scholarly journals Circuit Complexity Of Constraint Satisfaction Problems With Few Subpowers

2021 ◽  
Author(s):  
Amir El-Aooiti
Keyword(s):  
Polynomial Time ◽  
Constraint Satisfaction ◽  
Circuit Complexity ◽  
Constraint Satisfaction Problems ◽  
Solution Set ◽  
Alternative Method ◽  
Constraint Language ◽  
Np Complete

Although Constraint Satisfaction Problems (CSPs) are generally known to be NP-complete, placing restrictions on the constraint template can yield tractable subclasses. By studying the operations in the polymorphism of the constraint language, we can construct algorithms which solve our CSP in polynomial time. Previous results for CSPs with Mal’tsev [7] and generalized majority-minority operations [10] were improved to include CSPs with k-edge operations [15]. We present an alternative method to solve k-edge CSPs by utilizing Boolean trees placing the problem in the class NC2 . We do this by arranging the logical formulas describing the CSP into a Boolean tree where each leaf represents a constraint in the CSP. We take the conjunction of the constraint formulas yielding partial solutions at every step until we are left with a solution set at the root of the tree which satisfies all of the constraints.

scholarly journals Circuit Complexity Of Constraint Satisfaction Problems With Few Subpowers

2021 ◽  
Author(s):  
Amir El-Aooiti
Keyword(s):  
Polynomial Time ◽  
Constraint Satisfaction ◽  
Circuit Complexity ◽  
Constraint Satisfaction Problems ◽  
Solution Set ◽  
Alternative Method ◽  
Constraint Language ◽  
Np Complete

Although Constraint Satisfaction Problems (CSPs) are generally known to be NP-complete, placing restrictions on the constraint template can yield tractable subclasses. By studying the operations in the polymorphism of the constraint language, we can construct algorithms which solve our CSP in polynomial time. Previous results for CSPs with Mal’tsev [7] and generalized majority-minority operations [10] were improved to include CSPs with k-edge operations [15]. We present an alternative method to solve k-edge CSPs by utilizing Boolean trees placing the problem in the class NC2 . We do this by arranging the logical formulas describing the CSP into a Boolean tree where each leaf represents a constraint in the CSP. We take the conjunction of the constraint formulas yielding partial solutions at every step until we are left with a solution set at the root of the tree which satisfies all of the constraints.

scholarly journals Complexity Analysis of Generalized and Fractional Hypertree Decompositions

2021 ◽  
Vol 68 (5) ◽  
pp. 1-50
Author(s):  
Georg Gottlob ◽  
Matthias Lanzinger ◽  
Reinhard Pichler ◽  
Igor Razgon
Keyword(s):  
Polynomial Time ◽  
Constraint Satisfaction ◽  
Open Problem ◽  
Complexity Analysis ◽  
Decomposition Methods ◽  
Constraint Satisfaction Problems ◽  
Conjunctive Queries ◽  
Np Completeness ◽  
Np Complete ◽  
Efficient Approximation

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 .

scholarly journals A fine-grained arc-consistency algorithm for non-normalized constraint satisfaction problems

2011 ◽  
Vol 21 (4) ◽  
pp. 733-744 ◽  
Author(s):  
Marlene Arangú ◽  
Miguel Salido
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Programming ◽  
Real Life ◽  
Search Space ◽  
Constraint Satisfaction Problems ◽  
Fine Grained ◽  
Arc Consistency ◽  
Life Problems ◽  
Programming Techniques ◽  
Np Complete

A fine-grained arc-consistency algorithm for non-normalized constraint satisfaction problems Constraint programming is a powerful software technology for solving numerous real-life problems. Many of these problems can be modeled as Constraint Satisfaction Problems (CSPs) and solved using constraint programming techniques. However, solving a CSP is NP-complete so filtering techniques to reduce the search space are still necessary. Arc-consistency algorithms are widely used to prune the search space. The concept of arc-consistency is bidirectional, i.e., it must be ensured in both directions of the constraint (direct and inverse constraints). Two of the most well-known and frequently used arc-consistency algorithms for filtering CSPs are AC3 and AC4. These algorithms repeatedly carry out revisions and require support checks for identifying and deleting all unsupported values from the domains. Nevertheless, many revisions are ineffective, i.e., they cannot delete any value and consume a lot of checks and time. In this paper, we present AC4-OP, an optimized version of AC4 that manages the binary and non-normalized constraints in only one direction, storing the inverse founded supports for their later evaluation. Thus, it reduces the propagation phase avoiding unnecessary or ineffective checking. The use of AC4-OP reduces the number of constraint checks by 50% while pruning the same search space as AC4. The evaluation section shows the improvement of AC4-OP over AC4, AC6 and AC7 in random and non-normalized instances.

scholarly journals Constraint satisfaction problems in the logic LFP +rank

2021 ◽  
Author(s):  
Aklilu Habte
Keyword(s):  
Constraint Satisfaction ◽  
Gaussian Elimination ◽  
Natural Generalization ◽  
Constraint Satisfaction Problems ◽  
Theoretical Computer Science ◽  
Algebraic Operation ◽  
Finite Model ◽  
Theoretical Computer ◽  
Graph Homomorphisms ◽  
Np Complete

Constraint satisfaction problems (CSPs) are one of the central topics in theoretical computer science, in particular, in the area of artificial intelligence. Their computational complexity is due to relatively recent results from areas of mathematics, including finite-model-theory, algebra and graph homomorphisms. The main conjecture by Feder and Vardi states that any CSP over a finite relational template is either in P or is NP-complete. Further, it amounts to showing that every non NP-complete CSP can be expressed as an extension of first-order logic. A finite template is Mal'tsev, a compatible algebraic operation, which is closely related to an affine space over a finite field. The so-called Bulatov-Dalmau algorithm, a natural generalization of the Gaussian elimination on vector spaces, shows such CSPs are tractable. In this work, we prove that CSPs described over a finite template Mal'tsev are expressible in logic LFP+rnk, providing a logical proof that such CSPs are tractable.

scholarly journals On the Subexponential-Time Complexity of CSP

2015 ◽  
Vol 52 ◽  
pp. 203-234 ◽  
Author(s):  
Ronald De Haan ◽  
Iyad Kanj ◽  
Stefan Szeider
Keyword(s):  
Constraint Satisfaction ◽  
Time Complexity ◽  
Constraint Satisfaction Problem ◽  
Structural Parameters ◽  
Constraint Satisfaction Problems ◽  
Global Constraints ◽  
Brute Force ◽  
Subexponential Time ◽  
Complete Problems ◽  
Np Complete

Not all NP-complete problems share the same practical hardness with respect to exact computation. Whereas some NP-complete problems are amenable to efficient computational methods, others are yet to show any such sign. It becomes a major challenge to develop a theoretical framework that is more fine-grained than the theory of NP-completeness, and that can explain the distinction between the exact complexities of various NP-complete problems. This distinction is highly relevant for constraint satisfaction problems under natural restrictions, where various shades of hardness can be observed in practice. Acknowledging the NP-hardness of such problems, one has to look beyond polynomial time computation. The theory of subexponential-time complexity provides such a framework, and has been enjoying increasing popularity in complexity theory. An instance of the constraint satisfaction problem with n variables over a domain of d values can be solved by brute-force in dn steps (omitting a polynomial factor). In this paper we study the existence of subexponential-time algorithms, that is, algorithms running in do(n) steps, for various natural restrictions of the constraint satisfaction problem. We consider both the constraint satisfaction problem in which all the constraints are given extensionally as tables, and that in which all the constraints are given intensionally in the form of global constraints. We provide tight characterizations of the subexponential-time complexity of the aforementioned problems with respect to several natural structural parameters, which allows us to draw a detailed landscape of the subexponential-time complexity of the constraint satisfaction problem. Our analysis provides fundamental results indicating whether and when one can significantly improve on the brute-force search approach for solving the constraint satisfaction problem.

scholarly journals Compiling CSPs: A Complexity Map of (Non-Deterministic) Multivalued Decision Diagrams

2014 ◽  
Vol 23 (04) ◽  
pp. 1460015 ◽  
Author(s):  
Jérôme Amilhastre ◽  
Hélène Fargier ◽  
Alexandre Niveau ◽  
Cédric Pralet
Keyword(s):  
Polynomial Time ◽  
Constraint Satisfaction ◽  
Binary Decision Diagrams ◽  
Constraint Satisfaction Problems ◽  
Decision Diagrams ◽  
Np Hard ◽  
Ordered Binary Decision Diagrams ◽  
Binary Decision

Constraint Satisfaction Problems (CSPs) offer a powerful framework for representing a great variety of problems. The difficulty is that most of the requests associated with CSPs are NP-hard. When these requests have to be addressed online, Multivalued Decision Diagrams (MDDs) have been proposed as a way to compile CSPs. In the present paper, we draw a compilation map of MDDs, in the spirit of the NNF compilation map, analyzing MDDs according to their succinctness and to their tractable transformations and queries. Deterministic ordered MDDs are a generalization of ordered binary decision diagrams to non-Boolean domains: unsurprisingly, they have similar capabilities. More interestingly, our study puts forward the interest of non-deterministic ordered MDDs: when restricted to Boolean domains, they capture OBDDs and DNFs as proper subsets and have performances close to those of DNNFs. The comparison to classical, deterministic MDDs shows that relaxing the determinism requirement leads to an increase in succinctness and allows more transformations to be satisfied in polynomial time (typically, the disjunctive ones). Experiments on random problems confirm the gain in succinctness.

scholarly journals Generating Hard Satisfiable Formulas by Hiding Solutions Deceptively

2007 ◽  
Vol 28 ◽  
pp. 107-118 ◽  
Author(s):  
H. Jia ◽  
C. Moore ◽  
D. Strain
Keyword(s):  
Polynomial Time ◽  
Constraint Satisfaction ◽  
Search Algorithms ◽  
Constraint Satisfaction Problems ◽  
Experimental Results ◽  
New Method ◽  
Truth Assignment ◽  
Benchmark Instances ◽  
Problem Generator

To test incomplete search algorithms for constraint satisfaction problems such as 3-SAT, we need a source of hard, but satisfiable, benchmark instances. A simple way to do this is to choose a random truth assignment A, and then choose clauses randomly from among those satisfied by A. However, this method tends to produce easy problems, since the majority of literals point toward the "hidden'' assignment A. Last year, Achlioptas, Jia and Moore proposed a problem generator that cancels this effect by hiding both A and its complement. While the resulting formulas appear to be just as hard for DPLL algorithms as random 3-SAT formulas with no hidden assignment, they can be solved by WalkSAT in only polynomial time. Here we propose a new method to cancel the attraction to A, by choosing a clause with t > 0 literals satisfied by A with probability proportional to q^t for some q < 1. By varying q, we can generate formulas whose variables have no bias, i.e., which are equally likely to be true or false; we can even cause the formula to "deceptively'' point away from A. We present theoretical and experimental results suggesting that these formulas are exponentially hard both for DPLL algorithms and for incomplete algorithms such as WalkSAT.

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