Percolation of satisfiability in finite dimensions

The satisfiability and optimization of finite-dimensional Boolean formulas are studied using percolation theory, rare region arguments, and boundary effects. In contrast with mean-field results, there is no satisfiability transition, though there is a logical connectivity transition. In part of the disconnected phase, rare regions lead to a divergent running time for optimization algorithms. The thermodynamic ground state for the NP-hard two-dimensional maximum-satisfiability problem is typically unique. These results have implications for the computational study of disordered materials.Comment: 4 pages, 4 fig

Balanced Combinations of Solutions in Multi-Objective Optimization

For every list of integers x_1, ..., x_m there is some j such that x_1 + ... + x_j - x_{j+1} - ... - x_m \approx 0. So the list can be nearly balanced and for this we only need one alternation between addition and subtraction. But what if the x_i are k-dimensional integer vectors? Using results from topological degree theory we show that balancing is still possible, now with k alternations. This result is useful in multi-objective optimization, as it allows a polynomial-time computable balance of two alternatives with conflicting costs. The application to two multi-objective optimization problems yields the following results: - A randomized 1/2-approximation for multi-objective maximum asymmetric traveling salesman, which improves and simplifies the best known approximation for this problem. - A deterministic 1/2-approximation for multi-objective maximum weighted satisfiability

Approaching the ground states of the random maximum two-satisfiability problem by a greedy single-spin flipping process

In this brief report we explore the energy landscapes of two spin glass models using a greedy single-spin flipping process, {\tt Gmax}. The ground-state energy density of the random maximum two-satisfiability problem is efficiently approached by {\tt Gmax}. The achieved energy density $e(t)$ decreases with the evolution time $t$ as $e(t)-e(\infty)=h (\log_{10} t)^{-z}$ with a small prefactor $h$ and a scaling coefficient $z > 1$, indicating an energy landscape with deep and rugged funnel-shape regions. For the $\pm J$ Viana-Bray spin glass model, however, the greedy single-spin dynamics quickly gets trapped to a local minimal region of the energy landscape.Comment: 5 pages with 4 figures included. Accepted for publication in Physical Review E as a brief repor

Fuzzy Maximum Satisfiability

In this paper, we extend the Maximum Satisfiability (MaxSAT) problem to {\L}ukasiewicz logic. The MaxSAT problem for a set of formulae {\Phi} is the problem of finding an assignment to the variables in {\Phi} that satisfies the maximum number of formulae. Three possible solutions (encodings) are proposed to the new problem: (1) Disjunctive Linear Relations (DLRs), (2) Mixed Integer Linear Programming (MILP) and (3) Weighted Constraint Satisfaction Problem (WCSP). Like its Boolean counterpart, the extended fuzzy MaxSAT will have numerous applications in optimization problems that involve vagueness.Comment: 10 page

Pairs of SAT Assignment in Random Boolean Formulae

We investigate geometrical properties of the random K-satisfiability problem using the notion of x-satisfiability: a formula is x-satisfiable if there exist two SAT assignments differing in Nx variables. We show the existence of a sharp threshold for this property as a function of the clause density. For large enough K, we prove that there exists a region of clause density, below the satisfiability threshold, where the landscape of Hamming distances between SAT assignments experiences a gap: pairs of SAT-assignments exist at small x, and around x=1/2, but they donot exist at intermediate values of x. This result is consistent with the clustering scenario which is at the heart of the recent heuristic analysis of satisfiability using statistical physics analysis (the cavity method), and its algorithmic counterpart (the survey propagation algorithm). The method uses elementary probabilistic arguments (first and second moment methods), and might be useful in other problems of computational and physical interest where similar phenomena appear

A New General Method to Generate Random Modal Formulae for Testing Decision Procedures

The recent emergence of heavily-optimized modal decision procedures has highlighted the key role of empirical testing in this domain. Unfortunately, the introduction of extensive empirical tests for modal logics is recent, and so far none of the proposed test generators is very satisfactory. To cope with this fact, we present a new random generation method that provides benefits over previous methods for generating empirical tests. It fixes and much generalizes one of the best-known methods, the random CNF_[]m test, allowing for generating a much wider variety of problems, covering in principle the whole input space. Our new method produces much more suitable test sets for the current generation of modal decision procedures. We analyze the features of the new method by means of an extensive collection of empirical tests

The Satisfiability Threshold for a Seemingly Intractable Random Constraint Satisfaction Problem

We determine the exact threshold of satisfiability for random instances of a particular NP-complete constraint satisfaction problem (CSP). This is the first random CSP model for which we have determined a precise linear satisfiability threshold, and for which random instances with density near that threshold appear to be computationally difficult. More formally, it is the first random CSP model for which the satisfiability threshold is known and which shares the following characteristics with random k-SAT for k >= 3. The problem is NP-complete, the satisfiability threshold occurs when there is a linear number of clauses, and a uniformly random instance with a linear number of clauses asymptotically almost surely has exponential resolution complexity.Comment: This is the long version of a paper that will be published in the SIAM Journal on Discrete Mathematics. This long version includes an appendix and a computer program. The contents of the paper are unchanged in the latest version. The format of the arxiv submission was changed so that the computer program will appear as an ancillary file. Some comments in the computer program were update