On CNF formulas irredundant with respect to unit clause propagation

Max-SAT with cardinality constraint ( CC-Max-Sat ) is one of the classical NP-complete problems, that generalizes Maximum Coverage , Partial Vertex Cover , Max-2-SAT with bisection constraints, and has been extensively studied across all algorithmic paradigms. In this problem, we are given a CNF formula \(\Phi\) , and a positive integer \( k \) , and the goal is to find an assignment \(\beta\) with at most \( k \) variables set to true (also called a \( k \) -weight assignment) such that the number of clauses satisfied by \(\beta\) is maximized. The problem is known to admit an approximation algorithm with factor \(1-\frac{1}{e}\) , which is probably optimal. Furthermore, assuming Gap-Exponential Time Hypothesis (Gap-ETH), for any \(\epsilon > 0\) and any function \( h \) , no \(h(k)(n+m)^{o(k)}\) time algorithm can approximate Maximum Coverage (a monotone version of CC-Max-Sat ) with \( n \) elements and \( m \) sets to within a factor \((1-\frac{1}{e}+\epsilon)\) , even with a promise that there exist \( k \) sets that fully cover the whole universe. In fact, the problem is hard to approximate within 0.929, assuming Unique Games Conjecture, even when the input formula is 2-CNF. These intractable results lead us to explore families of formula, where we can circumvent these barriers. Toward this, we consider \(K_{d,d}\) -free formulas (that is, the clause-variable incidence bipartite graph of the formula excludes \(K_{d,d}\) as an induced subgraph). We show that for every \(\epsilon > 0\) , there exists an algorithm for CC-Max-Sat on \(K_{d,d}\) -free formulas with approximation ratio \((1-\epsilon)\) and running in time \(2^{{\mathcal{O}}((\frac{dk}{\epsilon})^{d})}(n+m)^{{\mathcal{O}}(1)}\) (these algorithms are called FPT-AS). For Maximum Coverage on \(K_{d,d}\) -free set families, we obtain FPT-AS with running time \((\frac{dk}{\epsilon})^{{\mathcal{O}}(dk)}n^{{\mathcal{O}}(1)}\) . Our second result considers “optimizing \( k \) ,” with fixed covering constraint for the Maximum Coverage problem. To explain our result, we first recast the Maximum Coverage problem as the Max Red Blue Dominating Set with Covering Constraint problem. Here, the input is a bipartite graph \(G=(A,B,E)\) , a positive integer \( t \) , and the objective is to find a minimum sized subset \(S\subseteq A\) , such that \(|N(S)|\) (the size of the set of neighbors of \( S \) ) is at least \( t \) . We design an additive approximation algorithm for Max Red Blue Dominating Set with Covering Constraint , on \(K_{d,d}\) -free bipartite graphs, running in FPT time. In particular, if \( k \) denotes the minimum size of \(S\subseteq A\) , such that \(|N(S)|\geq t\) , then our algorithm runs in time \((kd)^{{\mathcal{O}}(kd)}n^{{\mathcal{O}}{(1)}}\) and returns a set \(S^{\prime}\) such that \(|N(S^{\prime})|\geq t\) and \(|S^{\prime}|\leq k+1\) . This is in sharp contrast to the fact that, even a special case of our problem, namely, the Partial Vertex Cover problem (or Max \( k \) -VC ) is W[1]-hard, parameterized by \( k \) . Thus, we get the best possible parameterized approximation algorithm for the Maximum Coverage problem on \(K_{d,d}\) -free bipartite graphs.

The Approximate Degree of DNF and CNF Formulas

In the context of defining NP-completeness, a tableau represents a hypothetical accepting computation path p of a nondeterministic polynomial time Turing machine N on an input w. The tableau is encoded by the propositional logic formula ψ, defined as ψ=ψcell∧ψrest. The component ψcell enforces the constraint that each cell in the tableau contains exactly one symbol, while ψrest incorporates constraints governing the step-by-step behavior of N on w. Intuitively, ψrest appears to pose a much greater challenge for satisfiability. This raises the question of whether the distinction between ψcell being a 3cnf formula, rather than a cheap 2cnf formula, actually matters. We show that if, hypothetically, ψrest can be succinctly represented as a Horn formula, then satisfying ψ can be achieved efficiently in Kf(n,k) steps, where N operates within O(nk) steps and both k and K are constants. Asymptotically, f(n,k)≈n23k. Our method has the potential for iterative application. Technically, we trim ψcell down to a 2cnf–Horn formula, whose satisfiability allows for empty cells, or “holes”, in the tableau. This modified tableau represents exponentially many paths of N on w, rather than a single accepting path p. While a tableau with holes conceptualizes the satisfiability of ψtrim—a trimmed-down version of ψ—it does not directly address the satisfiability of ψ. Therefore, we introduce an external user who efficiently employs backtracking to fill in specific holes, ultimately verifying the satisfiability of the original ψ.

The 1 -in- 3 and N ot -A ll -E qual satisfiability problems for Boolean CNF formulas are two well-known NP -hard problems. In contrast, the promise 1 -in- 3 vs . N ot -A ll -E qual problem can be solved in polynomial time. In the present work, we investigate this constraint satisfaction problem in a regime where the promise is weakened from either side by a rainbow-free structure and establish a complexity dichotomy for the resulting class of computational problems.

This paper introduces two new compilation languages restricting weak decomposable negation normal form (wDNNF) circuits and integrates them into the knowledge compilation map. Positive (resp. negative) wDNNF circuits restrict wDNNF circuits so that each variable shared among the inputs of a conjunction node can only have positive (resp. negative) occurrences in that subcircuit. Unlike wDNNF circuits, pwDNNF (resp. nwDNNF) circuits satisfy the maximum (resp. minimum) cardinality query. We present a compiler for converting CNF formulae into pwDNNF and nwDNNF circuits by extending Bella - the state-of-the-art compiler for wDNNF circuits. We introduce a new caching scheme, called Cara, that exploits isomorphism. Using that scheme, we show a new compilation method based on copying subcircuits, which may significantly speed up compilations at the expense of increasing circuit sizes. Our experiments demonstrate that nwDNNF circuits are suitable for computing most probable explanations (MPEs) in two-layer Bayesian networks (BNs) with large domains.

Single-site dynamics are canonical Markov chain based algorithms for sampling from high-dimensional distributions, such as the Gibbs distributions of graphical models. We introduce a simple and generic parallel algorithm that faithfully simulates single-site dynamics. Under a much relaxed, asymptotic variant of the ℓ p -Dobrushin’s condition—where the Dobrushin’s influence matrix has a bounded ℓ p -induced operator norm for an arbitrary p ∈ [1, ∞]—our algorithm simulates N steps of single-site updates within a parallel depth of O ( N / n +log n ) on Õ( m ) processors, where n is the number of sites and m is the size of the graphical model. For Boolean-valued random variables, if the ℓ p -Dobrushin’s condition holds—specifically, if the ℓ p -induced operator norm of the Dobrushin’s influence matrix is less than 1—the parallel depth can be further reduced to O (log N + log n ), achieving an exponential speedup. These results suggest that single-site dynamics with near-linear mixing times can be parallelized into RNC sampling algorithms, independent of the maximum degree of the underlying graphical model, as long as the Dobrushin influence matrix maintains a bounded operator norm. We show the effectiveness of this approach with RNC samplers for the hardcore and Ising models within their uniqueness regimes, as well as an RNC SAT sampler for satisfying solutions of conjunctive normal form formulas in a local lemma regime. Furthermore, by employing non-adaptive simulated annealing, these RNC samplers can be transformed into RNC algorithms for approximate counting.

On MAX–SAT with cardinality constraint

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.

Backdoors for SAT, proposed by Williams et al. in 2003, are the sets of variables, the instantiation of which vastly simplifies the resulting subproblem. The focus of the present paper are ρ-backdoors — the probabilistic generalization of Strong Backdoor Sets. Unlike most kinds of backdoors, small ρ-backdoors with ρ > 0 are relatively easy to find and they can be found in many formulas. In the theoretical part of the paper, we show that there exists a connection between ρ-backdoors and the conflict information generated by CDCL SAT solvers. On the one hand, any set of variables appearing in some learnt clauses can be viewed as a ρ-backdoor (with ρ > 0) with respect to the Unit Propagation (UP) rule. On the other hand, surprisingly, ρ-backdoors can often be used to generate logical entailments of a formula, which can be viewed as learnt clauses, and we present several techniques and algorithms, that can be used to derive such clauses. We also show that a ρ-backdoor with ρ > 0 can be considered as a partial unsatisfiability certificate for a CNF formula as it proves that the formula is false for the fraction of at least ρ of all possible assignments. Therefore, from the practical viewpoint, finding ρ-backdoors with ρ close to 1 makes sense. To evaluate the proposed techniques, we implemented a proof-of-concept prototype, that interleaves the backdoor-based techniques with standard CDCL solving, and evaluated it on a variety of challenging benchmarks. The results of the experiments show that the proposed technique makes it possible to speed up the SAT solving for many hard SAT instances both from SAT Competitions and of industrial origin.

Polynomial calculus for optimization

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.

Parameterization above (or below) a guarantee is a successful concept in parameterized algorithms. The idea is that many computational problems admit “natural” guarantees bringing to algorithmic questions whether a better solution (above the guarantee) could be obtained efficiently. For example, for every boolean CNF formula on m clauses, there is an assignment that satisfies at least m/2 clauses. How difficult is it to decide whether there is an assignment satisfying more than m/2+k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m/2 +k$$\\end{document} clauses? Or, if an n-vertex graph has a perfect matching, then its vertex cover is at least n/2. Is there a vertex cover of size at least n/2+k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n/2 +k$$\\end{document} for some k≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 1$$\\end{document} and how difficult is it to find such a vertex cover? The above guarantee paradigm has led to several exciting discoveries in the areas of parameterized algorithms and kernelization. We argue that this paradigm could bring forth fresh perspectives on well-studied problems in approximation algorithms. Our example is the longest cycle problem. One of the oldest results in extremal combinatorics is the celebrated Dirac’s theorem from 1952. Dirac’s theorem provides the following guarantee on the length of the longest cycle: for every 2-connected n-vertex graph G with minimum degree δ(G)≤n/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta (G)\\le n/2$$\\end{document}, the length of a longest cycle L is at least 2δ(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\delta (G)$$\\end{document}. Thus the “essential” part in finding the longest cycle is in approximating the “offset” k=L-2δ(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k = L - 2 \\delta (G)$$\\end{document}. The main result of this paper is the above-guarantee approximation theorem for k. Informally, the theorem says that approximating the offset k is not harder than approximating the total length L of a cycle. In other words, for any (reasonably well-behaved) function f, a polynomial time algorithm constructing a cycle of length f(L) in an undirected graph with a cycle of length L, yields a polynomial time algorithm constructing a cycle of length 2δ(G)+Ω(f(k))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\delta (G)+\\Omega (f(k))$$\\end{document}.

We introduce backdoor DNFs, as a tool to measure the theoretical hardness of CNF formulas. Like backdoor sets and backdoor trees, backdoor DNFs are defined relative to a tractable class of CNF formulas. Each conjunctive term of a backdoor DNF defines a partial assignment that moves the input CNF formula into the base class. Backdoor DNFs are more expressive and potentially smaller than their predecessors backdoor sets and backdoor trees. We establish the fixed-parameter tractability of the backdoor DNF detection problem. Our results hold for the fundamental base classes Horn and 2CNF, and their combination. We complement our theoretical findings by an empirical study. Our experiments show that backdoor DNFs provide a significant improvement over their predecessors.

A CNF formula with each clause of length k and each variable occurring 4s times, where positive occurrences are 3s and negative occurrences are s, is a regular (3s + s, k)-CNF formula (F3s+s,k formula). The random regular exact (3s + s, k)-SAT problem is whether there exists a set of Boolean variable assignments such that exactly one literal is true for each clause in the F3s+s,k formula. By introducing a random instance generation model, the satisfiability phase transition of the solution is analyzed by using the first moment method, the second moment method, and the small subgraph conditioning method, which gives the phase transition point s* of the random regular exact (3s + s, k)-SAT problem for k≥3. When s < s*, F3s+s,k formula is satisfiable with high probability; when s > s*, F3s+s,k formula is unsatisfiable with high probability. Finally, through the experimental verification, the results show that the theoretical proofs are consistent with the experimental results.

This paper integrates weak decomposable negation normal form (wDNNF) circuits, introduced by Akshay et al. in 2018, into the knowledge compilation map. This circuit type generalises decomposable negation normal form (DNNF) circuits in such a way that they allow a restricted form of sharing variables among the inputs of a conjunction node. We show that wDNNF circuits have the same properties as DNNF circuits regarding the queries and transformations presented in the knowledge compilation map, whilst being strictly more succinct than DNNF circuits (that is, they can represent Boolean functions compactly). We also present and evaluate a knowledge compiler, called Bella, for converting CNF formulae into wDNNF circuits. Our experiments demonstrate that wDNNF circuits are suitable for configuration instances.

One question which has occupied mathematicians, engineers and scientists of this age is the problem of Boolean satisfiability (SAT) which asks whether a given CNF formula has at least a satisfying assignment. The goal of this work is to present a novel, efficient and general method of deciding SAT and finding all the satisfying assignments of CNF formulas. This method is powerful as it employs resolution identities in transforming clauses without a particular variable into clauses with that variable.

Exact satisfiability and phase transition analysis of the regular (k, d)-CNF formula

For several decades, much effort has been put into identifying classes of CNF formulas whose satisfiability can be decided in polynomial time. Classic results are the linear-time tractability of Horn formulas (Aspvall, Plass, and Tarjan, 1979) and Krom (i.e., 2CNF) formulas (Dowling and Gallier, 1984). Backdoors, introduced by Williams, Gomes and Selman (2003), gradually extend such a tractable class to all formulas of bounded distance to the class. Backdoor size provides a natural but rather crude distance measure between a formula and a tractable class. Backdoor depth, introduced by Mählmann, Siebertz, and Vigny (2021), is a more refined distance measure, which admits the utilization of different backdoor variables in parallel. We propose FPT approximation algorithms to compute backdoor depth into the classes Horn and Krom. This leads to a linear-time algorithm for deciding the satisfiability of formulas of bounded backdoor depth into these classes.

Hitting formulas, introduced by Iwama, are an unusual class of propositional CNF formulas. Not only is their satisfiability decidable in polynomial time, but even their models can be counted in closed form. This stands in stark contrast with other polynomial-time decidable classes, which usually have algorithms based on backtracking and resolution and for which model counting remains hard, like 2-SAT and Horn-SAT. However, those resolution-based algorithms usually easily imply an upper bound on resolution complexity, which is missing for hitting formulas. Are hitting formulas hard for resolution? In this paper we take the first steps towards answering this question. We show that the resolution complexity of hitting formulas is dominated by so-called irreducible hitting formulas, first studied by Kullmann and Zhao, that cannot be composed of smaller hitting formulas. However, by definition, large irreducible unsatisfiable hitting formulas are difficult to construct; it is not even known whether infinitely many exist. Building upon our theoretical results, we implement an efficient algorithm on top of the Nauty software package to enumerate all irreducible unsatisfiable hitting formulas with up to 14 clauses. We also determine the exact resolution complexity of the generated hitting formulas with up to 13 clauses by extending a known SAT encoding for our purposes. Our experimental results suggest that hitting formulas are indeed hard for resolution.