scholarly journals Supermodularity on chains and complexity of maximum constraint satisfaction

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Vladimir Deineko ◽  
Peter Jonsson ◽  
Mikael Klasson ◽  
Andrei Krokhin
Keyword(s):  
Polynomial Time ◽  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Finite Domain ◽  
Finite Collection ◽  
Simple Description ◽  
Weighted Constraints ◽  
Finite Set ◽  
International Audience ◽  
Overlapping Sets

International audience In the maximum constraint satisfaction problem ($\mathrm{Max \; CSP}$), 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 maximise the number (or the total weight) of satisfied constraints. This problem is $\mathrm{NP}$-hard in general so it is natural to study how restricting the allowed types of constraints affects the complexity of the problem. In this paper, we show that any $\mathrm{Max \; CSP}$ problem with a finite set of allowed constraint types, which includes all constants (i.e. constraints of the form $x=a$), is either solvable in polynomial time or is $\mathrm{NP}$-complete. Moreover, we present a simple description of all polynomial-time solvable cases of our problem. This description uses the well-known combinatorial property of supermodularity.

scholarly journals A Constraint Satisfaction Framework for Executing Perceptions and Actions in Diagrammatic Reasoning

2010 ◽  
Vol 39 ◽  
pp. 373-427 ◽  
Author(s):  
B. Banerjee ◽  
B. Chandrasekaran
Keyword(s):  
Problem Solving ◽  
Polynomial Time ◽  
Constraint Satisfaction ◽  
Complexity Analysis ◽  
Constraint Satisfaction Problem ◽  
Diagrammatic Reasoning ◽  
General Purpose ◽  
Spatial Problem ◽  
General Architecture ◽  
Spatial Problems

Diagrammatic reasoning (DR) is pervasive in human problem solving as a powerful adjunct to symbolic reasoning based on language-like representations. The research reported in this paper is a contribution to building a general purpose DR system as an extension to a SOAR-like problem solving architecture. The work is in a framework in which DR is modeled as a process where subtasks are solved, as appropriate, either by inference from symbolic representations or by interaction with a diagram, i.e., perceiving specified information from a diagram or modifying/creating objects in a diagram in specified ways according to problem solving needs. The perceptions and actions in most DR systems built so far are hand-coded for the specific application, even when the rest of the system is built using the general architecture. The absence of a general framework for executing perceptions/actions poses as a major hindrance to using them opportunistically -- the essence of open-ended search in problem solving. Our goal is to develop a framework for executing a wide variety of specified perceptions and actions across tasks/domains without human intervention. We observe that the domain/task-specific visual perceptions/actions can be transformed into domain/task-independent spatial problems. We specify a spatial problem as a quantified constraint satisfaction problem in the real domain using an open-ended vocabulary of properties, relations and actions involving three kinds of diagrammatic objects -- points, curves, regions. Solving a spatial problem from this specification requires computing the equivalent simplified quantifier-free expression, the complexity of which is inherently doubly exponential. We represent objects as configuration of simple elements to facilitate decomposition of complex problems into simpler and similar subproblems. We show that, if the symbolic solution to a subproblem can be expressed concisely, quantifiers can be eliminated from spatial problems in low-order polynomial time using similar previously solved subproblems. This requires determining the similarity of two problems, the existence of a mapping between them computable in polynomial time, and designing a memory for storing previously solved problems so as to facilitate search. The efficacy of the idea is shown by time complexity analysis. We demonstrate the proposed approach by executing perceptions and actions involved in DR tasks in two army applications.

scholarly journals On Maltsev Digraphs

10.37236/4419 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Catarina Carvalho ◽  
Laszlo Egri ◽  
Marcel Jackson ◽  
Todd Niven
Keyword(s):  
Polynomial Time ◽  
Constraint Satisfaction ◽  
Disjoint Union ◽  
Constraint Satisfaction Problem ◽  
Time Algorithm ◽  
Directed Path ◽  
Inductive Construction ◽  
Semilattice Operation ◽  
Directed Cycles

We study digraphs preserved by a Maltsev operation: Maltsev digraphs. We show that these digraphs retract either onto a directed path or to the disjoint union of directed cycles, showing in this way that the constraint satisfaction problem for Maltsev digraphs is in logspace, L. We then generalize results from Kazda (2011) to show that a Maltsev digraph is preserved not only by a majority operation, but by a class of other operations (e.g., minority, Pixley) and obtain a $O(|V_G|^4)$-time algorithm to recognize Maltsev digraphs. We also prove analogous results for digraphs preserved by conservative Maltsev operations which we use to establish that the list homomorphism problem for Maltsev digraphs is in L. We then give a polynomial time characterisation of Maltsev digraphs admitting a conservative 2-semilattice operation. Finally, we give a simple inductive construction of directed acyclic digraphs preserved by a Maltsev operation, and relate them with series parallel digraphs. 

scholarly journals Tiling a Rectangle with Polyominoes

2003 ◽  
Vol DMTCS Proceedings vol. AB,... (Proceedings) ◽  
Author(s):  
Olivier Bodini
Keyword(s):  
Polynomial Time ◽  
Finite Union ◽  
Finite Set ◽  
International Audience

International audience A polycube in dimension $d$ is a finite union of unit $d$-cubes whose vertices are on knots of the lattice $\mathbb{Z}^d$. We show that, for each family of polycubes $E$, there exists a finite set $F$ of bricks (parallelepiped rectangles) such that the bricks which can be tiled by $E$ are exactly the bricks which can be tiled by $F$. Consequently, if we know the set $F$, then we have an algorithm to decide in polynomial time if a brick is tilable or not by the tiles of $E$.

scholarly journals Constructions for Clumps Statistics.

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Frédérique Bassino ◽  
Julien Clément ◽  
Julien Fayolle ◽  
Pierre Nicodème
Keyword(s):  
Poisson Distribution ◽  
Unsolved Problem ◽  
Probabilistic Approach ◽  
Compound Poisson Distribution ◽  
Exact Results ◽  
Combinatorial Method ◽  
Asymptotic Results ◽  
Finite Set ◽  
International Audience ◽  
Overlapping Sets

International audience We consider a component of the word statistics known as clump; starting from a finite set of words, clumps are maximal overlapping sets of these occurrences. This object has first been studied by Schbath with the aim of counting the number of occurrences of words in random texts. Later work with similar probabilistic approach used the Chen-Stein approximation for a compound Poisson distribution, where the number of clumps follows a law close to Poisson. Presently there is no combinatorial counterpart to this approach, and we fill the gap here. We also provide a construction for the yet unsolved problem of clumps of an arbitrary finite set of words. In contrast with the probabilistic approach which only provides asymptotic results, the combinatorial method provides exact results that are useful when considering short sequences.

scholarly journals Algebraic Approach to Promise Constraint Satisfaction

2021 ◽  
Vol 68 (4) ◽  
pp. 1-66
Author(s):  
Libor Barto ◽  
Jakub Bulín ◽  
Andrei Krokhin ◽  
Jakub Opršal
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Algebraic Approach ◽  
Graph Colouring ◽  
Finite Domain ◽  
New Class ◽  
The Past ◽  
Open Questions ◽  
Constraint Language ◽  
Special Cases

The complexity and approximability of the constraint satisfaction problem (CSP) has been actively studied over the past 20 years. A new version of the CSP, the promise CSP (PCSP), has recently been proposed, motivated by open questions about the approximability of variants of satisfiability and graph colouring. The PCSP significantly extends the standard decision CSP. The complexity of CSPs with a fixed constraint language on a finite domain has recently been fully classified, greatly guided by the algebraic approach, which uses polymorphisms—high-dimensional symmetries of solution spaces—to analyse the complexity of problems. The corresponding classification for PCSPs is wide open and includes some long-standing open questions, such as the complexity of approximate graph colouring, as special cases. The basic algebraic approach to PCSP was initiated by Brakensiek and Guruswami, and in this article, we significantly extend it and lift it from concrete properties of polymorphisms to their abstract properties. We introduce a new class of problems that can be viewed as algebraic versions of the (Gap) Label Cover problem and show that every PCSP with a fixed constraint language is equivalent to a problem of this form. This allows us to identify a “measure of symmetry” that is well suited for comparing and relating the complexity of different PCSPs via the algebraic approach. We demonstrate how our theory can be applied by giving both general and specific hardness/tractability results. Among other things, we improve the state-of-the-art in approximate graph colouring by showing that, for any k ≥ 3, it is NP-hard to find a (2 k -1)-colouring of a given k -colourable graph.

scholarly journals Fine-Grained Time Complexity of Constraint Satisfaction Problems

2021 ◽  
Vol 13 (1) ◽  
pp. 1-32
Author(s):  
Peter Jonsson ◽  
Victor Lagerkvist ◽  
Biman Roy
Keyword(s):  
Constraint Satisfaction ◽  
Time Complexity ◽  
Constraint Satisfaction Problem ◽  
Constraint Satisfaction Problems ◽  
Finite Domain ◽  
Infinite Domain ◽  
Worst Case ◽  
Fine Grained ◽  
Restricted Problems ◽  
Np Complete

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.

scholarly journals Galois Connections for Patterns: An Algebra of Labelled Graphs

2021 ◽  
pp. 125-150
Author(s):  
David A. Cohen ◽  
Martin C. Cooper ◽  
Peter G. Jeavons ◽  
Stanislav Živný
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Galois Connection ◽  
Language Classes ◽  
Finite Set ◽  
Binary Constraint ◽  
Tree Width ◽  
To Come ◽  
Forbidden Patterns ◽  
Near Unanimity

AbstractA pattern is a generic instance of a binary constraint satisfaction problem (CSP) in which the compatibility of certain pairs of variable-value assignments may be unspecified. The notion of forbidden pattern has led to the discovery of several novel tractable classes for the CSP. However, for this field to come of age it is time for a theoretical study of the algebra of patterns. We present a Galois connection between lattices composed of sets of forbidden patterns and sets of generic instances, and investigate its consequences. We then extend patterns to augmented patterns and exhibit a similar Galois connection. Augmented patterns are a more powerful language than flat (i.e. non-augmented) patterns, as we demonstrate by showing that, for any $$k \ge 1$$ k ≥ 1 , instances with tree-width bounded by k cannot be specified by forbidding a finite set of flat patterns but can be specified by a finite set of augmented patterns. A single finite set of augmented patterns can also describe the class of instances such that each instance has a weak near-unanimity polymorphism of arity k (thus covering all tractable language classes).We investigate the power of forbidding augmented patterns and discuss their potential for describing new tractable classes.

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

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Robert Engström ◽  
Tommy Färnqvist ◽  
Peter Jonsson ◽  
Johan Thapper
Keyword(s):  
Graph Theory ◽  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Complete Graphs ◽  
Arbitrary Graph ◽  
Symmetric Relation ◽  
Binary Parameter ◽  
International Audience ◽  
Unique Games Conjecture ◽  
Unique Games

Graph Theory International audience 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.

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