Maximum Constraint Satisfaction Problem Research Articles

Optimal Polynomial-Time Compression for Boolean Max CSP

In the Boolean maximum constraint satisfaction problem— Max CSPΓ —one is given a collection of weighted applications of constraints from a finite constraint language Γ, over a common set of variables, and the goal is to assign Boolean values to the variables so that the total weight of satisfied constraints is maximized. There exists a concise dichotomy theorem providing a criterion on Γ for the problem to be polynomial-time solvable and stating that otherwise, it becomes NP-hard. We study the NP-hard cases through the lens of kernelization and provide a complete characterization of Max CSPΓ with respect to the optimal compression size. Namely, we prove that Max CSPΓ parameterized by the number of variables n is either polynomial-time solvable, or there exists an integer d ≥ 2 depending on Γ, such that: (1) An instance of Max CSPΓ can be compressed into an equivalent instance with 𝒪( n d log n ) bits in polynomial time, (2) Max CSPΓ does not admit such a compression to 𝒪( n d-ε ) bits unless NP ⊆ co-NP / poly. Our reductions are based on interpreting constraints as multilinear polynomials combined with the framework of “constraint implementations”, formerly used in the context of APX-hardness. As another application of our reductions, we reveal tight connections between optimal running times for solving Max CSPΓ . More precisely, we show that obtaining a running time of the form 𝒪(2 (1-ε) n ) for particular classes of Max CSP s is as hard as breaching this barrier for Max d -SAT for some d .

A new approach to solve the of maximum constraint satisfaction problem

<span lang="EN-US">The premature convergence of the simulated annealing algorithm, to solve many complex problems of artificial intelligence, refers to a failure mode where the process stops at a stable point that does not represent to an overall solution. Accelerating the speed of convergence and avoiding local solutions is the concern of this work. To overcome this weakness in order to improve the performance of the solution, a new hybrid approach is proposed. The new approach is able to take into consideration the state of the system during convergence via the use of Hopfield neural networks. To implement the proposed approach, the problem of maximum constraint satisfaction is modeled as a quadratic programming. This problem is solved via the use of the new approach. The approach is compared with other methods to show the effectiveness of the proposed approach.</span>

Assortment Optimization under a Single Transition Choice Model

In this study, we consider a new customer choice model which we call the single transition choice model. In this model, there is a universe of products and customers arrive at each product with a certain probability. If the arrived product is unavailable, then the seller can recommend a subset of available products and the customer will purchase one of the recommended products or choose not to purchase with certain transition probabilities. The distinguishing features of the model are that the seller can control which products to recommend depending on the arrived product, and each customer either purchases a product or leaves the market after one transition. We study the assortment optimization problem under this model. Particularly, we show that it is NP‐Hard even if the customer can transition from each product to at most two products. Despite the computational complexity, we provide polynomial time algorithms or approximation algorithms for several special cases, such as when the customer can only transition from each product to at most a given number of products and the size of each recommended set is bounded. Our approximation algorithms are developed by invoking the submodularity arguments, or connecting the problem with maximum constraint satisfaction problem and applying randomized rounding techniques to its semidefinite programming relaxation. We also provide a tight worst‐case performance bound for revenue‐ordered assortments. In addition, we propose a compact mixed‐integer program formulation, which is efficient for moderate size problems. Finally, we conduct numerical experiments to demonstrate the effectiveness of the proposed algorithms.

A new template concept to guide the local search: case of maximal constraint satisfaction problems

within constraint satisfaction problems (CSPs), our work proposes a new search heuristic. The latter is basically inspired by the template concept introduced by E.P.K. TSANG. It consists of a new template concept and could be classified as a min-conflict heuristic. In this paper we introduce a new guided local search that uses our heuristic in local search (LS) for Maximal constraint satisfaction problems (Max-CSPs). Compared to previous well known guided local search, our approach changes experimental results for the better.

A new template concept guided honey bee optimization for Max-CSPs

Constraint satisfaction problems (CSPs) are globally used to perform real life problems. They are dealt with many optimization complete and incomplete approaches. In this context nature inspired methods are very successful. In our work and within this context, we introduce a new guided honey bee optimization algorithm for Maximal constraint satisfaction problems (Max-CSPs). The guidance consists of a new template concept and could be classified as a min-conflict heuristic. Our new template concept overcomes the former template concept guidance problems. It makes it possible to apply a fair min conflict heuristic. Compared to previous well-known honeybee algorithms for Max-CSPs, our approach is clearly better in term of CPU time and in term of solution quality.

LP/SDP Hierarchy Lower Bounds for Decoding Random LDPC Codes

Random $(d_{v},d_{c})$ - regular low-density parity-check (LDPC) codes, where each variable is involved in $d_{v}$ parity checks and each parity check involves $d_{c}$ variables are well-known to achieve the Shannon capacity of the binary symmetric channel, for sufficiently large $d_{v}$ and $d_{c}$ , under exponential time decoding. However, polynomial time algorithms are only known to correct a much smaller fraction of errors. One of the most powerful polynomial-time algorithms with a formal analysis is the linear programming (LP) decoding algorithm of Feldman et al., which is known to correct an $\Omega (1/d_{c})$ fraction of errors. In this paper, we show that fairly powerful extensions of LP decoding, based on the Sherali–Adams and Lasserre hierarchies, fail to correct much more errors than the basic LP-decoder. In particular, we show that: 1) for any values of $d_{v}$ and $d_{c}$ , a linear number of rounds of the Sherali–Adams LP hierarchy cannot correct more than an $O(1/d_{c})$ fraction of errors on a random $(d_{v},d_{c})$ -regular LDPC code; and 2) for any value of $d_{v}$ and infinitely many values of $d_{c}$ , a linear number of rounds of the Lasserre SDP hierarchy cannot correct more than an $O(1/d_{c})$ fraction of errors on a random $(d_{v},d_{c})$ -regular LDPC code. Our proofs use a new stretching and collapsing technique that allows us to leverage recent progress in the study of the limitations of LP/SDP hierarchies for Maximum Constraint Satisfaction Problems (Max-CSPs). The problem then reduces to the construction of special balanced pairwise independent distributions for Sherali–Adams and special cosets of balanced pairwise independent subgroups for Lasserre. Our (algebraic) construction for the Lasserre hierarchy is based on designing sets of points in ${\mathbb F}_{q}^{d}$ (for $q$ any power of 2 and $d = 2,3$ ) with special hyperplane-incidence properties—constructions that may be of independent interest. An intriguing consequence of our work is that expansion seems to be both the strength and the weakness of random regular LDPC codes. Some of our techniques are more generally applicable to a large class of Boolean CSPs called Min-Ones. In particular, for $k$ -Hypergraph Vertex Cover, we obtain an improved integrality gap of $k-1-\epsilon $ that holds after a linear number of rounds of the Lasserre hierarchy, for any $k = q+1$ with $q$ an arbitrary prime power. The best previous gap for a linear number of rounds was equal to $2-\epsilon $ and due to Schoenebeck.

Approximation Resistance from Pairwise-Independent Subgroups

We show optimal (up to a constant factor) NP-hardness for a maximum constraint satisfaction problem with k variables per constraint (Max- k CSP) whenever k is larger than the domain size. This follows from our main result concerning CSPs given by a predicate: A CSP is approximation resistant if its predicate contains a subgroup that is balanced pairwise independent. Our main result is analogous to Austrin and Mossel’s, bypassing their Unique-Games Conjecture assumption whenever the predicate is an abelian subgroup. Our main ingredient is a new gap-amplification technique inspired by XOR lemmas. Using this technique, we also improve the NP-hardness of approximating Independent-Set on bounded-degree graphs, Almost-Coloring, Label-Cover, and various other problems.

DOC-BRelax: A new multi-agent system to solve Distributed Constraint Optimization Problems

Many problems in multi-agent systems can be described as Distributed Constraint Satisfaction Problems (DCSPs), where the goal is to find a set of assignments to variables that satisfies all constraints among agents. However, when real-life application problems are formalized as DCSPs, they are often over-constrained and have no solution that satisfies all constraints. Moreover, the globalization of the economy and democratization of the Internet, boosted by the huge growth in information and communication technologies, have largely contributed to the expansion of numerous distributed architectures. Thus this paper provides a new distributed management and decision support system suitable to these interdependencies and these complex environments. We present a Distributed Optimization under Constraints Basic Relax (DOC-BRelax) as a new framework for dealing with over-constrained situations. We also present a version of this framework called DOC-MaxRelax and a new algorithm for solving Distributed Maximal Constraint Satisfaction Problems (DMCSPs).

A family of honey-bee optimization algorithms for Max-CSPs

This paper presents new approaches for maximal constraint satisfaction problems (Max-CSPs). Inspired by the honey- bee marriage process, our approaches try to reach a solution that satisfies as many constraints as possible in a reasonable period of time. Our approaches are a centralized and distributed honey-bee optimization enhanced by a new parameter called local optimum detector. The latter allows detection of the local optimum. In this paper, newer algorithms and their experimental results are presented.

Genetic based discrete particle swarm optimization for Elderly Day Care Center timetabling

The timetabling problem of local Elderly Day Care Centers (EDCCs) is formulated into a weighted maximum constraint satisfaction problem (Max-CSP) in this study. The EDCC timetabling problem is a multi-dimensional assignment problem, where users (elderly) are required to perform activities that require different venues and timeslots, depending on operational constraints. These constraints are categorized into two: hard constraints, which must be fulfilled strictly, and soft constraints, which may be violated but with a penalty. Numerous methods have been successfully applied to the weighted Max-CSP; these methods include exact algorithms based on branch and bound techniques, and approximation methods based on repair heuristics, such as the min-conflict heuristic. This study aims to explore the potential of evolutionary algorithms by proposing a genetic-based discrete particle swarm optimization (GDPSO) to solve the EDCC timetabling problem. The proposed method is compared with the min-conflict random-walk algorithm (MCRW), Tabu search (TS), standard particle swarm optimization (SPSO), and a guided genetic algorithm (GGA). Computational evidence shows that GDPSO significantly outperforms the other algorithms in terms of solution quality and efficiency. Proposed GDPSO combines min-conflict strategy, random walk and genetic operators.Proposed GDPSO outperforms in stability and efficiency than MCRW, TS, GGA, and SPSO.Proposed GDPSO has a faster and consistent rate of convergence.Proposed GDPSO results in superior performance to solve complex timetabling problem.

An approximability-related parameter on graphs―-properties and applications

Graph Theory We introduce a binary parameter on optimisation problems called separation. The parameter is used to relate the approximation ratios of different optimisation problems; in other words, we can convert approximability (and non-approximability) result for one problem into (non)-approximability results for other problems. Our main application is the problem (weighted) maximum H-colourable subgraph (Max H-Col), which is a restriction of the general maximum constraint satisfaction problem (Max CSP) to a single, binary, and symmetric relation. Using known approximation ratios for Max k-cut, we obtain general asymptotic approximability results for Max H-Col for an arbitrary graph H. For several classes of graphs, we provide near-optimal results under the unique games conjecture. We also investigate separation as a graph parameter. In this vein, we study its properties on circular complete graphs. Furthermore, we establish a close connection to work by Šámal on cubical colourings of graphs. This connection shows that our parameter is closely related to a special type of chromatic number. We believe that this insight may turn out to be crucial for understanding the behaviour of the parameter, and in the longer term, for understanding the approximability of optimisation problems such as Max H-Col.

Guided Genetic Algorithm: A New Template Concept

Abstract Guided genetic algorithm and dynamic distributed double guided genetic algorithm are based on nature laws and by the Neo-Darwinism theory. These evolutionary approaches were very successful addressing Maximal Constraint Satisfaction Problems (Max-CSPs). Our work is inspired by a little mistake when dealing with in these two algorithms guidance. In fact these approaches are guided by the min-conflict heuristic and the template concept. The used template is distorted. So, we introduce a new template concept in order to allow a better guidance. We suggest considering the percentages of violated constraints in place of their number. This concept is, then, applied to guide the genetic algorithms. In this paper, we compare the latter guided genetic algorithm with our new template guided genetic algorithm. The experimentations show that our new template guidance improves the optimization process to find best solutions in better time.

GPU-PSO: Parallel Particle Swarm Optimization Approaches on Graphical Processing Unit for Constraint Reasoning: Case of Max-CSPs

Abstract Constraint Satisfaction Problems (CSPs) occur now in different domains. Several methods are used to solve them. In particular, Particle Swarm Optimization (PSO) allows to solve efficiently CSPs by significantly reducing the calculation time to explore the search space of solutions. However, this metaheuristic is excessively costing when facing large instances. In this paper we address the Maximal Constraint Satisfaction Problems (Max-CSPs). We introduce a new resolution approach that allows solving efficiently the Max-CSPs even with large instances. Our purpose is to implement a PSO based method by using the GPU architecture as a parallel computing framework. In particular, we focus on the implementation of two parallel novel approaches. The first one is a parallel GPU-PSO for Max-CSPs (GPU-PSO) and the second one is a GPU distributed PSO for Max-CSPs (GPU-DPSO). Our experimental results show the efficiency of the two proposed approaches and their ability to exploit GPU architecture.

An upper (lower) bound for Max (Min) CSP

The random constraint satisfaction problem (CSP) instances generated by Model RB have been widely used in the field of CSP and have some nice features. In this paper, we consider two optimization versions of CSP, i.e., the maximum constraint satisfaction problem (Max-CSP) and the minimum satisfaction problem (Min-CSP) of Model RB. The problem of the Max-CSP is how to find an assignment to all the variables such that the maximum number of constraints are satisfied and the problem of Min-CSP is how to find an assignment to all the variables such that the minimum number of constraints are satisfied. We use the first moment method to prove that when r > 2α(1/p−1) (or p > 2α/(2α+r)), an upper bound of Max-CSP can be derived. Similarly, we can prove that when r > 2α(1/p−1) (or p > 2α/(2α + r)), a lower bound of Min-CSP can be derived.

On the NP-Hardness of Max-Not-2

We prove that, for any $\epsilon>0$, given a satisfiable instance of Max-NTW (Not-2), it is NP-hard to find an assignment that satisfies a fraction $\frac 58 +\epsilon$ of the constraints. This, up to the existence of $\epsilon$, matches the approximation ratio obtained by the trivial algorithm that just picks an assignment at random, and thus the result is tight. Said equivalently, the result proves that Max-NTW is approximation resistant on satisfiable instances, and this makes complete our understanding of arity three maximum constraint satisfaction problems with regards to approximation resistance.

A New Hybrid Distributed Double Guided Genetic Swarm Algorithm for Optimization and Constraint Reasoning

In this paper the authors propose a new distributed double guided hybrid algorithm combining the particle swarm optimization (PSO) with genetic algorithms (GA) to resolve maximal constraint satisfaction problems (Max-CSPs). It consists on a multi-agent approach inspired by a centralized version of hybrid algorithm called Genetical Swarm Optimization (GSO). Their approach consists of a set of evolutionary agents dynamically created and cooperating in order to find an optimal solution. Each agent executes its own hybrid algorithm and it is able to compute its own parameters. The authors’ approach is compared to the GSO. It demonstrates its superiority. They reached these results thanks to the distribution using multi-agent systems, diversification and intensification mechanisms.

Load Balancing for the Dynamic Distributed Double Guided Genetic Algorithm for MAX-CSPs

The Dynamic Distributed Double Guided Genetic Algorithm (D3G2A) deals with Maximal Constraint Satisfaction Problems. The approach consists in creating agents cooperating together to solve problems. This paper aims to improve the D3G2A. The main purpose is to balance agent loads this distributed approach. The proposed approach will redistribute the load of Species agents more equally in order to improve the CPU time. This improvement allows not only reduction of the number of Species agents but also decreases communications agents cost. In this regard, a sub-population is composed of chromosomes violating a number of constraints in the same interval. Secondly, another proposed approach will redistribute the work load. This improvement allows diminution of inactive Species agents and it results in a balanced workloads. In fact, by analogy with social animal flocks, Species agents cooperate together to do all tasks in a reduced CPU time. Several comparisons are made about new approaches with the old version of the D3G2A. Results are promising.

Near-optimal algorithms for maximum constraint satisfaction problems

In this article, we present two approximation algorithms for the maximum constraint satisfaction problem with k variables in each constraint (MAX k -CSP). Given a (1 − ε) satisfiable 2CSP our first algorithm finds an assignment of variables satisfying a 1 − O (√ε) fraction of all constraints. The best previously known result, due to Zwick, was 1 − O (ε 1/3 ). The second algorithm finds a ck /2 k approximation for the MAX k -CSP problem (where c > 0.44 is an absolute constant). This result improves the previously best known algorithm by Hast, which had an approximation guarantee of Ω( k /(2 k log k )). Both results are optimal assuming the unique games conjecture and are based on rounding natural semidefinite programming relaxations. We also believe that our algorithms and their analysis are simpler than those previously known.

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.

The approximability of MAX CSP with fixed-value constraints

In the maximum constraint satisfaction problem (MAX CSP), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given finite domain to the variables so as to maximize the number (or the total weight, for the weighted case) of satisfied constraints. This problem is NP-hard in general, and, therefore, it is natural to study how restricting the allowed types of constraints affects the approximability of the problem. In this article, we show that any MAX CSP problem with a finite set of allowed constraint types, which includes all fixed-value constraints (i.e., constraints of the form x = a ), is either solvable exactly in polynomial time or else is APX-complete, even if the number of occurrences of variables in instances is bounded. Moreover, we present a simple description of all polynomial-time solvable cases of our problem. This description relies on the well-known algebraic combinatorial property of supermodularity.