Phase-transition in random SAT formulas is one of the properties best studied by theoretical SAT ... more Phase-transition in random SAT formulas is one of the properties best studied by theoretical SAT researchers. There exists a constant rk depending on k such that, if we choose randomly a k-SAT formula over n variables and m clauses, it will be satisfiable with high probability, if m/n < r, and unsatisfiable, otherwise. However, this criterion is useless in practice, because real-world or industrial instances have some properties not shown in random formulas. In the last years, several models of realistic random formulas have been proposed. Here we discuss about the phase transition in these models, and about the size of unsatisfiability proofs. We observe that in these models, like in real-world formulas, there is not a sharp phase transition, the transition occurs for smaller values of r, and the proofs on unsatisfiable formulas are smaller than in the classical random model. We also discuss about the strategies used by modern SAT solvers to exploit these properties.
We focus on the random generation of SAT instances that have properties similar to real-world ins... more We focus on the random generation of SAT instances that have properties similar to real-world instances. It is known that many industrial instances, even with a great number of variables, can be solved by a clever solver in a reasonable amount of time. This is not possible, in general, with classical randomly generated instances. We provide a different generation model of SAT instances, called scale-free random SAT instances. This is based on the use of a non-uniform probability distribution P(i)∼i−β to select variable i, where β is a parameter of the model. This results in formulas where the number of occurrences k of variables follows a power-law distribution P(k)∼k−δ, where δ=1+1/β. This property has been observed in most real-world SAT instances. For β=0, our model extends classical random SAT instances. We prove the existence of a SAT–UNSAT phase transition phenomenon for scale-free random 2-SAT instances with β<1/2 when the clause/variable ratio is m/n=1−2β(1−β)2. We also p...
The Maximum Satisfiability (MaxSAT) problem is an optimization variant of the Satisfiability (SAT... more The Maximum Satisfiability (MaxSAT) problem is an optimization variant of the Satisfiability (SAT) problem. Several combinatorial optimization problems can be translated into a MaxSAT formula. Among exact MaxSAT algorithms, SAT-based MaxSAT algorithms are the best performing approaches for real-world problems. We have extended the WPM2 algorithm by adding several improvements. In particular, we show that by solving some subproblems of the original MaxSAT instance we can dramatically increase the efficiency of WPM2. This led WPM2 to achieve the best overall results at the international MaxSAT Evaluation 2013 (MSE13) on industrial instances. Then, we present additional techniques and heuristics to further exploit the information retrieved from the resolution of the subproblems.We exhaustively analyze the impact of each improvement what contributes to our understanding of why they work. This architecture allows to convert exact algorithms into efficient incomplete algorithms. The resul...
Phase-transition in random SAT formulas is one of the properties best studied by theoretical SAT ... more Phase-transition in random SAT formulas is one of the properties best studied by theoretical SAT researchers. There exists a constant rk depending on k such that, if we choose randomly a k-SAT formula over n variables and m clauses, it will be satisfiable with high probability, if m/n < r, and unsatisfiable, otherwise. However, this criterion is useless in practice, because real-world or industrial instances have some properties not shown in random formulas. In the last years, several models of realistic random formulas have been proposed. Here we discuss about the phase transition in these models, and about the size of unsatisfiability proofs. We observe that in these models, like in real-world formulas, there is not a sharp phase transition, the transition occurs for smaller values of r, and the proofs on unsatisfiable formulas are smaller than in the classical random model. We also discuss about the strategies used by modern SAT solvers to exploit these properties.
We focus on the random generation of SAT instances that have properties similar to real-world ins... more We focus on the random generation of SAT instances that have properties similar to real-world instances. It is known that many industrial instances, even with a great number of variables, can be solved by a clever solver in a reasonable amount of time. This is not possible, in general, with classical randomly generated instances. We provide a different generation model of SAT instances, called scale-free random SAT instances. This is based on the use of a non-uniform probability distribution P(i)∼i−β to select variable i, where β is a parameter of the model. This results in formulas where the number of occurrences k of variables follows a power-law distribution P(k)∼k−δ, where δ=1+1/β. This property has been observed in most real-world SAT instances. For β=0, our model extends classical random SAT instances. We prove the existence of a SAT–UNSAT phase transition phenomenon for scale-free random 2-SAT instances with β<1/2 when the clause/variable ratio is m/n=1−2β(1−β)2. We also p...
The Maximum Satisfiability (MaxSAT) problem is an optimization variant of the Satisfiability (SAT... more The Maximum Satisfiability (MaxSAT) problem is an optimization variant of the Satisfiability (SAT) problem. Several combinatorial optimization problems can be translated into a MaxSAT formula. Among exact MaxSAT algorithms, SAT-based MaxSAT algorithms are the best performing approaches for real-world problems. We have extended the WPM2 algorithm by adding several improvements. In particular, we show that by solving some subproblems of the original MaxSAT instance we can dramatically increase the efficiency of WPM2. This led WPM2 to achieve the best overall results at the international MaxSAT Evaluation 2013 (MSE13) on industrial instances. Then, we present additional techniques and heuristics to further exploit the information retrieved from the resolution of the subproblems.We exhaustively analyze the impact of each improvement what contributes to our understanding of why they work. This architecture allows to convert exact algorithms into efficient incomplete algorithms. The resul...
We investigate a problem that lies at the intersection of three research areas, namely Automated ... more We investigate a problem that lies at the intersection of three research areas, namely Automated Negotiation, Vehicle Routing, and Multi-Objective Optimization. Specifically, we investigate the scenario that multiple competing logistics companies aim to cooperate by delivering truck loads for one another, in order to improve efficiency and reduce the distance they drive. In order to do so, these companies need to find ways to exchange their truck loads such that each of them individually benefits. We present a new heuristic algorithm that, given one set of orders to deliver for each company, tries to find the set of all order-exchanges that are Pareto-optimal and individually rational. Furthermore, we present experiments based on real-world test data from two major logistics companies, which show that our algorithm is able to find hundreds of solutions in a matter of minutes.
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Papers by Jordi Levy