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19
Satisfiability Solvers
, 2008
"... The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers. Despite the worstcase exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and h ..."
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Cited by 48 (0 self)
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The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers. Despite the worstcase exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and hardware verification [29–31, 228], automatic test pattern generation [138, 221], planning [129, 197], scheduling [103], and even challenging problems from algebra [238]. Annual SAT competitions have led to the development of dozens of clever implementations of such solvers [e.g. 13,
Short proofs may be spacious: An optimal separation of space and length in resolution
, 2008
"... A number of works have looked at the relationship between length and space of resolution proofs. A notorious question has been whether the existence of a short proof implies the existence of a proof that can be verified using limited space. In this paper we resolve the question by answering it negat ..."
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Cited by 22 (10 self)
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A number of works have looked at the relationship between length and space of resolution proofs. A notorious question has been whether the existence of a short proof implies the existence of a proof that can be verified using limited space. In this paper we resolve the question by answering it negatively in the strongest possible way. We show that there are families of 6CNF formulas of size n, for arbitrarily large n, that have resolution proofs of length O(n) but for which any proof requires space Ω(n / log n). This is the strongest asymptotic separation possible since any proof of length O(n) can always be transformed into a proof in space O(n / log n). Our result follows by reducing the space complexity of so called pebbling formulas over a directed acyclic graph to the blackwhite pebbling price of the graph. The proof is somewhat simpler than previous results (in particular, those reported in [Nordström 2006, Nordström and H˚astad 2008]) as it uses a slightly different flavor of pebbling formulas which allows for a rather straightforward reduction of proof space to standard blackwhite pebbling price.
Learning and Inference in WEIGHTED LOGIC WITH APPLICATION TO NATURAL LANGUAGE PROCESSING
, 2008
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PEBBLE GAMES, PROOF COMPLEXITY AND TIMESPACE TRADEOFFS
, 2010
"... Pebble games were extensively studied in the 1970s and 1980s in a number of different contexts. The last decade has seen a revival of interest in pebble games coming from the field of proof complexity. Pebbling has proven to be a useful tool for studying resolutionbased proof systems when compari ..."
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Cited by 17 (6 self)
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Pebble games were extensively studied in the 1970s and 1980s in a number of different contexts. The last decade has seen a revival of interest in pebble games coming from the field of proof complexity. Pebbling has proven to be a useful tool for studying resolutionbased proof systems when comparing the strength of different subsystems, showing bounds on proof space, and establishing sizespace tradeoffs. This is a survey of research in proof complexity drawing on results and tools from pebbling, with a focus on proof space lower bounds and tradeoffs between proof size and proof space.
Latent Class Models for Algorithm Portfolio Methods
"... Different solvers for computationally difficult problems such as satisfiability (SAT) perform best on different instances. Algorithm portfolios exploit this phenomenon by predicting solvers ’ performance on specific problem instances, then shifting computational resources to the solvers that appear ..."
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Cited by 15 (4 self)
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Different solvers for computationally difficult problems such as satisfiability (SAT) perform best on different instances. Algorithm portfolios exploit this phenomenon by predicting solvers ’ performance on specific problem instances, then shifting computational resources to the solvers that appear best suited. This paper develops a new approach to the problem of making such performance predictions: natural generative models of solver behavior. Two are proposed, both following from an assumption that problem instances cluster into latent classes: a mixture of multinomial distributions, and a mixture of Dirichlet compound multinomial distributions. The latter model extends the former to capture burstiness, the tendency of solver outcomes to recur. These models are integrated into an algorithm portfolio architecture and used to run standard SAT solvers on competition benchmarks. This approach is found competitive with the most prominent existing portfolio, SATzilla, which relies on domainspecific, handselected problem features; the latent class models, in contrast, use minimal domain knowledge. Their success suggests that these models can lead to more powerful and more general algorithm portfolio methods.
Towards an Optimal Separation of Space and Length in Resolution
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY
, 2008
"... Most stateoftheart satisfiability algorithms today are variants of the DPLL procedure augmented with clause learning. The main bottleneck for such algorithms, other than the obvious one of time, is the amount of memory used. In the field of proof complexity, the resources of time and memory corre ..."
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Cited by 14 (9 self)
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Most stateoftheart satisfiability algorithms today are variants of the DPLL procedure augmented with clause learning. The main bottleneck for such algorithms, other than the obvious one of time, is the amount of memory used. In the field of proof complexity, the resources of time and memory correspond to the length and space of resolution proofs. There has been a long line of research trying to understand these proof complexity measures, as well as relating them to the width of proofs, i.e., the size of the largest clause in the proof, which has been shown to be intimately connected with both length and space. While strong results have been proven for length and width, our understanding of space is still quite poor. For instance, it has remained open whether the fact that a formula is provable in short length implies that it is also provable in small space (which is the case for length versus width), or whether on the contrary these measures are completely unrelated in the sense that short proofs can be arbitrarily complex with respect to space. In this paper, we present some evidence that the true answer should be that the latter case holds and provide a possible roadmap for how such an optimal separation result could be obtained. We do this by proving a tight bound of Θ ( √ n) on the space needed for socalled pebbling contradictions over pyramid graphs of size n. This yields the first polynomial lower bound on space that is not a consequence of a corresponding lower bound on width, as well as an improvement of the weak separation of space and width in (Nordström 2006) from logarithmic to polynomial. Also, continuing the line of research initiated by (BenSasson 2002) into tradeoffs between different proof complexity measures, we present a simplified proof of the recent lengthspace tradeoff result in (Hertel and Pitassi 2007), and show how our ideas can be used to prove a couple of other exponential tradeoffs in resolution.
Exploiting Inference Rules to Compute Lower Bounds for MAXSAT Solving ∗
"... In this paper we present a general logical framework for (weighted) MAXSAT problem, and study properties of inference rules for branch and bound MAXSAT solver. Several rules, which are not equivalent but Λequivalent, are proposed, and we show that Λequivalent rules are also sound. As an example, ..."
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Cited by 14 (1 self)
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In this paper we present a general logical framework for (weighted) MAXSAT problem, and study properties of inference rules for branch and bound MAXSAT solver. Several rules, which are not equivalent but Λequivalent, are proposed, and we show that Λequivalent rules are also sound. As an example, we show how to exploit inference rules to achieve a new lower bound function for a MAX2SAT solver. Our new function is admissible and consistently better than the wellknown lower bound function. Based on the study of inference rules, we implement an efficient solver and the experimental results demonstrate that our solver outperforms the most efficient solver that has been implemented very recently [Heras and Larrosa, 2006], especially for large instances. 1
CIRCUMSPECT DESCENT PREVAILS IN SOLVING RANDOM CONSTRAINT SATISFACTION PROBLEMS
, 711
"... Abstract. We study the performance of stochastic local search algorithms for random instances of the Ksatisfiability (KSAT) problem. We introduce a new stochastic local search algorithm, ChainSAT, which moves in the energy landscape of a problem instance by never going upwards in energy. ChainSAT ..."
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Cited by 6 (0 self)
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Abstract. We study the performance of stochastic local search algorithms for random instances of the Ksatisfiability (KSAT) problem. We introduce a new stochastic local search algorithm, ChainSAT, which moves in the energy landscape of a problem instance by never going upwards in energy. ChainSAT is a focused algorithm in the sense that it considers only variables occurring in unsatisfied clauses. We show by extensive numerical investigations that ChainSAT and other focused algorithms solve large KSAT instances almost surely in linear time, up to high clausetovariable ratios α; for example, for K = 4 we observe lineartime performance well beyond the recently postulated clustering and condensation transitions in the solution space. The performance of ChainSAT is a surprise given that by design the algorithm gets trapped into the first local energy minimum it encounters, yet no such minima are encountered. We also study the geometry of the solution space as accessed by stochastic local search algorithms. 1.
On the Structure of Industrial SAT Instances
, 2009
"... During this decade, it has been observed that many realworld graphs, like the web and some social and metabolic networks, have a scalefree structure. These graphs are characterized by a big variability in the arity of nodes, that seems to follow a powerlaw distribution. This came as a big surpri ..."
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Cited by 5 (1 self)
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During this decade, it has been observed that many realworld graphs, like the web and some social and metabolic networks, have a scalefree structure. These graphs are characterized by a big variability in the arity of nodes, that seems to follow a powerlaw distribution. This came as a big surprise to researchers steeped in the tradition of classical random networks. SAT instances can also be seen as (bipartite) graphs. In this paper we study many families of industrial SAT instances used in SAT competitions, and show that most of them also present this scalefree structure. On the contrary, random SAT instances, viewed as graphs, are closer to the classical random graph model, where arity of nodes follows a Poisson distribution with small variability. This would explain their distinct nature. We also analyze what happens when we instantiate a fraction of the variables, at random or using some heuristics, and how the scalefree structure is modified by these instantiations. Finally, we study how the structure is modified during the execution of a SAT solver, concluding that the scalefree structure is preserved.
Efficient SAT techniques for absolute encoding of permutation problems: Application to Hamiltonian cycles
 IN: PROCEEDINGS SARA
, 2009
"... We study novel approaches for solving of hard combinatorial problems by translation to Boolean Satisfiability (SAT). Our focus is on combinatorial problems that can be represented as a permutation of n objects, subject to additional constraints. In the case of the Hamiltonian Cycle Problem (HCP), th ..."
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Cited by 4 (1 self)
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We study novel approaches for solving of hard combinatorial problems by translation to Boolean Satisfiability (SAT). Our focus is on combinatorial problems that can be represented as a permutation of n objects, subject to additional constraints. In the case of the Hamiltonian Cycle Problem (HCP), these constraints are that two adjacent nodes in a permutation should also be neighbors in the graph for which we search for a Hamiltonian cycle. We use the absolute SAT encoding of permutations, where for each of the n objects and each of its possible positions in a permutation, a predicate is defined to indicate whether the object is placed in that position. For implementation of this predicate, we compare the direct and logarithmic encodings that have been used previously, against 16 hierarchical parameterizable encodings of which we explore 416 instantiations. We propose the use of enumerative adjacency constraints—that enumerate the possible neighbors of a node in a permutation— instead of, or in addition to the exclusivity adjacency constraints—that exclude impossible neighbors, and that have been applied previously. We study 11 heuristics for efficiently choosing the first node in the Hamiltonian cycle, as well as 8 heuristics for static CNF variable ordering. We achieve at least 4 orders of magnitude average speedup on HCP benchmarks from the phase transition region, relative to the previously used encodings for solving of HCPs via SAT, such that the speedup is increasing with the size of the graphs.