Generalized Totalizer Encoding for Pseudo-Boolean Constraints

Pseudo-Boolean constraints, also known as 0-1 Integer Linear Constraints, are used to model many real-world problems. A common approach to solve these constraints is to encode them into a SAT formula. The runtime of the SAT solver on such formula is sensitive to the manner in which the given pseudo-Boolean constraints are encoded. In this paper, we propose generalized Totalizer encoding (GTE), which is an arc-consistency preserving extension of the Totalizer encoding to pseudo-Boolean constraints. Unlike some other encodings, the number of auxiliary variables required for GTE does not depend on the magnitudes of the coefficients. Instead, it depends on the number of distinct combinations of these coefficients. We show the superiority of GTE with respect to other encodings when large pseudo-Boolean constraints have low number of distinct coefficients. Our experimental results also show that GTE remains competitive even when the pseudo-Boolean constraints do not have this characteristic.Comment: 10 pages, 2 figures, 2 tables. To be published in 21st International Conference on Principles and Practice of Constraint Programming 201

Solving Linux Upgradeability Problems Using Boolean Optimization

Managing the software complexity of package-based systems can be regarded as one of the main challenges in software architectures. Upgrades are required on a short time basis and systems are expected to be reliable and consistent after that. For each package in the system, a set of dependencies and a set of conflicts have to be taken into account. Although this problem is computationally hard to solve, efficient tools are required. In the best scenario, the solutions provided should also be optimal in order to better fulfill users requirements and expectations. This paper describes two different tools, both based on Boolean satisfiability (SAT), for solving Linux upgradeability problems. The problem instances used in the evaluation of these tools were mainly obtained from real environments, and are subject to two different lexicographic optimization criteria. The developed tools can provide optimal solutions for many of the instances, but a few challenges remain. Moreover, it is our understanding that this problem has many similarities with other configuration problems, and therefore the same techniques can be used in other domains.Comment: In Proceedings LoCoCo 2010, arXiv:1007.083

On SAT representations of XOR constraints

We study the representation of systems S of linear equations over the two-element field (aka xor- or parity-constraints) via conjunctive normal forms F (boolean clause-sets). First we consider the problem of finding an "arc-consistent" representation ("AC"), meaning that unit-clause propagation will fix all forced assignments for all possible instantiations of the xor-variables. Our main negative result is that there is no polysize AC-representation in general. On the positive side we show that finding such an AC-representation is fixed-parameter tractable (fpt) in the number of equations. Then we turn to a stronger criterion of representation, namely propagation completeness ("PC") --- while AC only covers the variables of S, now all the variables in F (the variables in S plus auxiliary variables) are considered for PC. We show that the standard translation actually yields a PC representation for one equation, but fails so for two equations (in fact arbitrarily badly). We show that with a more intelligent translation we can also easily compute a translation to PC for two equations. We conjecture that computing a representation in PC is fpt in the number of equations.Comment: 39 pages; 2nd v. improved handling of acyclic systems, free-standing proof of the transformation from AC-representations to monotone circuits, improved wording and literature review; 3rd v. updated literature, strengthened treatment of monotonisation, improved discussions; 4th v. update of literature, discussions and formulations, more details and examples; conference v. to appear LATA 201