scholarly journals The Complexity of Promise SAT on Non-Boolean Domains

2021 ◽  
Vol 13 (4) ◽  
pp. 1-20
Author(s):  
Alex Brandts ◽  
Marcin Wrochna ◽  
Stanislav Živný
Keyword(s):  
Polynomial Time ◽  
Constraint Satisfaction ◽  
Algebraic Approach ◽  
Constraint Satisfaction Problems ◽  
Hardness Of Approximation ◽  
Np Hard ◽  
Cover Problem ◽  
Finite Domains ◽  
Np Hardness ◽  
Cnf Formulas

While 3-SAT is NP-hard, 2-SAT is solvable in polynomial time. Austrin et al. [SICOMP’17] proved a result known as “(2+ɛ)-SAT is NP-hard.” They showed that the problem of distinguishing k -CNF formulas that are g -satisfiable (i.e., some assignment satisfies at least g literals in every clause) from those that are not even 1-satisfiable is NP-hard if g/k < 1/2 and is in P otherwise. We study a generalisation of SAT on arbitrary finite domains, with clauses that are disjunctions of unary constraints, and establish analogous behaviour. Thus, we give a dichotomy for a natural fragment of promise constraint satisfaction problems ( PCSPs ) on arbitrary finite domains. The hardness side is proved using the algebraic approach via a new general NP-hardness criterion on polymorphisms, which is based on a gap version of the Layered Label Cover problem. We show that previously used criteria are insufficient—the problem hence gives an interesting benchmark of algebraic techniques for proving hardness of approximation in problems such as PCSPs.

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 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 Extended Formulation for CSP that is Compact for Instances of Bounded Treewidth

10.37236/5474 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Petr Kolman ◽  
Martin Koutecký
Keyword(s):  
Constraint Satisfaction ◽  
Upper Bounds ◽  
Constraint Satisfaction Problems ◽  
Np Hard ◽  
Extended Formulation ◽  
Bounded Treewidth ◽  
Extension Complexity ◽  
Constraint Graph ◽  
Hard Problems ◽  
Np Hard Problems

In this paper we provide an extended formulation for the class of constraint satisfaction problems and prove that its size is polynomial for instances whose constraint graph has bounded treewidth. This implies new upper bounds on extension complexity of several important NP-hard problems on graphs of bounded treewidth.

scholarly journals k-Approximate Quasiperiodicity Under Hamming and Edit Distance

Algorithmica ◽  
2021 ◽  
Author(s):  
Aleksander Kędzierski ◽  
Jakub Radoszewski
Keyword(s):  
Lower Bounds ◽  
Polynomial Time ◽  
Edit Distance ◽  
Hamming Distance ◽  
Efficient Algorithms ◽  
Np Hard ◽  
Total Cost ◽  
Polynomial Time Algorithms ◽  
Np Hardness

AbstractQuasiperiodicity in strings was introduced almost 30 years ago as an extension of string periodicity. The basic notions of quasiperiodicity are cover and seed. A cover of a text T is a string whose occurrences in T cover all positions of T. A seed of text T is a cover of a superstring of T. In various applications exact quasiperiodicity is still not sufficient due to the presence of errors. We consider approximate notions of quasiperiodicity, for which we allow approximate occurrences in T with a small Hamming, Levenshtein or weighted edit distance. In previous work Sim et al. (J Korea Inf Sci Soc 29(1):16–21, 2002) and Christodoulakis et al. (J Autom Lang Comb 10(5/6), 609–626, 2005) showed that computing approximate covers and seeds, respectively, under weighted edit distance is NP-hard. They, therefore, considered restricted approximate covers and seeds which need to be factors of the original string T and presented polynomial-time algorithms for computing them. Further algorithms, considering approximate occurrences with Hamming distance bounded by k, were given in several contributions by Guth et al. They also studied relaxed approximate quasiperiods. We present more efficient algorithms for computing restricted approximate covers and seeds. In particular, we improve upon the complexities of many of the aforementioned algorithms, also for relaxed quasiperiods. Our solutions are especially efficient if the number (or total cost) of allowed errors is small. We also show conditional lower bounds for computing restricted approximate covers and prove NP-hardness of computing non-restricted approximate covers and seeds under the Hamming distance.

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.

SUBMODULARITY-BASED DECOMPOSING FOR VALUED CSP

2013 ◽  
Vol 22 (02) ◽  
pp. 1350006
Author(s):  
MAHER HELAOUI ◽  
WADY NAANAA ◽  
BECHIR AYEB
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Satisfaction Problems ◽  
Combinatorial Problems ◽  
Submodular Functions ◽  
Problem Decomposition ◽  
Decomposition Scheme ◽  
Domain Partitioning ◽  
Np Hardness

Many combinatorial problems can be formulated as Valued Constraint Satisfaction Problems (VCSPs). In this framework, the constraints are defined by means of valuation functions to reflect several degrees of coherence. Despite the NP-hardness of the VCSP, tractable versions can be obtained by forcing the allowable valuation functions to have specific features. This is the case for submodular VCSPs, i.e. VCSPs that involve submodular valuation functions only. In this paper, we propose a problem decomposition scheme for binary VCSPs that takes advantage of submodular functions even when the studied problem is not submodular. The proposed scheme consists in decomposing the problem to be solved into a set of submodular, then tractable, subproblems. The decomposition scheme combines two techniques that where already used in the framework of constraint-based reasoning, but in separate manner. These techniques are domain partitioning and value permutation.

SHARED MEMORY IMPLEMENTATION OF CONSTRAINT SATISFACTION PROBLEM RESOLUTION

2001 ◽  
Vol 11 (04) ◽  
pp. 487-501 ◽  
Author(s):  
ZINEB HABBAS ◽  
MICHAËL KRAJECKI ◽  
DANIEL SINGER
Keyword(s):  
Load Balancing ◽  
Shared Memory ◽  
Constraint Satisfaction ◽  
Parallel Implementation ◽  
Constraint Satisfaction Problem ◽  
Search Tree ◽  
Constraint Satisfaction Problems ◽  
Memory Model ◽  
Problem Resolution ◽  
Finite Domains

Many problems in Computer Science, especially in Artificial Intelligence, can be formulated as Constraint Satisfaction Problems (CSP). This paper presents a parallel implementation of the Forward-Checking algorithm for solving a binary CSP over finite domains. Its main contribution is to use a simple decomposition strategy in order to distribute dynamically the search tree among machines. The feasibility and benefit of this approach are studied for a Shared Memory model. An implementation is drafted using the new emergent standard OpenMP library for shared memory, thus controlling load balancing. We mainly highlight satisfactory efficiencies without using any tricky load balancing policy. All the experiments were carried out running on the Sillicon Graphics Origin 2000 parallel machine.

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