Solving Constraint Satisfaction Problems Research Articles

Solving Boolean Satisfiability Problems With The Quantum Approximate Optimization Algorithm

One of the most prominent application areas for quantum computers is solving hard constraint satisfaction and optimization problems. However, detailed analyses of the complexity of standard quantum algorithms have suggested that outperforming classical methods for these problems would require extremely large and powerful quantum computers. The quantum approximate optimization algorithm (QAOA) is designed for near-term quantum computers, yet previous work has shown strong limitations on the ability of QAOA to outperform classical algorithms for optimization problems. Here we instead apply QAOA to hard constraint satisfaction problems, where both classical and quantum algorithms are expected to require exponential time. We analytically characterize the average success probability of QAOA on a constraint satisfaction problem commonly studied using statistical physics methods: random k-SAT at the threshold for satisfiability, as the number of variables n goes to infinity. We complement these theoretical results with numerical experiments on the performance of QAOA for small n, which match the limiting theoretical bounds closely. We then compare QAOA with leading classical solvers. For random 8-SAT, we find that for more than 14 quantum circuit layers, QAOA achieves more efficient scaling than the highest-performance classical solver we tested, WalkSATlm. Our results suggest that near-term quantum algorithms for solving constraint satisfaction problems may outperform their classical counterparts. Published by the American Physical Society 2024

Limiting the memory consumption of caching for detecting subproblem dominance in constraint problems

Solving constraint satisfaction problems often involves a large amount of redundant exploration stemming from the existence of subproblems whose information can be reused for other subproblems. Subproblem dominance is a general notion of reusability that arises when one subproblem imposes more constraints on the remaining part of the search than another subproblem and allows the search to reuse the solutions of the dominating subproblem for the dominated subproblem. The search can exploit subproblem dominance by storing the subproblems that have been explored in a cache and abandoning the current subproblem whenever the cache contains a subproblem that dominates it. While using caching makes it possible to solve problems where subproblem dominance arises orders of magnitude faster, storing all of these subproblems can require a substantial amount of memory, making it impractical in many cases. This paper analyses the dominance between different subproblems for various constraint problems, revealing that only a relatively small number of subproblems dominate other subproblems. Based on these findings, two types of strategies are proposed for reducing the number of subproblems stored in the cache: limiting the number of subproblems that can be stored in the cache and periodically cleaning up the cache. An experimental evaluation demonstrates that these strategies provide an effective instrument for reducing the memory consumption of caching, allowing it to be used on a larger scale. However, there is a trade-off between saving memory and reducing redundant exploration, as removing subproblems from the cache may prevent dominance from being detected for certain subproblems.

Highly reproducible and CMOS-compatible VO2-based oscillators for brain-inspired computing.

With remarkable electrical and optical switching properties induced at low power and near room temperature (68°C), vanadium dioxide (VO2) has sparked rising interest in unconventional computing among the phase-change materials research community. The scalability and the potential to compute beyond the von Neumann model make VO2 especially appealing for implementation in oscillating neural networks for artificial intelligence applications, to solve constraint satisfaction problems, and for pattern recognition. Its integration into large networks of oscillators on a Silicon platform still poses challenges associated with the stabilization in the correct oxidation state and the ability to fabricate a structure with predictable electrical behavior showing very low variability. In this work, the role played by the different annealing parameters applied by three methods (slow thermal annealing, flash annealing, and rapid thermal annealing), following the vanadium oxide atomic layer deposition, on the formation of VO2 grains is studied and an optimal substrate stack configuration that minimizes variability between devices is proposed. Material and electrical characterizations are performed on the different films and a step-by-step recipe to build reproducible VO2-based oscillators is presented, which is argued to be made possible thanks to the introduction of a hafnium oxide (HfO2) layer between the silicon substrate and the vanadium oxide layer. Up to seven nearly identical VO2-based devices are contacted simultaneously to create a network of oscillators, paving the way for large-scale implementation of VO2 oscillating neural networks.

DeepSaDe: Learning Neural Networks That Guarantee Domain Constraint Satisfaction

As machine learning models, specifically neural networks, are becoming increasingly popular, there are concerns regarding their trustworthiness, specially in safety-critical applications, e.g. actions of an autonomous vehicle must be safe. There are approaches that can train neural networks where such domain requirements are enforced as constraints, but they either cannot guarantee that the constraint will be satisfied by all possible predictions (even on unseen data) or they are limited in the type of constraints that can be enforced. In this paper, we present an approach to train neural networks which can enforce a wide variety of constraints and guarantee that the constraint is satisfied by all possible predictions. The approach builds on earlier work where learning linear models is formulated as a constraint satisfaction problem (CSP). To make this idea applicable to neural networks, two crucial new elements are added: constraint propagation over the network layers, and weight updates based on a mix of gradient descent and CSP solving. Evaluation on various machine learning tasks demonstrates that our approach is flexible enough to enforce a wide variety of domain constraints and is able to guarantee them in neural networks.

基于边际概率分布重新进行单变量选取的置信传播算法求解约束满足问题

基于边际概率分布重新进行单变量选取的置信传播算法求解约束满足问题

Constraint satisfaction algorithms: edition of timetables in the license-master-doctorate system

In this paper, we studied some algorithms for solving constraint satisfaction problem (CSP) and then applied them to solve the problem of generating schedules in a university setting. In other words, we studied the genetic algorithm, the simulated annealing, the hill climbing, a hybridization of the genetic algorithm and the simulated annealing as well as a hybridization of the genetic algorithm and the hill climbing. These algorithms have been tested on the problem of scheduling in a university environment. The hybrid uses hill climbing or simulated annealing to improve each individual in the starting population to a certain stopping point. These individuals are then sent to the genetic algorithm. Our results show that the hybridization of the genetic algorithm with a metaheuristic gives better execution time and performs better as the problem size increases compared to the classical genetic algorithm.

Constraint satisfaction algorithms: edition of timetables in the license-master-doctorate system

In this paper, we studied some algorithms for solving constraint satisfaction problem (CSP) and then applied them to solve the problem of generating schedules in a university setting. In other words, we studied the genetic algorithm, the simulated annealing, the hill climbing, a hybridization of the genetic algorithm and the simulated annealing as well as a hybridization of the genetic algorithm and the hill climbing. These algorithms have been tested on the problem of scheduling in a university environment. The hybrid uses hill climbing or simulated annealing to improve each individual in the starting population to a certain stopping point. These individuals are then sent to the genetic algorithm. Our results show that the hybridization of the genetic algorithm with a metaheuristic gives better execution time and performs better as the problem size increases compared to the classical genetic algorithm.

Finite-rate sparse quantum codes aplenty

We introduce a methodology for generating random multi-qubit stabilizer codes based on solving a constraint satisfaction problem (CSP) on random bipartite graphs. This framework allows us to enforce stabilizer commutation, X/Z balancing, finite rate, sparsity, and maximum-degree constraints simultaneously in a CSP that we can then solve numerically. Using a state-of-the-art CSP solver, we obtain convincing evidence for the existence of a satisfiability threshold. Furthermore, the extent of the satisfiable phase increases with the number of qubits. In that phase, finding sparse codes becomes an easy problem. Moreover, we observe that the sparse codes found in the satisfiable phase practically achieve the channel capacity for erasure noise. Our results show that intermediate-size finite-rate sparse quantum codes are easy to find, while also demonstrating a flexible methodology for generating good codes with custom properties. We therefore establish a complete and customizable pipeline for random quantum code discovery.

Graph constraints refined for transitive relations

As a powerful modelling tool, Constraint Satisfaction Problems (CSPs) can be solved efficiently using various solving paradigms and can encapsulate various types of constraints in a single model. Graph constraints are one such type of constraint. Existing graph constraints are dedicated to solve classical problems, such as constrained path problems and subgraph extraction problems. However, in some scenarios modelled with CSPs, there are relations where a formal path is formed by selecting a set of vertices on a graph, and the shortest path between two neighbouring vertices on that path needs to be restricted. Such relations are known as transitive relations on the graph, which cannot be expressed by existing graph constraints. Although expressive table constraints can be used to express transitive relations, this approach significantly reduces the performance of using table constraints. Therefore, to express transitive relations on a graph that can be efficiently modelled and reasoned by various types of CSP solvers, we propose two new constraints and present propagation algorithms based on generalized arc consistency. Finally, we perform comparative experiments to demonstrate the correctness and efficiency of the proposed approach. The results of the experiments not only prove the correctness of the propagation algorithms but also show that our proposed approach of modelling directly using new constraints and then reasoning using proprietary propagation algorithms, is more efficient than translating relations into table constraints and then reasoning.

Incremental Updates of Generalized Hypertree Decompositions

Structural decomposition methods, such as generalized hypertree decompositions, have been successfully used for solving constraint satisfaction problems (CSPs). As decompositions can be reused to solve CSPs with the same constraint scopes, investing resources in computing good decompositions is beneficial, even though the computation itself is hard. Unfortunately, current methods need to compute a completely new decomposition, even if the scopes change only slightly. In this article, we make the first steps toward solving the problem of updating the decomposition of a CSP P so that it becomes a valid decomposition of a new CSP P ' produced by some modification of P . Even though the problem is hard in theory, we propose and implement a framework for effectively updating generalized hypertree decompositions. The experimental evaluation of our algorithm strongly suggests practical applicability.

HSJ-Solver: a new method based on GHD for answering conjunctive queries and solving constraint satisfaction problems

HSJ-Solver: a new method based on GHD for answering conjunctive queries and solving constraint satisfaction problems

Realization of c-Inference as a SAT Problem

Semantically based on Spohn’s ranking functions, c-representations are special ranking models obtainedby assigning individual integer impacts to the conditionals in a knowledge base R and by defining the rank of eachpossible world as the sum of the impacts of falsified conditionals. c-Inference is the inference relation taking allc-representations of a given knowledge base R into account. In this paper, we show that c-inference can be realizedas a boolean satisfiability problem (SAT), which in turn allows c-inference to be implemented by a SAT solver. Weprovide a stepwise transformation of the characterization of c-inference as as constraint satisfaction problem (CSP),into a solvable-equivalent SAT problem. We present a SAT-based implementation of c-inference using the SMTsolver Z3, demonstrating the feasibility of the approach. Up to now, there has been only one previous implementationof c-inference; this previous implementation utilizes a Prolog-based CSP solver. First evaluation results demonstratethat our SAT-based implementation outperforms the previous CSP-based implementation, providing a promising basis forfurther developing this approach.

Learning variable ordering heuristics for solving Constraint Satisfaction Problems

Backtracking search algorithms are often used to solve the Constraint Satisfaction Problem (CSP), which is widely applied in various domains such as automated planning and scheduling. The efficiency of backtracking search depends greatly on the variable ordering heuristics. Currently, the most commonly used heuristics are hand-crafted based on expert knowledge. In this paper, we propose a deep reinforcement learning based approach to automatically discover new variable ordering heuristics that are better adapted for a given class of CSP instances, without the need of relying on hand-crafted features and heuristics. We show that directly optimizing the search tree size is not convenient for learning, and propose to optimize the expected cost of reaching a leaf node in the search tree. To capture the complex relations among the variables and constraints, we design a representation scheme based on Graph Neural Network that can process CSP instances with different sizes and constraint arities. Experimental results on random CSP instances show that on small and medium sized instances, the learned policies outperform classical hand-crafted heuristics with smaller search tree (up to 10.36% reduction). Moreover, without further training, our policies directly generalize to instances of larger sizes and much harder to solve than those in training, with even larger reduction in the search tree size (up to 18.74%).

Complexity Analysis of Generalized and Fractional Hypertree Decompositions

Hypertree decompositions (HDs), as well as the more powerful generalized hypertree decompositions (GHDs), and the yet more general fractional hypertree decompositions (FHDs) are hypergraph decomposition methods successfully used for answering conjunctive queries and for solving constraint satisfaction problems. Every hypergraph H has a width relative to each of these methods: its hypertree width hw(H) , its generalized hypertree width ghw(H) , and its fractional hypertree width fhw(H) , respectively. It is known that hw(H)≤ k can be checked in polynomial time for fixed k , while checking ghw(H)≤ k is NP-complete for k ≥ 3 . The complexity of checking fhw(H)≤ k for a fixed k has been open for over a decade. We settle this open problem by showing that checking fhw(H)≤ k is NP-complete, even for k=2 . The same construction allows us to prove also the NP-completeness of checking ghw(H)≤ k for k=2 . After that, we identify meaningful restrictions that make checking for bounded ghw or fhw tractable or allow for an efficient approximation of the fhw .

Partial Domain Search Tree for Constraint-Satisfaction Problems

The traditional approach for solving Constraint satisfaction Problems (CSPs) is searching the Assignment Space in which each state represents an assignment to some variables. This paper suggests a new search space formalization for CSPs, the Partial Domain Search Tree (PDST). In each PDST node aunique subset of the original domain is considered, values are excluded from the domains in each node to insure that a given set of constraints is satisfied. We provide theoretical analysis of this new approach showing that searching the PDST is beneficial for loosely constrained problems. Experimental results show that this new formalization is a promising direction for future research. In some cases searching the PDST outperforms the traditional approach by an order of magnitude. Furthermore, PDST can enhance Local Search techniques resulting in solutions that violate up to 30% less constraints.

Opposition-Inspired synergy in sub-colonies of ants: The case of Focused Ant Solver

Recently, Opposition-Inspired Learning strategies were proposed to improve the search process of ant-based algorithms to solve combinatorial problems. In this paper, we propose a collaborative framework of these strategies called Multiple Opposite Synergic Strategy for Ants (MOSSA). Because of each strategy has a different goal, we expect that the ants algorithm will benefit from their collaboration. The algorithm strongly uses the pheromone matrix for accomplishing stigmergy. To evaluate our framework, we use a recently proposed algorithm to solve Constraint Satisfaction Problems named Focused Ant Solver. Results and statistical analysis show that using MOSSA, Focused Ant Solver is able to solve more problems from the transition phase.

HyperBench

To cope with the intractability of answering Conjunctive Queries (CQs) and solving Constraint Satisfaction Problems (CSPs), several notions of hypergraph decompositions have been proposed—giving rise to different notions of width, noticeably, plain, generalized, and fractional hypertree width (hw, ghw, and fhw). Given the increasing interest in using such decomposition methods in practice, a publicly accessible repository of decomposition software, as well as a large set of benchmarks, and a web-accessible workbench for inserting, analyzing, and retrieving hypergraphs are called for. We address this need by providing (i) concrete implementations of hypergraph decompositions (including new practical algorithms), (ii) a new, comprehensive benchmark of hypergraphs stemming from disparate CQ and CSP collections, and (iii) HyperBench, our new web-interface for accessing the benchmark and the results of our analyses. In addition, we describe a number of actual experiments we carried out with this new infrastructure.

Conflict history based heuristic for constraint satisfaction problem solving

The variable ordering heuristic is an important module in algorithms dedicated to solve Constraint Satisfaction Problems (CSP), while it impacts the efficiency of exploring the search space and the size of the search tree. It also exploits, often implicitly, the structure of the instances. In this paper, we propose Conflict-History Search (CHS), a dynamic and adaptive variable ordering heuristic for CSP solving. It is based on the search failures and considers the temporality of these failures throughout the solving steps. The exponential recency weighted average is used to estimate the evolution of the hardness of constraints throughout the search. The experimental evaluation on XCSP3 instances shows that integrating CHS to solvers based on MAC (Maintaining Arc Consistency) and BTD (Backtracking with Tree Decomposition) achieves competitive results and improvements compared to the state-of-the-art heuristics. Beyond the decision problem, we show empirically that the solving of the constraint optimization problem (COP) can also take advantage of this heuristic.

A framework for step-wise explaining how to solve constraint satisfaction problems

We explore the problem of step-wise explaining how to solve constraint satisfaction problems, with a use case on logic grid puzzles. More specifically, we study the problem of explaining the inference steps that one can take during propagation, in a way that is easy to interpret for a person. Thereby, we aim to give the constraint solver explainable agency, which can help in building trust in the solver by being able to understand and even learn from the explanations. The main challenge is that of finding a sequence of simple explanations, where each explanation should aim to be as cognitively easy as possible for a human to verify and understand. This contrasts with the arbitrary combination of facts and constraints that the solver may use when propagating. We propose the use of a cost function to quantify how simple an individual explanation of an inference step is, and identify the explanation-production problem of finding the best sequence of explanations of a CSP. Our approach is agnostic of the underlying constraint propagation mechanisms, and can provide explanations even for inference steps resulting from combinations of constraints. In case multiple constraints are involved, we also develop a mechanism that allows to break the most difficult steps up and thus gives the user the ability to zoom in on specific parts of the explanation. Our proposed algorithm iteratively constructs the explanation sequence by using an optimistic estimate of the cost function to guide the search for the best explanation at each step. Our experiments on logic grid puzzles show the feasibility of the approach in terms of the quality of the individual explanations and the resulting explanation sequences obtained.

Bayes DistNet - A Robust Neural Network for Algorithm Runtime Distribution Predictions

Randomized algorithms are used in many state-of-the-art solvers for constraint satisfaction problems (CSP) and Boolean satisfiability (SAT) problems. For many of these problems, there is no single solver which will dominate others. Having access to the underlying runtime distributions (RTD) of these solvers can allow for better use of algorithm selection, algorithm portfolios, and restart strategies. Previous state-of-the-art methods directly try to predict a fixed parametric distribution that the input instance follows. In this paper, we extend RTD prediction models into the Bayesian setting for the first time. This new model achieves robust predictive performance in the low observation setting, as well as handling censored observations. This technique also allows for richer representations which cannot be achieved by the classical models which restrict their output representations. Our model outperforms the previous state-of-the-art model in settings in which data is scarce, and can make use of censored data such as lower bound time estimates, where that type of data would otherwise be discarded. It can also quantify its uncertainty in its predictions, allowing for algorithm portfolio models to make better informed decisions about which algorithm to run on a particular instance.