George Katsirelos

A compression algorithm for large arity extensional constraints

We present an algorithm for compressing table constraints representing allowed or disallowed tuples. This type of constraint is used for example in configuration problems, where the satisfying tuples are read from a database. The arity of these constraints may be large. A generic GAC algorithm for such a constraint requires time exponential in the arity of the constraint to maintain GAC, but Bessiere and Regin showed in [1] that for the case of allowed tuples, GAC can be enforced in time proportional to the number of allowed tuples, using the algorithm GAC-Schema. We introduce a more compact representation for a set of tuples, which allows a potentially exponential reduction in the space needed to represent the satisfying tuples and exponential reduction in the time needed to enforce GAC. We show that this representation can be constructed from a decision tree that represents the original tuples and demonstrate that it does in practice produce a significantly shorter description of the constraint. We also show that this representation can be efficiently used in existing algorithms and can be used to improve GAC-Schema further. Finally, we show that this method can be used to improve the complexity of enforcing GAC on a table constraint defined in terms of forbidden tuples.

A new framework for computational protein design through cost function network optimization

MOTIVATION The main challenge for structure-based computational protein design (CPD) remains the combinatorial nature of the search space. Even in its simplest fixed-backbone formulation, CPD encompasses a computationally difficult NP-hard problem that prevents the exact exploration of complex systems defining large sequence-conformation spaces. RESULTS We present here a CPD framework, based on cost function network (CFN) solving, a recent exact combinatorial optimization technique, to efficiently handle highly complex combinatorial spaces encountered in various protein design problems. We show that the CFN-based approach is able to solve optimality a variety of complex designs that could often not be solved using a usual CPD-dedicated tool or state-of-the-art exact operations research tools. Beyond the identification of the optimal solution, the global minimum-energy conformation, the CFN-based method is also able to quickly enumerate large ensembles of suboptimal solutions of interest to rationally build experimental enzyme mutant libraries. AVAILABILITY The combined pipeline used to generate energetic models (based on a patched version of the open source solver Osprey 2.0), the conversion to CFN models (based on Perl scripts) and CFN solving (based on the open source solver toulbar2) are all available at http://genoweb.toulouse.inra.fr/~tschiex/CPD

Computational protein design as an optimization problem

Proteins are chains of simple molecules called amino acids. The three-dimensional shape of a protein and its amino acid composition define its biological function. Over millions of years, living organisms have evolved a large catalog of proteins. By exploring the space of possible amino acid sequences, protein engineering aims at similarly designing tailored proteins with specific desirable properties. In Computational Protein Design (CPD), the challenge of identifying a protein that performs a given task is defined as the combinatorial optimization of a complex energy function over amino acid sequences. In this paper, we introduce the CPD problem and some of the main approaches that have been used by structural biologists to solve it, with an emphasis on the exact method embodied in the dead-end elimination/A? algorithm (DEE/A?). The CPD problem is a specific form of binary Cost Function Network (CFN, aka Weighted CSP). We show how DEE algorithms can be incorporated and suitably modified to be maintained during search, at reasonable computational cost. We then evaluate the efficiency of CFN algorithms as implemented in our solver toulbar2, on a set of real CPD instances built in collaboration with structural biologists. The CPD problem can be easily reduced to 0/1 Linear Programming, 0/1 Quadratic Programming, 0/1 Quadratic Optimization, Weighted Partial MaxSAT and Graphical Model optimization problems. We compare toulbar2 with these different approaches using a variety of solvers. We observe tremendous differences in the difficulty that each approach has on these instances. Overall, the CFN approach shows the best efficiency on these problems, improving by several orders of magnitude against the exact DEE/A? approach. The introduction of dead-end elimination before or during search allows to further improve these results.

Resolution and parallelizability: barriers to the efficient parallelization of SAT solvers

Recent attempts to create versions of Satisfiability (SAT) solvers that exploit parallel hardware and information sharing have met with limited success. In fact, the most successful parallel solvers in recent competitions were based on portfolio approaches with little to no exchange of information between processors. This experience contradicts the apparent parallelizability of exploring a combinatorial search space. We present evidence that this discrepancy can be explained by studying SAT solvers through a proof complexity lens, as resolution refutation engines. Starting with the observation that a recently studied measure of resolution proofs, namely depth, provides a (weak) upper bound to the best possible speedup achievable by such solvers, we empirically show the existence of bottlenecks to parallelizability that resolution proofs typically generated by SAT solvers exhibit. Further, we propose a new measure of parallelizability based on the best-case makespan of an offline resource constrained scheduling problem. This measure explicitly accounts for a bounded number of parallel processors and appears to empirically correlate with parallel speedups observed in practice. Our findings suggest that efficient parallelization of SAT solvers is not simply a matter of designing the right clause sharing heuristics; even in the best case, it can be -- and indeed is -- hindered by the structure of the resolution proofs current SAT solvers typically produce.

Combining Symmetry Breaking and Global Constraints

We propose a new family of constraints which combine together lexicographical ordering constraints for symmetry breaking with other common global constraints. We give a general purpose propagator for this family of constraints, and show how to improve its complexity by exploiting properties of the included global constraints.

The weighted Grammar constraint

We introduce the WeightedGrammar constraint and propose propagation algorithms based on the CYK parser and the Earley parser. We show that the traces of these algorithms can be encoded as a weighted negation normal form (WNNF), a generalization of NNF that allows nodes to carry weights. Based on this connection, we prove the correctness and complexity of these algorithms. Specifically, these algorithms enforce domain consistency on the WeightedGrammar constraint in time O(n3). Further, we propose that the WNNF constraint can be decomposed into a set of primitive arithmetic constraint without hindering propagation.

Complexity of and algorithms for the manipulation of Borda, Nanson's and Baldwin's voting rules

We investigate manipulation of the Borda voting rule, as well as two elimination style voting rules, Nansons and Baldwins voting rules, which are based on Borda voting. We argue that these rules have a number of desirable computational properties. For unweighted Borda voting, we prove that it is NP-hard for a coalition of two manipulators to compute a manipulation. This resolves a long-standing open problem in the computational complexity of manipulating common voting rules. We prove that manipulation of Baldwins and Nansons rules is computationally more difficult than manipulation of Borda, as it is NP-hard for a single manipulator to compute a manipulation. In addition, for Baldwins and Nansons rules with weighted votes, we prove that it is NP-hard for a coalition of manipulators to compute a manipulation with a small number of candidates.Because of these NP-hardness results, we compute manipulations using heuristic algorithms that attempt to minimise the number of manipulators. We propose several new heuristic methods. Experiments show that these methods significantly outperform the previously best known heuristic method for the Borda rule. Our results suggest that, whilst computing a manipulation of the Borda rule is NP-hard, computational complexity may provide only a weak barrier against manipulation in practice. In contrast to the Borda rule, our experiments with Baldwins and Nansons rules demonstrate that both of them are often more difficult to manipulate in practice. These results suggest that elimination style voting rules deserve further study.

Multi-language evaluation of exact solvers in graphical model discrete optimization

By representing the constraints and objective function in factorized form, graphical models can concisely define various NP-hard optimization problems. They are therefore extensively used in several areas of computer science and artificial intelligence. Graphical models can be deterministic or stochastic, optimize a sum or product of local functions, defining a joint cost or probability distribution. Simple transformations exist between these two types of models, but also with MaxSAT or linear programming. In this paper, we report on a large comparison of exact solvers which are all state-of-the-art for their own target language. These solvers are all evaluated on deterministic and probabilistic graphical models coming from the Probabilistic Inference Challenge 2011, the Computer Vision and Pattern Recognition OpenGM2 benchmark, the Weighted Partial MaxSAT Evaluation 2013, the MaxCSP 2008 Competition, the MiniZinc Challenge 2012 & 2013, and the CFLib (a library of Cost Function Networks). All 3026 instances are made publicly available in five different formats and seven formulations. To our knowledge, this is the first evaluation that encompasses such a large set of related NP-complete optimization frameworks, despite their tight connections. The results show that a small number of evaluated solvers are able to perform well on multiple areas. By exploiting the variability and complementarity of solver performances, we show that a simple portfolio approach can be very effective. This portfolio won the last UAI Evaluation 2014 (MAP task).

Using Minimal Correction Sets to More Efficiently Compute Minimal Unsatisfiable Sets

An unsatisfiable set is a set of formulas whose conjunction is unsatisfiable. Every unsatisfiable set can be corrected, i.e., made satisfiable, by removing a subset of its members. The subset whose removal yields satisfiability is called a correction subset. Given an unsatisfiable set \({\mathcal {F}} \) there is a well known hitting set duality between the unsatisfiable subsets of \({\mathcal {F}} \) and the correction subsets of \({\mathcal {F}} \): every unsatisfiable subset hits (has a non-empty intersection with) every correction subset, and, dually, every correction subset hits every unsatisfiable subset. An important problem with many applications in practice is to find a minimal unsatisfiable subset (mus) of \({\mathcal {F}} \), i.e., an unsatisfiable subset all of whose proper subsets are satisfiable. A number of algorithms for this important problem have been proposed. In this paper we present new algorithms for finding a single mus and for finding all muses. Our algorithms exploit in a new way the duality between correction subsets and unsatisfiable subsets. We show that our algorithms advance the state of the art, enabling more effective computation of muses.

The seqbin constraint revisited

We revisit the SeqBin constraint [1]. This meta-constraint subsumes a number of important global constraints like Change [2], Smooth [3] and IncreasingNValue [4]. We show that the previously proposed filtering algorithm for SeqBin has two drawbacks even under strong restrictions: it does not detect bounds disentailment and it is not idempotent. We identify the cause for these problems, and propose a new propagator that overcomes both issues. Our algorithm is based on a connection to the problem of finding a path of a given cost in a restricted n-partite graph. Our propagator enforces domain consistency in O(nd2) and, for special cases of SeqBin that include Change, Smooth and IncreasingNValue in O(nd) time.