Hans van Maaren

A two-phase algorithm for solving a class of hard satisfiability problemsfn1fn1Supported by the Dutch Organization for Scientific Research (NWO) under grant SION 612-33-001.

The DIMACS suite of satisfiability (SAT) benchmarks contains a set of instances that are very hard for existing algorithms. These instances arise from learning the parity function on 32bits. In this paper we develop a two-phase algorithm that is capable of solving these instances. In the first phase, a polynomially solvable subproblem is identified and solved. Using the solution to this problem, we can considerably restrict the size of the search space in the second phase of the algorithm, which is an extension of the well-known Davis-Putnam-Logemann-Loveland algorithm. We conclude with reporting on our computational results on the parity instances.

March_eq: implementing additional reasoning into an efficient look-ahead SAT solver

This paper discusses several techniques to make the look- ahead architecture for satisfiability (Sat) solvers more competitive. Our contribution consists of reduction of the computational costs to perform look-ahead and a cheap integration of both equivalence reasoning and local learning. Most proposed techniques are illustrated with experimental results of their implementation in our solver march_eq.

Finding guaranteed MUSes fast

We introduce an algorithm for finding a minimal unsatisfiable subset (MUS) of a CNF formula. We have implemented and evaluated the algorithm and found that its performance is very competitive on a wide range of benchmarks, including both formulas that are close to minimal unsatisfiable and formulas containing MUSes that are only a small fraction of the formula size. In our simple but effective algorithm we associate assignments with clauses. The notion of associated assignment has emerged from our work on a Brouwers fixed point approximation algorithm applied to satisfiability. There, clauses are regarded to be entities that order the set of assignments and that can select an assignment to be associated with them, resulting in a Pareto optimal agreement. In this presentation we abandon all terminology from this theory which is superfluous with respect to the recent objective and make the paper self contained.

Aligning CNF- and equivalence-reasoning

Structural logical formulas sometimes yield a substantial fraction of so called equivalence clauses after translation to CNF. Probably the best known example of this is the parity-family. Large instances of such CNF formulas cannot be solved in reasonable time if no detection of, and extra reasoning with, these clauses is incorporated. That is, in solving these formulas, there is a more or less separate algorithmic device dealing with the equivalence clauses, called equivalence reasoning, and another dealing with the remaining clauses. In this paper we propose a way to align these two reasoning devices by introducing parameters for which we establish optimal values over a variety of existing benchmarks. We obtain a truly convincing speed-up in solving such formulas with respect to the best solving methods existing so far.

Generation of classes of robust periodic railway timetables

In this paper we discuss the problem of randomly sampling classes of fixed-interval railway timetables from a so-called timetable structure. Using a standard model for the timetable structure, we introduce a natural partitioning of the set of feasible timetables into classes. We then define a new probability distribution where the probability of each class depends on the robustness of the timetables in that class. Due to the difficulty of sampling directly from this distribution, we propose a heuristic sampling method and illustrate using practical data that our method indeed favors classes containing robust timetables over others.

A short note on some tractable cases of the satisfiability problem

It is shown that the tractable class of CNF formulas solvable by linear autarkies properly contains the class of q-Horn formulas and that it is incomparable with SLUR.

Elliptic approximations of propositional formulae

Abstract A propositional formula can be approximated by a concave quadratic function. This approximation is obtained as a second-order Taylor expansion of a concave smooth model. It is shown that in the 3-SAT case, the involved parameters can be set to such values that yield optimal discriminative properties. Two concentric (generally elliptic) quadratic convex regions are established, the inner one containing only satisfiable assignments and the outer one excluding the average non-satisfiable assignment and including all satisfiable assignments.

Dynamic Symmetry Breaking by Simulating Zykov Contraction

We present a new method to break symmetry in graph coloring problems. While most alternative techniques add symmetry breaking predicates in a pre-processing step, we developed a learning scheme that translates each encountered conflict into one conflict clause which covers equivalent conflicts arising from any permutation of the colors. Our technique introduces new Boolean variables during the search. For many problems the size of the resolution refutation can be significantly reduced by this technique. Although this is shown for various hand-made refutations, it is rarely used in practice, because it is hard to determine which variables to introduce defining useful predicates. In case of graph coloring, the reason for each conflicting coloring can be expressed as a node in the Zykov-tree, that stems from merging some vertices and adding some edges. So, we focus on variables that represent the Boolean expression that two vertices can be merged (if set to true), or that an edge can be placed between them (if set to false). Further, our algorithm reduces the number of introduced variables by reusing them. We implemented our technique in the state-of-the-art solver minisat. It is competitive with alternative SAT based techniques for graph coloring problems. Moreover, our technique can be used on top of other symmetry breaking techniques. In fact, combined with adding symmetry breaking predicates, huge performance gains are realized.

Solving satisfiability problems using elliptic approximations - effective branching rules

Abstract An elliptic approximation of 3SAT problems is derived. It is used to derive branching rules for application in a Davis–Putnam–Logemann–Loveland branching & backtracking algorithm. Using the ellipsoid several well-known branching rules are rediscovered, but now they are obtained with a geometrical motivation. In fact, these rules can be considered to be approximations of the new rules we obtain, that make full use of the elliptic structure. These rules are more effective than the ‘old’ branching rules in terms of node counts. Extensive computational results are provided.

Effective incorporation of double look-ahead procedures

We introduce an adaptive algorithm to control the use of the double look-ahead procedure. This procedure sometimes enhances the performance of look-ahead based satisfiability solvers. Current use of this procedure is driven by static heuristics. Experiments show that over a wide variety of instances, different parameter settings result in optimal performance. Moreover, a strategy that yields fast performance on one particular class of instances may cause a significant slowdown on other families. Using a single adaptive strategy, we accomplish performances close to the optimal performances reached by the various static settings. On some families, we clearly outperform even the fastest performance based on static heuristics. This paper provides a description of the algorithm and a comparison with the static strategies. This method is incorporated in march_dl, satz, and kcnfs. Also, the dynamic behavior of the algorithm is illustrated by adaptation plots on various benchmarks.