constraint satisfaction problems (CSPs) are a class of problems that are ubiquitous in science and engineering. They feature a collection of constraints specified over subsets of variables. A CSP can be solved either ...
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constraint satisfaction problems (CSPs) are a class of problems that are ubiquitous in science and engineering. They feature a collection of constraints specified over subsets of variables. A CSP can be solved either directly or by reducing it to other problems. This paper introduces the Julia ecosystem for solving and analyzing CSPs with a focus on the programming practices. We introduce some important CSPs and show how these problems are reduced to each other. We also show how to transform CSPs into tensor networks, how to optimize the tensor network contraction orders, and how to extract the solution space properties by contracting the tensor networks with generic element types. Examples are given, which include computing the entropy constant, analyzing the overlap gap property, and the reduction between CSPs.
Spiking neural networks (SNNs) offer an effective approach to solving constraint satisfaction problems (CSPs) by leveraging their temporal, event-driven dynamics. Moreover, neuromorphic hardware platforms provide the ...
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Spiking neural networks (SNNs) offer an effective approach to solving constraint satisfaction problems (CSPs) by leveraging their temporal, event-driven dynamics. Moreover, neuromorphic hardware platforms provide the potential for achieving significant energy efficiency in implementing such models. Building upon these foundations, we present an enhanced, fully spiking pipeline for solving CSPs on the SpiNNaker neuromorphic hardware platform. Focusing on the use case of Sudoku puzzles, we demonstrate that the adoption of a constraint stabilization strategy, coupled with a neuron idling mechanism and a built-in validation process, enables this application to be realized through a series of additional layers of neurons capable of performing control logic operations, verifying solutions, and memorizing the network’s state. Simulations conducted in the GPU-enhanced Neuronal Networks (GeNN) environment validate the contributions of each pipeline component before deployment on SpiNNaker. This approach offers three key advantages: i) Improved success rates for solving CSPs, particularly for challenging instances from the hard class, surpassing state-of-the-art SNN-based solvers. ii) Reduced data transmission overhead by transmitting only the final activity state from SpiNNaker instead of all generated spikes. iii) Substantially decreased spike extraction time. Compared to previous work focused on the same use case, our approach achieves a significant reduction in the number of extracted spikes (54.63% to 99.98%) and extraction time (88.56% to 96.41%). Impact Statement—This work presents a novel approach to address constraint satisfaction problems through Spiking Neural Networks (SNNs) utilising neuromorphic tools like the GeNN framework and the SpiNNaker platform. We propose a new fully spiking pipeline that incorporates a constraint stabilization strategy, a neuron idling mechanism, and a built-in validation procedure. Our pipeline targets efficiency and performance of SNN
Recent progress in o1-like models has significantly enhanced the reasoning abilities of Large Language Models (LLMs), empowering them to tackle increasingly complex tasks through reflection capabilities, such as makin...
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The Feder-Vardi dichotomy conjecture for constraint satisfaction problems (CSPs) with finite templates, confirmed independently by Bulatov and Zhuk, has a counterpart for infinite templates due to Bodirsky and Pinsker...
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We examine the complexity of constraint satisfaction problems that consist of a set of AllDiff constraints. Such CSPs naturally model a wide range of real-world and combinatorial problems, like scheduling, frequency a...
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We examine the complexity of constraint satisfaction problems that consist of a set of AllDiff constraints. Such CSPs naturally model a wide range of real-world and combinatorial problems, like scheduling, frequency allocations, and graph coloring problems. As this problem is known to be NP-complete, we investigate under which further assumptions it becomes tractable. We observe that a crucial property seems to be the convexity of the variable domains and constraints. Our main contribution is an extensive study of the complexity of Multiple AllDiff CSPs for a set of natural parameters, like maximum domain size and maximum size of the constraint scopes. We show that, depending on the parameter, convexity can make the problem tractable even though it is provably intractable in general. Interestingly, the convexity of constraints is the key property in achieving fixed parameter tractability, while the convexity of domains does not usually make the problem easier. (C) 2012 Elsevier B.V. All rights reserved.
In a constraintsatisfaction problem (CSP) the goal is to find an assignment of a given set of variables subject to specified constraints. A global cardinality constraint is an additional requirement that prescribes h...
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In a constraintsatisfaction problem (CSP) the goal is to find an assignment of a given set of variables subject to specified constraints. A global cardinality constraint is an additional requirement that prescribes how many variables must be assigned a certain value. We study the complexity of the problem CCSP(Gamma), the CSP with global cardinality constraints that allows only relations from the set Gamma. The main result of this paper characterizes sets Gamma that give rise to problems solvable in polynomial time, and states that the remaining such problems are NP-complete.
For n >= 3, let (H-n, E) denote the nth Henson graph, i.e., the unique countable homogeneous graph with exactly those finite graphs as induced subgraphs that do not embed the complete graph on n vertices. We show t...
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For n >= 3, let (H-n, E) denote the nth Henson graph, i.e., the unique countable homogeneous graph with exactly those finite graphs as induced subgraphs that do not embed the complete graph on n vertices. We show that for all structures Gamma with domain H-n whose relations are first-order definable in (H-n, E) the constraintsatisfaction problem for F either is in P or is NP-complete. We moreover show a similar complexity dichotomy for all structures whose relations are first-order definable in a homogeneous graph whose reflexive closure is an equivalence relation. Together with earlier results, in particular for the random graph, this completes the complexity classification of constraint satisfaction problems of structures first-order definable in countably infinite homogeneous graphs: all such problems are either in P or NP-complete.
We study the complexity of Boolean constraint satisfaction problems (CSPs) when the assignment must have Hamming weight in some congruence class modulo M, for various choices of the modulus M. Due to the known classif...
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We study the complexity of Boolean constraint satisfaction problems (CSPs) when the assignment must have Hamming weight in some congruence class modulo M, for various choices of the modulus M. Due to the known classification of tractable Boolean CSPs, this mainly reduces to the study of three cases: 2-SAT, HORN-SAT, and LIN-2 (linear equations mod 2). We classify the moduli M for which these respective problems are polynomial time solvable, and when they are not (assuming the exponential time hypothesis). Our study reveals that this modular constraint lends a surprising richness to these classic, well-studied problems, with interesting broader connections to complexity theory and coding theory. The HORN-SAT case is connected to the covering complexity of polynomials representing the NAND function mod M. The LIN-2 case is tied to the sparsity of polynomials representing the OR function mod M, which in turn has connections to modular weight distribution properties of linear codes and locally decodable codes. In both cases, the analysis of our algorithm as well as the hardness reduction rely on these polynomial representations, highlighting an Interesting algebraic common ground between hard cases for our algorithms and the gadgets which show hardness. These new complexity measures of polynomial representations merit further study. The inspiration for our study comes from a recent work by Nagele, Sudakov, and Zenklusen on submodular minimization with a global congruence constraint. Our algorithm for HORN-SAT has strong similarities to their algorithm, and in particular identical kinds of set systems arise in both cases. Our connection to polynomial representations leads to a simpler analysis of such set systems and also sheds light on (but does not resolve) the complexity of submodular minimization with a congruency requirement modulo a composite M.
We study which constraint satisfaction problems (CSPs) are solvable in NL. In particular, we identify a general condition called bounded path duality, that explains all the families of CSI's previously known to be...
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ISBN:
(纸本)3540438645
We study which constraint satisfaction problems (CSPs) are solvable in NL. In particular, we identify a general condition called bounded path duality, that explains all the families of CSI's previously known to be in NL. Bounded path duality captures the class of constraint satisfaction problems that can be solved by linear Datalog programs, i.e., Datalog programs with at most one IDB in the body of each rule. We obtain several alternative characterizations of bounded path duality. We also address the problem of deciding which constraint satisfaction problems have bounded path duality. In this direction we identify a subclass of bounded path duality problems, called (1,k)-path duality problems for which membership is decidable. Finally, we study which closure operations guarantee bounded path duality. We show that closure under any operation in the pseudovariety generated by the class of dual discriminator operations is a sufficient condition for bounded path duality.
A distance constraintsatisfaction problem is a constraintsatisfaction problem (CSP) whose constraint language consists of relations that are first-order definable over (Z;succ), i.e., over the integers with the succ...
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ISBN:
(纸本)9783662476727;9783662476710
A distance constraintsatisfaction problem is a constraintsatisfaction problem (CSP) whose constraint language consists of relations that are first-order definable over (Z;succ), i.e., over the integers with the successor function. Our main result says that every distance CSP is in P or NP-complete, unless it can be formulated as a finite domain CSP in which case the computational complexity is not known in general.
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