Search

Pseudo-Boolean proof logging has been used successfully to provide certificates of optimality from a variety of constraint- and satisifability-style solvers that combine reasoning with a backtracking or clause-learning search. Another paradigm, occurring in dynamic programming and decision diagram solving, instead reasons about partial states and possible transitions between them. We describe a framework for generating clean and efficient pseudo-Boolean proofs for these kinds of algorithm, and use it to produce certifying algorithms for knapsack, longest path, and interval scheduling. Because we use a common proof system, we can also reason about hybrid solving algorithms: we demonstrate this by providing proof logging for a dynamic programming based knapsack propagator inside a constraint programming solver.

Proof logging has long been the established method to certify correctness of Boolean satisfiability (SAT) solvers, but has only recently been introduced for SAT-based optimization (MaxSAT). The focus of this paper is solution-improving search (SIS), in which a SAT solver is iteratively queried for increasingly better solutions until an optimal one is found. A challenging aspect of modern SIS solvers is that they make use of complex "without loss of generality" arguments that are quite involved to understand even at a human meta-level, let alone to express in a simple, machine-verifiable proof. In this work, we develop pseudo-Boolean proof logging methods for solution-improving MaxSAT solving, and use them to produce a certifying version of the state-of-the-art solver Pacose with VeriPB proofs. Our experimental evaluation demonstrates that this approach works in practice. We hope that this is yet another step towards general adoption of proof logging in MaxSAT solving.

Proof logging has long been the established method to certify correctness of Boolean satisfiability (SAT) solvers, but has only recently been introduced for SAT-based optimization (MaxSAT). The focus of this paper is solution-improving search (SIS), in which a SAT solver is iteratively queried for increasingly better solutions until an optimal one is found. A challenging aspect of modern SIS solvers is that they make use of complex "without loss of generality"arguments that are quite involved to understand even at a human meta-level, let alone to express in a simple, machine-verifiable proof. In this work, we develop pseudo-Boolean proof logging methods for solution-improving MaxSAT solving, and use them to produce a certifying version of the state-of-the-art solver Pacose with VeriPB proofs. Our experimental evaluation demonstrates that this approach works in practice. We hope that this is yet another step towards general adoption of proof logging in MaxSAT solving.

Pseudo-Boolean proof logging has been used successfully to provide certificates of optimality from a variety of constraint- and satisifability-style solvers that combine reasoning with a backtracking or clause-learning search. Another paradigm, occurring in dynamic programming and decision diagram solving, instead reasons about partial states and possible transitions between them. We describe a framework for generating clean and efficient pseudo-Boolean proofs for these kinds of algorithm, and use it to produce certifying algorithms for knapsack, longest path, and interval scheduling. Because we use a common proof system, we can also reason about hybrid solving algorithms: we demonstrate this by providing proof logging for a dynamic programming based knapsack propagator inside a constraint programming solver.

Modern subgraph-finding algorithm implementations consist of thousands of lines of highly optimized code, and this complexity raises questions about their trustworthiness. Recently, some state-of-the-art subgraph solvers have been enhanced to output machine-verifiable proofs that their results are correct. While this significantly improves reliability, it is not a fully satisfactory solution, since end-users have to trust both the proof checking algorithms and the translation of the high-level graph problem into a low-level 0-1 integer linear program (ILP) used for the proofs. In this work, we present the first formally verified toolchain capable of full end-to-end verification for subgraph solving, which closes both of these trust gaps. We have built encoder frontends for various graph problems together with a 0-1 ILP (a.k.a. pseudo-Boolean) proof checker, all implemented and formally verified in the CAKEML ecosystem. This toolchain is flexible and extensible, and we use it to build verified proof checkers for both decision and optimization graph problems, namely, subgraph isomorphism, maximum clique, and maximum common (connected) induced subgraph. Our experimental evaluation shows that end-to-end formal verification is now feasible for a wide range of hard graph problems.

Building on the progress in Boolean satisfiability (SAT) solving over the last decades, maximum satisfiability (MaxSAT) has become a viable approach for solving NP-hard optimization problems, but ensuring correctness of MaxSAT solvers has remained an important concern. For SAT, this is largely a solved problem thanks to the use of proof logging, meaning that solvers emit machine-verifiable proofs of (un)satisfiability to certify correctness. However, for MaxSAT, proof logging solvers have started being developed only very recently. Moreover, these nascent efforts have only targeted the core solving process, ignoring the preprocessing phase where input problem instances can be substantially reformulated before being passed on to the solver proper. In this work, we demonstrate how pseudo-Boolean proof logging can be used to certify the correctness of a wide range of modern MaxSAT preprocessing techniques. By combining and extending the VeriPB and CakePB tools, we provide formally verified, end-to-end proof checking that the input and preprocessed output MaxSAT problem instances have the same optimal value. An extensive evaluation on applied MaxSAT benchmarks shows that our approach is feasible in practice.

This report documents the program and the outcomes of Dagstuhl Seminar 22411 "Theory and Practice of SAT and Combinatorial Solving". The purpose of this workshop was to explore the Boolean satisfiability (SAT) problem, which plays a fascinating dual role in computer science. By the theory of NP-completeness, this problem captures thousands of important applications in different fields, and a rich mathematical theory has been developed showing that all these problems are likely to be infeasible to solve in the worst case. But real-world problems are typically not worst-case, and in recent decades exceedingly efficient algorithms based on so-called conflict-driven clause learning (CDCL) have turned SAT solvers into highly practical tools for solving large-scale real-world problems in a wide range of application areas. Analogous developments have taken place for problems beyond NP such as SAT-based optimization (MaxSAT), pseudo-Boolean optimization, satisfiability modulo theories (SMT) solving, quantified Boolean formula (QBF) solving, constraint programming, and mixed integer programming, where the conflict-driven paradigm has sometimes been added to other powerful techniques. The current state of the art in combinatorial solving presents a host of exciting challenges at the borderline between theory and practice. Can we gain a deeper scientific understanding of the techniques and heuristics used in modern combinatorial solvers and why they are so successful? Can we develop tools for rigorous analysis of the potential and limitations of these algorithms? Can computational complexity theory be extended to shed light on real-world settings that go beyond worst case? Can more powerful methods of reasoning developed in theoretical research be harnessed to yield improvements in practical performance? And can state-of-the-art combinatorial solvers be enhanced to not only solve problems, but also provide verifiable proofs of correctness for the solutions they produce? This w

Conflict analysis has been successfully generalized from Boolean satisfiability (SAT) solving to mixed integer programming (MIP) solvers, but although MIP solvers operate with general linear inequalities, the conflict analysis in MIP has been limited to reasoning with the more restricted class of clausal constraint. This is in contrast to how conflict analysis is performed in so-called pseudo-Boolean solving, where solvers can reason directly with 0-1 integer linear inequalities rather than with clausal constraints extracted from such inequalities. In this work, we investigate how pseudo-Boolean conflict analysis can be integrated in MIP solving, focusing on 0-1 integer linear programs (0-1 ILPs). Phrased in MIP terminology, conflict analysis can be understood as a sequence of linear combinations and cuts. We leverage this perspective to design a new conflict analysis algorithm based on mixed integer rounding (MIR) cuts, which theoretically dominates the state-of-the-art division-based method in pseudo-Boolean solving. We also report results from a first proof-of-concept implementation of different pseudo-Boolean conflict analysis methods in the open-source MIP solver SCIP. When evaluated on a large and diverse set of 0-1 ILP instances from MIPLIB2017, our new MIR-based conflict analysis outperforms both previous pseudo-Boolean methods and the clause-based method used in MIP. Our conclusion is that pseudo-Boolean conflict analysis in MIP is a promising research direction that merits further study, and that it might also make sense to investigate the use of such conflict analysis to generate stronger no-goods in constraint programming.

We consider the graph $k$-colouring problem encoded as a set of polynomial equations in the standard way over $0/1$-valued variables. We prove that there are bounded-degree graphs that do not have legal $k$-colourings but for which the polynomial calculus proof system defined in [Clegg et al '96, Alekhnovich et al '02] requires linear degree, and hence exponential size, to establish this fact. This implies a linear degree lower bound for any algorithms based on Gr\"{o}bner bases solving graph $k$-colouring using this encoding. The same bound applies also for the algorithm studied in a sequence of papers [De Loera et al '08,'09,'11,'15] based on Hilbert's Nullstellensatz proofs for a slightly different encoding, thus resolving an open problem mentioned in [De Loera et al '08,'09,'11] and [Li '16]. We obtain our results by combining the polynomial calculus degree lower bound for functional pigeonhole principle (FPHP) formulas over bounded-degree bipartite graphs in [Mik\v{s}a and Nordstr\"{o}m '15] with a reduction from FPHP to $k$-colouring derivable by polynomial calculus in constant degree.

We consider the pebble game on DAGs with bounded fan-in introduced in [Paterson and Hewitt '70] and the reversible version of this game in [Bennett '89], and study the question of how hard it is to decide exactly or approximately the number of pebbles needed for a given DAG in these games. We prove that the problem of eciding whether $s$~pebbles suffice to reversibly pebble a DAG $G$ is PSPACE-complete, as was previously shown for the standard pebble game in [Gilbert, Lengauer and Tarjan '80]. Via two different graph product constructions we then strengthen these results to establish that both standard and reversible pebbling space are PSPACE-hard to approximate to within any additive constant. To the best of our knowledge, these are the first hardness of approximation results for pebble games in an unrestricted setting (even for polynomial time). Also, since [Chan '13] proved that reversible pebbling is equivalent to the games in [Dymond and Tompa '85] and [Raz and McKenzie '99], our results apply to the Dymond--Tompa and Raz--McKenzie games as well, and from the same paper it follows that resolution depth is PSPACE-hard to determine up to any additive constant. We also obtain a multiplicative logarithmic separation between reversible and standard pebbling space. This improves on the additive logarithmic separation previously known and could plausibly be tight, although we are not able to prove this. We leave as an interesting open problem whether our additive hardness of approximation result could be strengthened to a multiplicative bound if the computational resources are decreased from polynomial space to the more common setting of polynomial time.

Conflict analysis has been successfully generalized from Boolean satisfiability (SAT) solving to mixed integer programming (MIP) solvers, but although MIP solvers operate with general linear inequalities, the conflict analysis in MIP has been limited to reasoning with the more restricted class of clausal constraint. This is in contrast to how conflict analysis is performed in so-called pseudo-Boolean solving, where solvers can reason directly with 0-1 integer linear inequalities rather than with clausal constraints extracted from such inequalities. In this work, we investigate how pseudo-Boolean conflict analysis can be integrated in MIP solving, focusing on 0-1 integer linear programs (0-1 ILPs). Phrased in MIP terminology, conflict analysis can be understood as a sequence of linear combinations and cuts. We leverage this perspective to design a new conflict analysis algorithm based on mixed integer rounding (MIR) cuts, which theoretically dominates the state-of-the-art division-based method in pseudo-Boolean solving. We also report results from a first proof-of-concept implementation of different pseudo-Boolean conflict analysis methods in the open-source MIP solver SCIP. When evaluated on a large and diverse set of 0-1 ILP instances from MIPLIB 2017, our new MIR-based conflict analysis outperforms both previous pseudo-Boolean methods and the clause-based method used in MIP. Our conclusion is that pseudo-Boolean conflict analysis in MIP is a promising research direction that merits further study, and that it might also make sense to investigate the use of such conflict analysis to generate stronger no-goods in constraint programming.

We prove that polynomial calculus (and hence also Nullstellensatz) over any field requires linear degree to refute that sparse random regular graphs, as well as sparse Erdős-Rényi random graphs, are 3-colourable. Using the known relation between size and degree for polynomial calculus proofs, this implies strongly exponential lower bounds on proof size

In the last couple of decades, developments in SAT-based optimization have led to highly efficient maximum satisfiability (MaxSAT) solvers, but in contrast to the SAT solvers on which MaxSAT solving rests, there has been little parallel development of techniques to prove the correctness of MaxSAT results. We show how pseudo-Boolean proof logging can be used to certify state-of-the-art core-guided MaxSAT solving, including advanced techniques like structure sharing, weight-aware core extraction and hardening. Our experimental evaluation demonstrates that this approach is viable in practice. We are hopeful that this is the first step towards general proof logging techniques for MaxSAT solvers.

Symmetry and dominance breaking can be crucial for solving hard combinatorial search and optimisation problems, but the correctness of these techniques sometimes relies on subtle arguments. For this reason, it is desirable to produce efficient, machine-verifiable certificates that solutions have been computed correctly. Building on the cutting planes proof system, we develop a certification method for optimisation problems in which symmetry and dominance breaking is easily expressible. Our experimental evaluation demonstrates that we can efficiently verify fully general symmetry breaking in Boolean satisfiability (SAT) solving, thus providing, for the first time, a unified method to certify a range of advanced SAT techniques that also includes cardinality and parity (XOR) reasoning. In addition, we apply our method to maximum clique solving and constraint programming as a proof of concept that the approach applies to a wider range of combinatorial problems.

Conflict analysis has been successfully generalized from Boolean satisfiability (SAT) solving to mixed integer programming (MIP) solvers, but although MIP solvers operate with general linear inequalities, the conflict analysis in MIP has been limited to reasoning with the more restricted class of clausal constraint. This is in contrast to how conflict analysis is performed in so-called pseudo-Boolean solving, where solvers can reason directly with 0-1 integer linear inequalities rather than with clausal constraints extracted from such inequalities. In this work, we investigate how pseudo-Boolean conflict analysis can be integrated in MIP solving, focusing on 0-1 integer linear programs (0-1 ILPs). Phrased in MIP terminology, conflict analysis can be understood as a sequence of linear combinations and cuts. We leverage this perspective to design a new conflict analysis algorithm based on mixed integer rounding (MIR) cuts, which theoretically dominates the state-of-the-art division-based method in pseudo-Boolean solving. We also report results from a first proof-of-concept implementation of different pseudo-Boolean conflict analysis methods in the open-source MIP solver SCIP. When evaluated on a large and diverse set of 0-1 ILP instances from MIPLIB 2017, our new MIR-based conflict analysis outperforms both previous pseudo-Boolean methods and the clause-based method used in MIP. Our conclusion is that pseudo-Boolean conflict analysis in MIP is a promising research direction that merits further study, and that it might also make sense to investigate the use of such conflict analysis to generate stronger no-goods in constraint programming.

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.

The dramatic improvements in Boolean satisfiability (SAT) solving since the turn of the millennium have made it possible to leverage state-of-the-art conflict-driven clause learning (CDCL) solvers for many combinatorial problems in academia and industry, and the use of proof logging has played a crucial role in increasing the confidence that the results these solvers produce are correct. However, the fact that SAT proof logging is performed in conjunctive normal form (CNF) clausal format means that it has not been possible to extend guarantees of correctness to the use of SAT solvers for more expressive combinatorial paradigms, where the first step is an unverified translation of the input to CNF. In this work, we show how cutting-planes-based reasoning can provide proof logging for solvers that translate pseudo-Boolean (a.k.a. 0-1 integer linear) decision problems to CNF and then run CDCL. To support a wide range of encodings, we provide a uniform and easily extensible framework for proof logging of CNF translations. We are hopeful that this is just a first step towards providing a unified proof logging approach that will also extend to maximum satisfiability (MaxSAT) solving and pseudo-Boolean optimization in general.

We describe the design and implementation of a new constraint programming solver that can produce an auditable record of what problem was solved and how the solution was reached. As well as a solution, this solver provides an independently verifiable proof log demonstrating that the solution is correct. This proof log uses the VeriPB proof system, which is based upon cutting planes reasoning with extension variables. We explain how this system can support global constraints, variables with large domains, and reformulation, despite not natively understanding any of these concepts.

Symmetry and dominance breaking can be crucial for solving hard combinatorial search and optimisation problems, but the correctness of these techniques sometimes relies on subtle arguments. For this reason, it is desirable to produce efficient, machine-verifiable certificates that solutions have been computed correctly. Building on the cutting planes proof system, we develop a certification method for optimisation problems in which symmetry and dominance breaking are easily expressible. Our experimental evaluation demonstrates that we can efficiently verify fully general symmetry breaking in Boolean satisfiability (SAT) solving, thus providing, for the first time, a unified method to certify a range of advanced SAT techniques that also includes XOR and cardinality reasoning. In addition, we apply our method to maximum clique solving and constraint programming as a proof of concept that the approach applies to a wider range of combinatorial problems., Comment: This paper was published in the Journal of Artificial Intelligence Research https://doi.org/10.1613/jair.1.14296 It is an extended version of our (equally-named) paper to appear in the proceedings of AAAI 2022 https://ojs.aaai.org/index.php/AAAI/article/view/20283

The dramatic improvements in Boolean satisfiability (SAT) solving since the turn of the millennium have made it possible to leverage state-of-the-art conflict-driven clause learning (CDCL) solvers for many combinatorial problems in academia and industry, and the use of proof logging has played a crucial role in increasing the confidence that the results these solvers produce are correct. However, the fact that SAT proof logging is performed in conjunctive normal form (CNF) clausal format means that it has not been possible to extend guarantees of correctness to the use of SAT solvers for more expressive combinatorial paradigms, where the first step is an unverified translation of the input to CNF. In this work, we show how cutting-planes-based reasoning can provide proof logging for solvers that translate pseudo-Boolean (a.k.a. 0-1 integer linear) decision problems to CNF and then run CDCL. To support a wide range of encodings, we provide a uniform and easily extensible framework for proof logging of CNF translations. We are hopeful that this is just a first step towards providing a unified proof logging approach that will also extend to maximum satisfiability (MaxSAT) solving and pseudo-Boolean optimization in general.

We describe the design and implementation of a new constraint programming solver that can produce an auditable record of what problem was solved and how the solution was reached. As well as a solution, this solver provides an independently verifiable proof log demonstrating that the solution is correct. This proof log uses the VeriPB proof system, which is based upon cutting planes reasoning with extension variables. We explain how this system can support global constraints, variables with large domains, and reformulation, despite not natively understanding any of these concepts.