Constraint satisfaction problems in the logic LFP +rank

Algebraic Operation ◽

Finite Model ◽

Theoretical Computer ◽

Graph Homomorphisms ◽

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.

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

International Journal of Applied Mathematics and Computer Science ◽

10.2478/v10006-011-0058-2 ◽

2011 ◽

Vol 21 (4) ◽

pp. 733-744 ◽

Cited By ~ 1

Author(s):

Marlene Arangú ◽

Miguel Salido

Keyword(s):

Constraint Satisfaction ◽

Constraint Programming ◽

Real Life ◽

Search Space ◽

Fine Grained ◽

Arc Consistency ◽

Life Problems ◽

Programming Techniques ◽

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.

Mathematical Foundations of Computer Science 2007 - Lecture Notes in Computer Science ◽

Combinatorial Proof that Subprojective Constraint Satisfaction Problems are NP-Complete

10.1007/978-3-540-74456-6_16 ◽

2007 ◽

pp. 159-170 ◽

Cited By ~ 1

Author(s):

Jaroslav Nešetřil ◽

Mark Siggers

Keyword(s):

Constraint Satisfaction ◽

Combinatorial Proof ◽

Circuit Complexity Of Constraint Satisfaction Problems With Few Subpowers

10.32920/ryerson.14649945 ◽

2021 ◽

Author(s):

Amir El-Aooiti

Keyword(s):

Polynomial Time ◽

Constraint Satisfaction ◽

Circuit Complexity ◽

Solution Set ◽

Alternative Method ◽

Constraint Language ◽

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.

Open k-monopolies in graphs: complexity and related concepts

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.654 ◽

2016 ◽

Vol Vol. 18 no. 3 (Graph Theory) ◽

Author(s):

Dorota Kuziak ◽

Iztok Peterin ◽

Ismael G. Yero

Keyword(s):

Positive Integer ◽

Computer Science ◽

Long Range ◽

Chordal Graphs ◽

Discrete Mathematics ◽

Voting Systems ◽

Minimum Cardinality ◽

Theoretical Computer ◽

Closed monopolies in graphs have a quite long range of applications in several problems related to overcoming failures, since they frequently have some common approaches around the notion of majorities, for instance to consensus problems, diagnosis problems or voting systems. We introduce here open $k$-monopolies in graphs which are closely related to different parameters in graphs. Given a graph $G=(V,E)$ and $X\subseteq V$, if $\delta_X(v)$ is the number of neighbors $v$ has in $X$, $k$ is an integer and $t$ is a positive integer, then we establish in this article a connection between the following three concepts: - Given a nonempty set $M\subseteq V$ a vertex $v$ of $G$ is said to be $k$-controlled by $M$ if $\delta_M(v)\ge \frac{\delta_V(v)}{2}+k$. The set $M$ is called an open $k$-monopoly for $G$ if it $k$-controls every vertex $v$ of $G$. - A function $f: V\rightarrow \{-1,1\}$ is called a signed total $t$-dominating function for $G$ if $f(N(v))=\sum_{v\in N(v)}f(v)\geq t$ for all $v\in V$. - A nonempty set $S\subseteq V$ is a global (defensive and offensive) $k$-alliance in $G$ if $\delta_S(v)\ge \delta_{V-S}(v)+k$ holds for every $v\in V$. In this article we prove that the problem of computing the minimum cardinality of an open $0$-monopoly in a graph is NP-complete even restricted to bipartite or chordal graphs. In addition we present some general bounds for the minimum cardinality of open $k$-monopolies and we derive some exact values. Comment: 18 pages, Discrete Mathematics & Theoretical Computer Science (2016)

E. Grädel, P.G. Kolaitis, L. Libkin, M. Marx, J. Spencer, M.Y. Vardi, Y. Venema and S. Weinstein. Finite model theory and its applications. Texts in Theoretical Computer Science. Springer, Berlin, 2007, xiii + 437 pp.

Bulletin of Symbolic Logic ◽

10.1017/s1079898600000846 ◽

2010 ◽

Vol 16 (3) ◽

pp. 406-407

Author(s):

Stephan Kreutzer

Keyword(s):

Computer Science ◽

Model Theory ◽

Finite Model ◽

Theoretical Computer ◽

Finite Model Theory

On the Subexponential-Time Complexity of CSP

Journal of Artificial Intelligence Research ◽

10.1613/jair.4540 ◽

2015 ◽

Vol 52 ◽

pp. 203-234 ◽

Cited By ~ 2

Author(s):

Ronald De Haan ◽

Iyad Kanj ◽

Stefan Szeider

Keyword(s):

Constraint Satisfaction ◽

Time Complexity ◽

Constraint Satisfaction Problem ◽

Structural Parameters ◽

Global Constraints ◽

Brute Force ◽

Subexponential Time ◽

Complete Problems ◽

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.

Fine-Grained Time Complexity of Constraint Satisfaction Problems

ACM Transactions on Computation Theory ◽

10.1145/3434387 ◽

2021 ◽

Vol 13 (1) ◽

pp. 1-32

Author(s):

Peter Jonsson ◽

Victor Lagerkvist ◽

Biman Roy

Keyword(s):

Constraint Satisfaction ◽

Time Complexity ◽

Constraint Satisfaction Problem ◽

Finite Domain ◽

Infinite Domain ◽

Worst Case ◽

Fine Grained ◽

Restricted Problems ◽

We study the constraint satisfaction problem (CSP) parameterized by a constraint language Γ (CSPΓ) and how the choice of Γ affects its worst-case time complexity. Under the exponential-time hypothesis (ETH), we rule out the existence of subexponential algorithms for finite-domain NP-complete CSPΓ problems. This extends to certain infinite-domain CSPs and structurally restricted problems. For CSPs with finite domain D and where all unary relations are available, we identify a relation S D such that the time complexity of the NP-complete problem CSP({ S D }) is a lower bound for all NP-complete CSPs of this kind. We also prove that the time complexity of CSP({ S D }) strictly decreases when |D| increases (unless the ETH is false) and provide stronger complexity results in the special case when |D|=3.

Complexity Results for 1-safe Nets

DAIMI Report Series ◽

10.7146/dpb.v22i455.6773 ◽

1993 ◽

Vol 22 (455) ◽

Cited By ~ 2

Author(s):

Allan Cheng ◽

Javier Esparza ◽

Jens Palsberg

Keyword(s):

Petri Nets ◽

Computer Science ◽

Free Choice ◽

Theoretical Computer ◽

Software Technology ◽

Complexity Results ◽

We study the complexity of several standard problems for 1-safe Petri nets and some of its subclasses. We prove that reachability, liveness, and deadlock are all PSPACE-complete for 1-safe nets. We also prove that deadlock is NP-complete for free-choice nets and for 1-safe free-choice nets. Finally, we prove that for arbitrary Petri nets, deadlock is equivalent to reachability and liveness. This paper is to be presented at FST & TCS, Foundations of Software Technology & Theoretical Computer Science, to be held 15-17 December 1993, in Bombay, India.A version of the paper with most proofs omitted is to appear in the proceedings.

Complexity Analysis of Generalized and Fractional Hypertree Decompositions

Journal of the ACM ◽

10.1145/3457374 ◽

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 ◽

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 .