does not tell the full story, and we complement that evaluation with experiments based on a variety of assumptions about queries and the underlying knowledge base.

Horn Upper Bounds and Renaming1

Selman and Kautz proposed an approach to compute a Horn CNF approximation of CNF formulas. We extend this approach and propose a new algorithm for the Horn least upper bound that involves renaming variables. Although we provide negative results for the quality of approximation, experimental results for random CNF demonstrate that the proposed algorithm improves both computational efficiency and approximation quality. We observe an interesting behavior in the Horn approximation sizes and errors which we call the “Horn bump”: Maxima occur in an intermediate range of densities below the satisfiability threshold.

A spectral technique for random satisfiable 3CNF formulas

AbstractLet I be a random 3CNF formula generated by choosing a truth assignment ϕ for variables x1, xn uniformly at random and including every clause with i literals set true by ϕ with probability pi, independently. We show that for any constants 0 ≤ η2,η3 ≤ 1 there is a constant dmin so that for all d ≥ dmin a spectral algorithm similar to the graph coloring algorithm of Alon and Kahale will find a satisfying assignment with high probability for p1 = d/n2, p2 = η2d/n2, and p3 = η3d/n2. Appropriately setting the ηi's yields natural distributions on satisfiable 3CNFs, not‐all‐equal‐sat 3CNFs, and exactly‐one‐sat 3CNFs. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008

It’s All about the Support: A New Perspective on the Satisfiability Problem

We study a new approach to the satisfiability problem, which we call the Support Paradigm. Given a CNF formula F and an assignment ψ to its variables we say that a literal x supports a clause C in F w.r.t. ψ if x is the only literal that evaluates to true in C. Our focus in this work will be on heuristics that obey the following general template: start at some assignment to the variables, then iteratively, using some predefined (greedy) rule, try to minimize the number of unsatisfied clauses (or the distance from some satisfying assignment) until a satisfying assignment is reached. We say that such a heuristic is part of the Support Paradigm if the greedy rule uses the support as its main criterion. We present a new algorithm in the Support Paradigm and rigorously prove its effectiveness for a certain distribution over satisfiable k-CNF formulas known as the planted distribution. One motivation for this work is recent experimental results showing that some simple variants of the RWalkSAT algorithm, which base their greedy rule on the support, seem to remain effective for random 3CNF formulas in the “hard” near-threshold regime, while for example RWalkSAT, which disregards the support, is already inefficient in a much earlier stage.

Easily refutable subformulas of large random 3CNF formulas.

A simple nonconstructive argument shows that most 3CNF formulas with cn clauses (where c is a large enough constant) are not satisfiable. It is an open question whether there is an efficient refutation algorithm that for most formulas with cn clauses proves that they are not satisfiable. We present a polynomial time algorithm that for most 3CNF formulas with cn 3/2 clauses (where c is a large enough constant) finds a subformula with O(c 2 n) clauses and then proves that this subformula is not satisfiable (and hence that the original formula is not satisfiable). Previously, it was only known how to efficiently certify the unsatisfiability of random 3CNF formulas with at least poly(log(n)) · n 3/2 clauses. Our algorithm is simple enough to run in practice. We present some preliminary experimental results.

Correlations between Horn fractions, satisfiability and solver performance for fixed density random 3-CNF instances

An enhanced concept of sub-optimal reverse Horn fraction of a CNF-formula was introduced in [18]. It was shown that this fraction is very useful in effectively (almost) separating 3-colorable random graphs with fixed node-edge density from the non-3-colorable ones. A correlation between this enhanced sub-optimal reverse Horn fraction and satisfiability of random 3-SAT instances with a fixed density was observed. In this paper, we present experimental evidence that this correlation scales to larger-sized instances and that it extends to solver performances as well, both of complete and incomplete solvers. Furthermore, we give a motivation for various phases in the algorithm aHS, establishing the enhanced sub-optimal reverse Horn fraction, and we present clear evidence for the fact that the observed correlations are stronger than correlations between satisfiability and sub-optimal MAXSAT-fractions established similarly to the enhanced sub-optimal reverse Horn fraction. The latter observation is noteworthy because the correlation between satisfiability and the optimal MAXSAT-fraction is obviously 100%.