221 research outputs found

    Guarantees and Limits of Preprocessing in Constraint Satisfaction and Reasoning

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    We present a first theoretical analysis of the power of polynomial-time preprocessing for important combinatorial problems from various areas in AI. We consider problems from Constraint Satisfaction, Global Constraints, Satisfiability, Nonmonotonic and Bayesian Reasoning under structural restrictions. All these problems involve two tasks: (i) identifying the structure in the input as required by the restriction, and (ii) using the identified structure to solve the reasoning task efficiently. We show that for most of the considered problems, task (i) admits a polynomial-time preprocessing to a problem kernel whose size is polynomial in a structural problem parameter of the input, in contrast to task (ii) which does not admit such a reduction to a problem kernel of polynomial size, subject to a complexity theoretic assumption. As a notable exception we show that the consistency problem for the AtMost-NValue constraint admits a polynomial kernel consisting of a quadratic number of variables and domain values. Our results provide a firm worst-case guarantees and theoretical boundaries for the performance of polynomial-time preprocessing algorithms for the considered problems.Comment: arXiv admin note: substantial text overlap with arXiv:1104.2541, arXiv:1104.556

    Global Constraint Catalog, 2nd Edition (revision a)

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    This report presents a catalogue of global constraints where each constraint is explicitly described in terms of graph properties and/or automata and/or first order logical formulae with arithmetic. When available, it also presents some typical usage as well as some pointers to existing filtering algorithms

    Global Constraint Catalog, 2nd Edition

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    This report presents a catalogue of global constraints where each constraint is explicitly described in terms of graph properties and/or automata and/or first order logical formulae with arithmetic. When available, it also presents some typical usage as well as some pointers to existing filtering algorithms

    XCSP3-core: A Format for Representing Constraint Satisfaction/Optimization Problems

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    In this document, we introduce XCSP3-core, a subset of XCSP3 that allows us to represent constraint satisfaction/optimization problems. The interest of XCSP3-core is multiple: (i) focusing on the most popular frameworks (CSP and COP) and constraints, (ii) facilitating the parsing process by means of dedicated XCSP3-core parsers written in Java and C++ (using callback functions), (iii) and defining a core format for comparisons (competitions) of constraint solvers.Comment: arXiv admin note: substantial text overlap with arXiv:1611.0339

    Global constraints as graph properties on structured network of elementary constraints of the same type

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    This report introduces a classification scheme for the global constraints. This classification is based on four basic ingredients from which one can generate almost all existing global constraints and come up with new interesting constraints. Global constraints are defined in a very concise way, in term of graph properties that have to hold, where the graph is a structured network of same elementary constraints. Since this classification is based on the internal structure of the global constraints it is also a strong hint for the pruning algorithms of the global constraints

    Fixed-parameter tractability of Directed Multicut with three terminal pairs parameterized by the size of the cutset: twin-width meets flow-augmentation

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    We show fixed-parameter tractability of the Directed Multicut problem with three terminal pairs (with a randomized algorithm). This problem, given a directed graph GG, pairs of vertices (called terminals) (s1,t1)(s_1,t_1), (s2,t2)(s_2,t_2), and (s3,t3)(s_3,t_3), and an integer kk, asks to find a set of at most kk non-terminal vertices in GG that intersect all s1t1s_1t_1-paths, all s2t2s_2t_2-paths, and all s3t3s_3t_3-paths. The parameterized complexity of this case has been open since Chitnis, Cygan, Hajiaghayi, and Marx proved fixed-parameter tractability of the 2-terminal-pairs case at SODA 2012, and Pilipczuk and Wahlstr\"{o}m proved the W[1]-hardness of the 4-terminal-pairs case at SODA 2016. On the technical side, we use two recent developments in parameterized algorithms. Using the technique of directed flow-augmentation [Kim, Kratsch, Pilipczuk, Wahlstr\"{o}m, STOC 2022] we cast the problem as a CSP problem with few variables and constraints over a large ordered domain.We observe that this problem can be in turn encoded as an FO model-checking task over a structure consisting of a few 0-1 matrices. We look at this problem through the lenses of twin-width, a recently introduced structural parameter [Bonnet, Kim, Thomass\'{e}, Watrigant, FOCS 2020]: By a recent characterization [Bonnet, Giocanti, Ossona de Mendes, Simon, Thomass\'{e}, Toru\'{n}czyk, STOC 2022] the said FO model-checking task can be done in FPT time if the said matrices have bounded grid rank. To complete the proof, we show an irrelevant vertex rule: If any of the matrices in the said encoding has a large grid minor, a vertex corresponding to the ``middle'' box in the grid minor can be proclaimed irrelevant -- not contained in the sought solution -- and thus reduced

    Fixed-Parameter Tractability of Directed Multicut with Three Terminal Pairs Parameterized by the Size of the Cutset: Twin-width Meets Flow-Augmentation

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    We show fixed-parameter tractability of the Directed Multicut problem withthree terminal pairs (with a randomized algorithm). This problem, given adirected graph GG, pairs of vertices (called terminals) (s1,t1)(s_1,t_1),(s2,t2)(s_2,t_2), and (s3,t3)(s_3,t_3), and an integer kk, asks to find a set of at mostkk non-terminal vertices in GG that intersect all s1t1s_1t_1-paths, alls2t2s_2t_2-paths, and all s3t3s_3t_3-paths. The parameterized complexity of thiscase has been open since Chitnis, Cygan, Hajiaghayi, and Marx provedfixed-parameter tractability of the 2-terminal-pairs case at SODA 2012, andPilipczuk and Wahlstr\"{o}m proved the W[1]-hardness of the 4-terminal-pairscase at SODA 2016. On the technical side, we use two recent developments in parameterizedalgorithms. Using the technique of directed flow-augmentation [Kim, Kratsch,Pilipczuk, Wahlstr\"{o}m, STOC 2022] we cast the problem as a CSP problem withfew variables and constraints over a large ordered domain.We observe that thisproblem can be in turn encoded as an FO model-checking task over a structureconsisting of a few 0-1 matrices. We look at this problem through the lenses oftwin-width, a recently introduced structural parameter [Bonnet, Kim,Thomass\'{e}, Watrigant, FOCS 2020]: By a recent characterization [Bonnet,Giocanti, Ossona de Mendes, Simon, Thomass\'{e}, Toru\'{n}czyk, STOC 2022] thesaid FO model-checking task can be done in FPT time if the said matrices havebounded grid rank. To complete the proof, we show an irrelevant vertex rule: Ifany of the matrices in the said encoding has a large grid minor, a vertexcorresponding to the ``middle'' box in the grid minor can be proclaimedirrelevant -- not contained in the sought solution -- and thus reduced.<br

    Target oriented relational model finding

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    Lecture Notes in Computer Science 8411, 2014Model finders are becoming useful in many software engineering problems. Kodkod is one of the most popular, due to its support for relational logic (a combination of first order logic with relational algebra operators and transitive closure), allowing a simpler specification of constraints, and support for partial instances, allowing the specification of a priori (exact, but potentially partial) knowledge about a problem's solution. However, in some software engineering problems, such as model repair or bidirectional model transformation, knowledge about the solution is not exact, but instead there is a known target that the solution should approximate. In this paper we extend Kodkod's partial instances to allow the specification of such targets, and show how its model finding procedure can be adapted to support them (using both PMax-SAT solvers or SAT solvers with cardinality constraints). Two case studies are also presented, including a careful performance evaluation to assess the effectiveness of the proposed extension.(undefined

    Dynamically weakened constraints in bounded search for constraint optimisation problems

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    Combinatorial optimisation problems, where the goal is to an optimal solution from the set of solutions of a problem involving resources, constraints on how these resources can be used, and a ranking of solutions are of both theoretical and practical interest. Many real world problems (such as routing vehicles or planning timetables) can be modelled as constraint optimisation problems, and solved via a variety of solver technologies which rely on differing algorithms for search and inference. The starting point for the work presented in this thesis is two existing approaches to solving constraint optimisation problems: constraint programming and decision diagram branch and bound search. Constraint programming models problems using variables which have domains of values and valid value assignments to variables are restricted by constraints. Constraint programming is a mature approach to solving optimisation problems, and typically relies on backtracking search algorithms combined with constraint propagators (which infer from incomplete solutions which values can be removed from the domains of variables which are yet to be assigned a value). Decision diagram branch and bound search is a less mature approach which solves problems modelled as dynamic programming models using width restricted decision diagrams to provide bounds during search. The main contribution of this thesis is adapting decision diagram branch and bound to be the search scheme in a general purpose constraint solver. To achieve this we propose a method in which we introduce a new algorithm for each constraint that we wish to include in our solver and these new algorithms weaken individual constraints, so that they respect the problem relaxations introduced while using decision diagram branch and bound as the search algorithm in our solver. Constraints are weakened during search based on the problem relaxations imposed by the search algorithm: before search begins there is no way of telling which relaxations will be introduced. We attempt to provide weakening algorithms which require little to no changes to existing propagation algorithms. We provide weakening algorithms for a number of built-in constraints in the Flatzinc specifi- cation, as well as for global constraints and symmetry reduction constraints. We implement a solver in Go and empirically verify the competitiveness of our approach. We show that our solver can be parallelised using Goroutines and channels and that our approach scales well. Finally, we also provide an implementation of our approach in a solver which is tailored towards solving extremal graph problems. We use the forbidden subgraph problem to show that our approach of using decision diagram branch and bound as a search scheme in a constraint solver can be paired with canonical search. Canonical search is a technique for graph search which ensures that no two isomorphic graphs are returned during search. We pair our solver with the Nauty graph isomorphism algorithm to achieve this, and explore the relationship between branch and bound and canonical search
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