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In this paper, we study the computational complexity of the Maximum Satisfiability problem in terms of the length L of a given formula. We present an algorithm with running time O(1.0927^L), hence, improving the previously known best upper bound O(1.1058^L) developed more than 20 years ago by Bansal and Raman. Theoretically speaking, our algorithm increases the length of solvable formulas by 13.3% (compare this to the recent breakthrough result for Maximum Satisfiability problem with respect to the number of clauses by Xu et al. in 2019 giving a 7.5% improvement). Besides, we propose a significantly simpler algorithm with running time O(1.1049^L). The algorithm outperforms Bansal's and Raman's algorithm in simplicity and running time.
We study the setting in which a mobile searcher must locate a hidden target in a bounded or unbounded search domain, with no information about the hider's position. In particular, we consider online search, in which the performance of the search strategy is evaluated by its worst case competitive ratio. We introduce a multi-criteria search problem in which the searcher has a budget on its allotted search time, and the objective is to design strategies that are competitively efficient, respect the budget, and maximize the total searched ground. We give analytically optimal strategies for the line and the star domains, and efficient heuristics for general networks.
Given an unsatisfiable set of constraints F, a maximal satisfiable subset (MSS) is a maximal subset of constraints C ⊆ F such that C is satisfiable. Over the past two decades, the steady improvement in runtime performance of algorithms for finding MSS has led to an increased adoption of MSS-based techniques in wide variety of domains. Motivated by the progress in finding an MSS, the past decade has witnessed a surge of interest in design of algorithmic techniques to enumerate all the MSSes, which has subsequently led to discovery of new applications utilizing enumeration of MSSes. The development of techniques for finding and enumeration of MSSes mirrors a similar phenomenon of finding and enumeration of SAT solutions in the early 2000s, which subsequently motivated design of algorithmic techniques for model counting. In a similar spirit, we undertake study to investigate the feasibility of MSS counting techniques. In particular, the focus point of our investigation is to answer whether one can design efficient MSS counting techniques that do not rely on explicit MSS enumeration. The primary contribution of this work is an affirmative answer to the above question. Our tool, CountMSS, uses a novel architecture of a wrapper W and a remainder R such that the desired MSS count can be expressed as |W| − |R|. CountMSS relies on the advances in projected model counting to efficiently compute |W| and |R|. Our empirical evaluation demonstrates that CountMSS is able to scale to instances clearly beyond the reach of enumeration-based techniques.
Many practical applications require the solution of numerically challenging linear programs (LPs) and mixed integer programs (MIPs). Scaling is a widely used preconditioning technique that aims at reducing the error propagation of the involved linear systems, thereby improving the numerical behavior of the dual simplex algorithm and, consequently, LP-based branch-and-bound. A reliable scaling method often makes the difference whether these problems can be solved correctly or not. In this paper, we investigate the use of machine learning to choose at the beginning of the solution process between two common scaling methods: Standard scaling and Curtis-Reid scaling. The latter often, but not always, leads to a more robust solution process, but may suffer from longer solution times. Rather than training for overall solution time, we propose to use the attention level of a MIP solution process as a learning label. We evaluate the predictive power of a random forest approach and a linear regressor that learns the (square-root of the) difference in attention level. It turns out that the resulting classification not only reduces various types of numerical errors by large margins, but it also improves the performance of the dual simplex algorithm. The learned model has been implemented within the FICO Xpress MIP solver and it is used by default since release 8.9, May 2020, to determine the scaling algorithm Xpress applies before solving an LP or a MIP.
In 1989, computer searches by Lam, Thiel, and Swiercz experimentally resolved Lam's problem from projective geometry—the long-standing problem of determining if a projective plane of order ten exists. Both the original search and an independent verification in 2011 discovered no such projective plane. However, these searches were each performed using highly specialized custom-written code and did not produce nonexistence certificates. In this paper, we resolve Lam's problem by translating the problem into Boolean logic and use satisfiability (SAT) solvers to produce nonexistence certificates that can be verified by a third party. Our work uncovered consistency issues in both previous searches—highlighting the difficulty of relying on special-purpose search code for nonexistence results.
Combinatorial optimization has found applications in numerous fields, from aerospace to transportation planning and economics. The goal is to find an optimal solution among a finite set of possibilities. The well-known challenge one faces with combinatorial optimization is the state-space explosion problem: the number of possibilities grows exponentially with the problem size, which makes solving intractable for large problems. In the last years, deep reinforcement learning (DRL) has shown its promise for designing good heuristics dedicated to solve NP-hard combinatorial optimization problems. However, current approaches have an important shortcoming: they only provide an approximate solution with no systematic ways to improve it or to prove optimality. In another context, constraint programming (CP) is a generic tool to solve combinatorial optimization problems. Based on a complete search procedure, it will always find the optimal solution if we allow an execution time large enough. A critical design choice, that makes CP non-trivial to use in practice, is the branching decision, directing how the search space is explored. In this work, we propose a general and hybrid approach, based on DRL and CP, for solving combinatorial optimization problems. The core of our approach is based on a dynamic programming formulation, that acts as a bridge between both techniques. We experimentally show that our solver is efficient to solve three challenging problems: the traveling salesman problem with time windows, the 4-moments portfolio optimization problem, and the 0-1 knapsack problem. Results obtained show that the framework introduced outperforms the stand-alone RL and CP solutions, while being competitive with industrial solvers.
Recent work introduced XDP and XUP priority functions for best-first bounded-suboptimal search that do not need to perform state re-expansions as long as the search heuristic is consistent. However, that work had several limitations that are rectified here. This paper analyzes the sufficiency and necessity of the conditions used to formulate XDP and XUP. The analysis presents a simpler proof and generalizes the result in three aspects: (1) the priority function no longer has to be differentiable everywhere, (2) the quality of the solution does not have to be bounded by a constant factor, and (3) directed graphs are handled correctly. These results allow the introduction of more priority functions, such as piecewise linear functions, and more variants of bounded-suboptimal search, such as constant suboptimality. Several new priority functions are presented in this paper that, according to empirical results, can significantly outperform existing approaches including XDP.
Stochastic Boolean Satisfiability (SSAT) is a powerful representation for the concise encoding of quantified decision problems with uncertainty. While it shares commonalities with quantified Boolean formula (QBF) satisfiability and has the same PSPACE-complete complexity, SSAT solving tends to be more challenging as it involves expensive model counting, a.k.a. Sharp-SAT. To date, SSAT solvers, especially those imposing no restrictions on quantification levels, remain much lacking. In this paper, we present a new SSAT solver based on the framework of clause selection and cube distribution previously proposed for QBF solving. With model counting integrated and learning techniques strengthened, our solver is general and effective. Experimental results demonstrate the overall superiority of the proposed algorithm in both solving performance and memory usage compared to the state-of-the-art solvers on a number of benchmark formulas.
We show that the CNF satisfiability problem can be solved O^*(1.2226^m) time, where m is the number of clauses in the formula, improving the known upper bounds O^*(1.234^m) given by Yamamoto 15 years ago and O^*(1.239^m) given by Hirsch 22 years ago. By using an amortized technique and careful case analysis, we successfully avoid the bottlenecks in previous algorithms and get the improvement.
The constraint satisfaction problem (CSP) has important applications in computer science and AI. In particular, infinite-domain CSPs have been intensively used in subareas of AI such as spatio-temporal reasoning. Since constraint satisfaction is a computationally hard problem, much work has been devoted to identifying restricted problems that are efficiently solvable. One way of doing this is to restrict the interactions of variables and constraints, and a highly successful approach is to bound the treewidth of the underlying primal graph. Bodirsky & Dalmau [J. Comput. System. Sci., 79(1), 2013] and Huang et al. [Artif. Intell., 195, 2013] proved that CSP(Γ) can be solved in n^(f(w)) time (where n is the size of the instance, w is the treewidth of the primal graph and f is a computable function) for certain classes of constraint languages Γ. We improve this bound to f(w)n^(O(1)), where the function f only depends on the language Γ, for CSPs whose basic relations have the patchwork property. Hence, such problems are fixed-parameter tractable and our algorithm is asymptotically faster than the previous ones. Additionally, our approach is not restricted to binary constraints, so it is applicable to a strictly larger class of problems than that of Huang et al. However, there exist natural problems that are covered by Bodirsky & Dalmau's algorithm but not by ours.
The disjunctive temporal problem (DTP) is an expressive temporal formalism that extends Dechter et al.'s simple temporal problem. The DTP is well studied in the literature and has many important applications. It is known that deciding satisfiability of DTPs is NP-hard and that, in many cases, single-exponential algorithms (running in O(c^n) time) do not exist under the Exponential-Time Hypothesis. The computational hardness makes it worthwhile to identify restricted problems that are efficiently solvable. One way of doing this is to restrict the interactions of variables and constraints. We show that instances of DTP of any arity with integers bounded by poly(n) can be solved in n^{f(w)} time, where n denotes the problem size, w is the treewidth of the incidence graph and f is a computable function; in other words, this problem is in the complexity class XP and it can be solved in polynomial time whenever w is fixed. We complement this result by showing that binary DTPs that only involve the integers 0 and 1 are not fixed-parameter tractable with respect to treewidth, i.e. they do not admit a f(w)poly(n)$ time algorithm for any computable function f, under standard complexity assumptions. For instances with unbounded integers, we show that even binary DTPs parameterized by treewidth cannot be in XP, unless P = NP.
Nonlinear metrics, such as the F1-score, Matthews correlation coefficient, and Fowlkes–Mallows index, are often used to evaluate the performance of machine learning models, in particular, when facing imbalanced datasets that contain more samples of one class than the other. Recent optimal decision tree algorithms have shown remarkable progress in producing trees that are optimal with respect to linear criteria, such as accuracy, but unfortunately nonlinear metrics remain a challenge. To address this gap, we propose a novel algorithm based on bi-objective optimisation, which treats misclassifications of each binary class as a separate objective. We show that, for a large class of metrics, the optimal tree lies on the Pareto frontier. Consequently, we obtain the optimal tree by using our method to generate the set of all nondominated trees. To the best of our knowledge, this is the first method to compute provably optimal decision trees for nonlinear metrics. Our approach leads to a trade-off when compared to optimising linear metrics: the resulting trees may be more desirable according to the given nonlinear metric at the expense of higher runtimes. Nevertheless, the experiments illustrate that runtimes are reasonable for majority of the tested datasets.
Adding constraint support in Machine Learning has the potential to address outstanding issues in data-driven AI systems, such as safety and fairness. Existing approaches typically apply constrained optimization techniques to ML training, enforce constraint satisfaction by adjusting the model design, or use constraints to correct the output. Here, we investigate a different, complementary, strategy based on "teaching" constraint satisfaction to a supervised ML method via the direct use of a state-of-the-art constraint solver: this enables taking advantage of decades of research on constrained optimization with limited effort. In practice, we use a decomposition scheme alternating master steps (in charge of enforcing the constraints) and learner steps (where any supervised ML model and training algorithm can be employed). The process leads to approximate constraint satisfaction in general, and convergence properties are difficult to establish; despite this fact, we found empirically that even a naive setup of our approach performs well on ML tasks with fairness constraints, and on classical datasets with synthetic constraints.
Core-guided techniques have revolutionized Boolean satisfiability approaches to optimization problems (MaxSAT), but the process at the heart of these methods, strengthening bounds on solutions by repeatedly adding cardinality constraints, remains a bottleneck. Cardinality constraints require significant work to be re-encoded to SAT, and SAT solvers are notoriously weak at cardinality reasoning. In this work, we lift core-guided search to pseudo-Boolean (PB) solvers, which deal with more general PB optimization problems and operate natively with cardinality constraints. The cutting planes method used in such solvers allows us to derive stronger cardinality constraints, which yield better updates to solution bounds, and the increased efficiency of objective function reformulation also makes it feasible to switch repeatedly between lower-bounding and upper- bounding search. A thorough evaluation on applied and crafted benchmarks shows that our core-guided PB solver significantly improves on the state of the art in pseudo-Boolean optimization.
Electrical Muscle Stimulation (EMS) has become a popular interaction technology in Human-Computer Interaction; allowing the computer to take direct control of the user's body. To date, however, the explorations have been limited to coarse, toy examples, due to the low resolution of achievable control. To increase this resolution, the EMS needs to increase significantly in complexity - using large numbers of electrodes in complex patterns. The calibration of such a system remains an unsolved challenge. We present a new SAT-based black-box calibration method, which requires no spatial information about muscular or electrode positioning. The method encodes domain knowledge and observations in a constraint model, and uses these to prune the space of feasible control signals. In a simulated environment we find this method can scale reliably to large arrays while requiring only a modest number of trials, and preliminary tests on real hardware show we can effectively calibrate an electrode array in a few minutes.
The dramatic improvements in combinatorial optimization algorithms over the last decades have had a major impact in artificial intelligence, operations research, and beyond, but the output of current state-of-the-art solvers is often hard to verify and is sometimes wrong. For Boolean satisfiability (SAT) solvers proof logging has been introduced as a way to certify correctness, but the methods used seem hard to generalize to stronger paradigms. What is more, even for enhanced SAT techniques such as parity (XOR) reasoning, cardinality detection, and symmetry handling, it has remained beyond reach to design practically efficient proofs in the standard DRAT format. In this work, we show how to instead use pseudo-Boolean inequalities with extension variables to concisely justify XOR reasoning. Our experimental evaluation of a SAT solver integration shows a dramatic decrease in proof logging and verification time compared to existing DRAT methods. Since our method is a strict generalization of DRAT, and readily lends itself to expressing also 0-1 programming and even constraint programming problems, we hope this work points the way towards a unified approach for efficient machine-verifiable proofs for a rich class of combinatorial optimization paradigms.
Mathematical modeling is a standard approach to solve many real-world problems and diversity of solutions is an important issue, emerging in applying solutions obtained from mathematical models to real-world problems. Many studies have been devoted to finding diverse solutions. Baste et al. (Algorithms 2019, IJCAI 2020) recently initiated the study of computing diverse solutions of combinatorial problems from the perspective of fixed-parameter tractability. They considered problems of finding r solutions that maximize some diversity measures (the minimum or sum of the pairwise Hamming distances among them) and gave some fixed-parameter tractable algorithms for the diverse version of several well-known problems, such as Vertex Cover, Feedback Vertex Set, d-Hitting Set}, and problems on bounded-treewidth graphs. In this work, we further investigate the (fixed-parameter) tractability of problems of finding diverse spanning trees, paths, and several subgraphs. In particular, we show that, given a graph G and an integer r, the problem of computing r spanning trees of G maximizing the sum of the pairwise Hamming distances among them can be solved in polynomial time. To the best of the authors' knowledge, this is the first polynomial-time solvable case for finding diverse solutions of unbounded size.
Formal verification of neural networks is an active topic of research, and recent advances have significantly increased the size of the networks that verification tools can handle. However, most methods are designed for verification of an idealized model of the actual network which works over real arithmetic and ignores rounding imprecisions. This idealization is in stark contrast to network quantization, which is a technique that trades numerical precision for computational efficiency and is, therefore, often applied in practice. Neglecting rounding errors of such low-bit quantized neural networks has been shown to lead to wrong conclusions about the network's correctness. Thus, the desired approach for verifying quantized neural networks would be one that takes these rounding errors into account. In this paper, we show that verifying the bit-exact implementation of quantized neural networks with bit-vector specifications is PSPACE-hard, even though verifying idealized real-valued networks and satisfiability of bit-vector specifications alone are each in NP. Furthermore, we explore several practical heuristics toward closing the complexity gap between idealized and bit-exact verification. In particular, we propose three techniques for making SMT-based verification of quantized neural networks more scalable. Our experiments demonstrate that our proposed methods allow a speedup of up to three orders of magnitude over existing approaches.
The research field of stochastic matching has yielded many developments for various applications. In most stochastic matching problems, the probability distributions inherent in the nodes and edges are set a priori, and are not controllable. However, many matching services have options, which we call control variables, that affect the probability distributions and thus what constitutes an optimum matching. Although several methods for optimizing the values of the control variables have been developed, their optimization in consideration of the matching problem is still in its infancy. In this paper, we formulate an optimization problem for determining the values of the control variables so as to maximize the expected value of matching weights. Since this problem involves hard to evaluate objective values and is non-convex, we construct an approximation algorithm via a minimum-cost flow algorithm that can find 3-approximation solutions rapidly. Simulations on real data from a ride-hailing platform and a crowd-sourcing market show that the proposed method can find solutions with high profits of the service provider in practical time.
Machine learning (ML) is ubiquitous in modern life. Since it is being deployed in technologies that affect our privacy and safety, it is often crucial to understand the reasoning behind its decisions, warranting the need for explainable AI. Rule-based models, such as decision trees, decision lists, and decision sets, are conventionally deemed to be the most interpretable. Recent work uses propositional satisfiability (SAT) solving (and its optimization variants) to generate minimum-size decision sets. Motivated by limited practical scalability of these earlier methods, this paper proposes a novel approach to learn minimum-size decision sets by enumerating individual rules of the target decision set independently of each other, and then solving a set cover problem to select a subset of rules. The approach makes use of modern maximum satisfiability and integer linear programming technologies. Experiments on a wide range of publicly available datasets demonstrate the advantage of the new approach over the state of the art in SAT-based decision set learning.
We consider the problem of minimizing a smooth, Lipschitz, convex function over a compact, convex set using sub-zeroth-order oracles: an oracle that outputs the sign of the directional derivative for a given point and a given direction, an oracle that compares the function values for a given pair of points, and an oracle that outputs a noisy function value for a given point. We show that the sample complexity of optimization using these oracles is polynomial in the relevant parameters. The optimization algorithm that we provide for the comparator oracle is the first algorithm with a known rate of convergence that is polynomial in the number of dimensions. We also give an algorithm for the noisy-value oracle that incurs sublinear regret in the number of queries and polynomial regret in the number of dimensions.
Identifying discrete patterns in binary data is an important dimensionality reduction tool in machine learning and data mining. In this paper, we consider the problem of low-rank binary matrix factorisation (BMF) under Boolean arithmetic. Due to the hardness of this problem, most previous attempts rely on heuristic techniques. We formulate the problem as a mixed integer linear program and use a large scale optimisation technique of column generation to solve it without the need of heuristic pattern mining. Our approach focuses on accuracy and on the provision of optimality guarantees. Experimental results on real world datasets demonstrate that our proposed method is effective at producing highly accurate factorisations and improves on the previously available best known results for 15 out of 24 problem instances.
We describe a compilation language of backdoor decomposable monotone circuits (BDMCs) which generalizes several concepts appearing in the literature, e.g. DNNFs and backdoor trees. A C-BDMC sentence is a monotone circuit which satisfies decomposability property (such as in DNNF) in which the inputs (or leaves) are associated with CNF encodings from a given base class C. We consider the class of propagation complete (PC) encodings as a base class and we show that PC-BDMCs are polynomially equivalent to PC encodings. Additionally, we use this to determine the properties of PC-BDMCs and PC encodings with respect to the knowledge compilation map including the list of efficient operations on the languages.
We explore the potential of continuous local search (CLS) in SAT solving by proposing a novel approach for finding a solution of a hybrid system of Boolean constraints. The algorithm is based on CLS combined with belief propagation on binary decision diagrams (BDDs). Our framework accepts all Boolean constraints that admit compact BDDs, including symmetric Boolean constraints and small-coefficient pseudo-Boolean constraints as interesting families. We propose a novel algorithm for efficiently computing the gradient needed by CLS. We study the capabilities and limitations of our versatile CLS solver, GradSAT, by applying it on many benchmark instances. The experimental results indicate that GradSAT can be a useful addition to the portfolio of existing SAT and MaxSAT solvers for solving Boolean satisfiability and optimization problems.
The past two decades have seen the significant improvements of the scalability of practical model counters, which have been quite influential in many applications from artificial intelligence to formal verification. While most of exact counters fall into two categories, search-based and compilation-based, Huang and Darwiche's remarkable observation ties these two categories: the trace of a search-based exact model counter corresponds to a Decision-DNNF formula. Taking advantage of literal equivalences, this paper designs an efficient model counting technique such that its trace is a generalization of Decision-DNNF formula. We first propose a generalization of Decision-DNNF, called CCDD, to capture literal equivalences, then show that CCDD supports model counting in linear time, and finally design a model counter, called ExactMC, whose trace corresponds to CCDD. We perform an extensive experimental evaluation over a comprehensive set of benchmarks and conduct performance comparison of ExactMC vis-a-vis the state of the art counters, c2d, Dsharp, miniC2D, D4, ADDMC, and Ganak. Our empirical evaluation demonstrates ExactMC can solve 885 instances while the prior state of the art could solve only 843 instances, representing a significant improvement of 42 instances.