Hard 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

Polynomial Time Linear Programs for Hard Constraint Satisfaction Problems

Polynomial Time Linear Programs for Hard Constraint Satisfaction Problems

Polynomial Time Linear Programs for Hard Constraint Satisfaction Problems

Polynomial Time Linear Programs for Hard Constraint Satisfaction Problems

A Variable Depth Search Algorithm for Binary Constraint Satisfaction Problems

The constraint satisfaction problem (CSP) is a popular used paradigm to model a wide spectrum of optimization problems in artificial intelligence. This paper presents a fast metaheuristic for solving binary constraint satisfaction problems. The method can be classified as a variable depth search metaheuristic combining a greedy local search using a self-adaptive weighting strategy on the constraint weights. Several metaheuristics have been developed in the past using various penalty weight mechanisms on the constraints. What distinguishes the proposed metaheuristic from those developed in the past is the update ofkvariables during each iteration when moving from one assignment of values to another. The benchmark is based on hard random constraint satisfaction problems enjoying several features that make them of a great theoretical and practical interest. The results show that the proposed metaheuristic is capable of solving hard unsolved problems that still remain a challenge for both complete and incomplete methods. In addition, the proposed metaheuristic is remarkably faster than all existing solvers when tested on previously solved instances. Finally, its distinctive feature contrary to other metaheuristics is the absence of parameter tuning making it highly suitable in practical scenarios.

The Chaos Within Sudoku

The mathematical structure of Sudoku puzzles is akin to hard constraint satisfaction problems lying at the basis of many applications, including protein folding and the ground-state problem of glassy spin systems. Via an exact mapping of Sudoku into a deterministic, continuous-time dynamical system, here we show that the difficulty of Sudoku translates into transient chaotic behavior exhibited by this system. We also show that the escape rate κ, an invariant of transient chaos, provides a scalar measure of the puzzle's hardness that correlates well with human difficulty ratings. Accordingly, η = −log10 κ can be used to define a “Richter”-type scale for puzzle hardness, with easy puzzles having 0 < η ≤ 1, medium ones 1 < η ≤ 2, hard with 2 < η ≤ 3 and ultra-hard with η > 3. To our best knowledge, there are no known puzzles with η > 4.

Correlation-based decimation in constraint satisfaction problems

We study hard constraint satisfaction problems using some decimation algorithms based on mean-field approximations. The message-passing approach is used to estimate, beside the usual one-variable marginals, the pair correlation functions. The identification of strongly correlated pairs allows to use a new decimation procedure, where the relative orientation of a pair of variables is fixed. We apply this novel decimation to locked occupation problems, a class of hard constraint satisfaction problems where the usual belief-propagation guided decimation performs poorly. The pair-decimation approach provides a significant improvement.

Decimation flows in constraint satisfaction problems

We study hard constraint satisfaction problems with a decimation approach based onmessage passing algorithms. Decimation induces a renormalization flow in the space ofproblems, and we exploit the fact that this flow transforms some of the constraints intolinear constraints over GF(2). In particular, when the flow hits the subspace of linearproblems, one can stop the decimation and use Gaussian elimination. We introduce a newdecimation algorithm which uses this linear structure and shows a strongly improvedperformance with respect to the usual decimation methods for some of the hardest lockedoccupation problems.

Hard constraint satisfaction problems have hard gaps at location 1

An instance of the maximum constraint satisfaction problem ( Max CSP) is a finite collection of constraints on a set of variables, and the goal is to assign values to the variables that maximises the number of satisfied constraints. Max CSP captures many well-known problems (such as Max k - SAT and Max Cut) and is consequently NP-hard. Thus, it is natural to study how restrictions on the allowed constraint types (or constraint language) affect the complexity and approximability of Max CSP. The PCP theorem is equivalent to the existence of a constraint language for which Max CSP has a hard gap at location 1; i.e. it is NP-hard to distinguish between satisfiable instances and instances where at most some constant fraction of the constraints are satisfiable. All constraint languages, for which the CSP problem (i.e., the problem of deciding whether all constraints can be satisfied) is currently known to be NP-hard, have a certain algebraic property. We prove that any constraint language with this algebraic property makes Max CSP have a hard gap at location 1 which, in particular, implies that such problems cannot have a PTAS unless P = NP . We then apply this result to Max CSP restricted to a single constraint type; this class of problems contains, for instance, Max Cut and Max DiCut. Assuming P ≠ NP, we show that such problems do not admit PTAS except in some trivial cases. Our results hold even if the number of occurrences of each variable is bounded by a constant. Finally, we give some applications of our results.

Constraint satisfaction problems and neural networks: A statistical physics perspective

A new field of research is rapidly expanding at the crossroad between statistical physics, information theory and combinatorial optimization. In particular, the use of cutting edge statistical physics concepts and methods allow one to solve very large constraint satisfaction problems like random satisfiability, coloring, or error correction. Several aspects of these developments should be relevant for the understanding of functional complexity in neural networks. On the one hand the message passing procedures which are used in these new algorithms are based on local exchange of information, and succeed in solving some of the hardest computational problems. On the other hand some crucial inference problems in neurobiology, like those generated in multi-electrode recordings, naturally translate into hard constraint satisfaction problems. This paper gives a non-technical introduction to this field, emphasizing the main ideas at work in message passing strategies and their possible relevance to neural networks modelling. It also introduces a new message passing algorithm for inferring interactions between variables from correlation data, which could be useful in the analysis of multi-electrode recording data.

Constraint satisfaction problems with isolated solutions are hard

We study the phase diagram and the algorithmic hardness of the random ‘locked’constraint satisfaction problems, and compare them to the commonly studied ‘non-locked’problems like satisfiability of Boolean formulae or graph coloring. The special property ofthe locked problems is that clusters of solutions are isolated points. This simplifiessignificantly the determination of the phase diagram, which makes the locked problemsparticularly appealing from the mathematical point of view. On the other hand, weshow empirically that the clustered phase of these problems is extremely hardfrom the algorithmic point of view: the best known algorithms all fail to findsolutions. Our results suggest that the easy/hard transition (for currently knownalgorithms) in the locked problems coincides with the clustering transition. Theseshould thus be regarded as new benchmarks of really hard constraint satisfactionproblems.

Random Subcubes as a Toy Model for Constraint Satisfaction Problems

We present an exactly solvable random-subcube model inspired by the structure of hard constraint satisfaction and optimization problems. Our model reproduces the structure of the solution space of the random k-satisfiability and k-coloring problems, and undergoes the same phase transitions as these problems. The comparison becomes quantitative in the large-k limit. Distance properties, as well the x-satisfiability threshold, are studied. The model is also generalized to define a continuous energy landscape useful for studying several aspects of glassy dynamics.

Solving partial constraint satisfaction problems with tree decomposition

AbstractIn this paper, we describe a computational study to solve hard partial constraint satisfaction problems (PCSPs) to optimality. The PCSP is a general class of problems that contains a diversity of problems, such as generalized subgraph problems, MAX‐SAT, Boolean quadratic programs, and assignment problems like coloring and frequency planning. We present a dynamic programming algorithm that solves PCSPs based on the structure (tree decomposition) of the underlying constraint graph. With the use of dominance and bounding techniques, we are able to solve small and medium‐size instances of the problem to optimality and to obtain good lower bounds for large‐size instances within reasonable time and memory limits. © 2002 Wiley Periodicals, Inc.

IMPROVING EVOLUTIONARY ALGORITHMS FOR EFFICIENT CONSTRAINT SATISFACTION

Hard or large-scale constraint satisfaction and optimization problems, occur widely in artificial intelligence and operations research. These problems are often difficult to solve with global search methods, but many of them can be efficiently solved by local search methods. Evolutionary algorithms are local search methods which have considerable success in tackling difficult, or ill-defined optimization problems. In contrast they have not been so successful in tackling constraint satisfaction problems. Other local search methods, in particular GENET and EGENET are designed specifically for constraint satisfaction problems, and have demonstrated remarkable success in solving hard examples of these problems. In this paper we examine how we can transfer the mechanisms that were so successful in (E)GENET to evolutionary algorithms, in order to tackle constraint satisfaction algorithms efficiently. An empirical comparison of our evolutionary algorithm improved by mechanisms from EGENET and shows how it can markedly improve on the efficiency of EGENET in solving certain hard instances of constraint satisfaction problems.