One of the key research interests in the area of Constraint Satisfaction Problem (CSP) is to identify tractable classes of constraints and develop efficient solutions for them. In this paper, we introduce generalized staircase (GS) constraints which is an important generalization of one such tractable class found in the literature, namely, staircase constraints. GS constraints are of two kinds, down staircase (DS) and up staircase (US). We first examine several properties of GS constraints, and then show that arc consistency is sufficient to determine a solution to a CSP over DS constraints. Further, we propose an optimal O(cd) time and space algorithm to compute arc consistency for GS constraints where c is the number of constraints and d is the size of the largest domain. Next, observing that arc consistency is not necessary for solving a DSCSP, we propose a more efficient algorithm for solving it. With regard to US constraints, arc consistency is not known to be sufficient to determine a solution, and therefore, methods such as path consistency or variable elimination are required. Since arc consistency acts as a subroutine for these existing methods, replacing it by our optimal O(cd) arc consistency algorithm produces a more efficient method for solving a USCSP.
GS constraints are a subclass of CRC constraints. While path consistency is sufficient to solve a CSP over CRC constraints, we show that arc consistency is sufficient to solve a CSP over DS constraints (please refer to Theorem ). However, it is not known that whether US constraints are solvable using arc consistency. For the class of CRC constraints, a path consistency based algorithm was proposed in @cite_8 that runs in @math time. Based on variable-elimination, @cite_31 stated an @math time algorithm for CRC constraints where @math is the degree of elimination of the triangulated graph of the underlying constraint network. Both these algorithms for CRC constraints use arc consistency as a subroutine which takes @math time. US constraints being a subclass of CRC constraints can be solved by either of these techniques mentioned above. Our arc consistency algorithm for GS constraints that runs in @math time, acting as a subroutine, thus improves the time complexity of the algorithm by @cite_31 to @math (which is linear in @math ), when applied to US constraints.
@cite_31 @cite_8
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This paper studies constraint satisfaction over connected row-convex (CRC) constraints. It shows that CRC constraints are closed under composition, intersection, and transposition, the basic operations of path-consistency algorithms. This establishes that path consistency over CRC constraints produces a minimal and decomposable network and is thus a polynomial-time decision procedure for CRC networks. This paper also presents a new path-consistency algorithm for CRC constraints running in time O(n(3)d(2)) and space O(n(2)d), where n is the number of variables and d is the size of the largest domain, improving the traditional time and space complexity by orders of magnitude. The paper also shows how to construct CRC constraints by conjunction and disjunction of a set of basic CRC constraints, highlighting how CRC constraints generalize monotone constraints and presenting interesting subclasses of CRC constraints. Experimental results show that the algorithm behaves well in practice. (C) 1999 Elsevier Science B.V. All rights reserved. We study the maximum flow problem subject to binary disjunctive constraints in a directed graph: A negative disjunctive constraint states that a certain pair of arcs in a digraph cannot be simultaneously used for sending flow in a feasible solution. In contrast to this, positive disjunctive constraints force that for certain pairs of arcs at least one arc has to carry flow in a feasible solution. It is convenient to represent the negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the arcs of the underlying graph, and whose edges encode the constraints. Analogously we represent the positive disjunctive constraints by a so-called forcing graph. For conflict graphs we prove that the maximum flow problem is strongly @math -hard, even if the conflict graph consists only of unconnected edges. This result still holds if the network consists only of disjoint paths of length three. For forcing graphs we also provide a sharp line between polynomially solvable and strongly @math -hard instances for the case where the flow values are required to be integral. Moreover, our hardness results imply that no polynomial time approximation algorithm can exist for both problems. In contrast to this we show that the maximum flow problem with a forcing graph can be solved efficiently if fractional flow values are allowed. Let @math be a nontrivial @math -ary predicate. Consider a random instance of the constraint satisfaction problem @math on @math variables with @math constraints, each being @math applied to @math randomly chosen literals. Provided the constraint density satisfies @math , such an instance is unsatisfiable with high probability. The problem is to efficiently find a proof of unsatisfiability. We show that whenever the predicate @math supports a @math - probability distribution on its satisfying assignments, the sum of squares (SOS) algorithm of degree @math (which runs in time @math ) refute a random instance of @math . In particular, the polynomial-time SOS algorithm requires @math constraints to refute random instances of CSP @math when @math supports a @math -wise uniform distribution on its satisfying assignments. Together with recent work of [LRS15], our result also implies that polynomial-size semidefinite programming relaxation for refutation requires at least @math constraints. Our results (which also extend with no change to CSPs over larger alphabets) subsume all previously known lower bounds for semialgebraic refutation of random CSPs. For every constraint predicate @math , they give a three-way hardness tradeoff between the density of constraints, the SOS degree (hence running time), and the strength of the refutation. By recent algorithmic results of [AOW15] and [RRS16], this full three-way tradeoff is , up to lower-order factors. Many real-life Constraint Satisfaction Problems (CSPs) involve some constraints similar to the alldifferent constraints. These constraints are called constraints of difference. They are defined on a subset of variables by a set of tuples for which the values occuring in the same tuple are all different. In this paper, a new filtering algorithm for these constraints is presented. It achieves the generalized arc-consistency condition for these non-binary constraints. It is based on matching theory and its complexity is low. In fact, for a constraint defined on a subset of p variables having domains of cardinality at most d, its space complexity is O(pd) and its time complexity is O(p2d2). This filtering algorithm has been successfully used in the system RESYN ( 1992), to solve the subgraph isomorphism problem. Constraint satisfaction problems (CSP) are a class of combinatorial problems that can be solved efficiently by combining consistency methods such as arc-consistency together with a backtracking search. However these techniques are not adapted to symmetrical CSP. In fact one can exhibit rather small CSP that cannot be solved with consistency techniques. The relevance of this symmetry problem to real world applications is very strong since it can prevent a CSP solver to solve even small instances of real world problems. This paper describes a general solution for this kind of problems. Both a theoretical study and experimental results using the constraint-based library PECOS are provided. Abstract Constraint satisfaction networks have been shown to be a very useful tool for knowledge representation in Artificial Intelligence applications. These networks often utilize local constraint propagation techniques to achieve local consistency (consistent labeling in vision). Such methods have been used extensively in the context of image understanding and interpretation, as well as planning, natural language analysis and truth maintenance systems. In this paper we study the parallel complexity of discrete relaxation, one of the most commonly used constraint propagation techniques. Since the constraint propagation procedures such as discrete relaxation appear to operate locally, it has been previously believed that the relaxation approach for achieving local consistency has a natural parallel solution. Our analysis suggests that a parallel solution is unlikely to improve the known sequential solutions by much. Specifically, we prove that the problem solved by discrete relaxation (arc consistency) is log-space complete for P (the class of polynomial-time deterministic sequential algorithms). Intuitively, this implies that discrete relaxation is inherently sequential and it is unlikely that we can solve the polynomial-time version of the consistent labeling problem in logarithmic time by using only a polynomial number of processors. Some practical implications of our result are discussed. We also provide a two-way transformation between AND OR graphs, propositional Horn satisfiability and local consistency in constraint networks that allows us to develop optimal linear-time algorithms for local consistency in constraint networks. We perform a comprehensive study of mappings between constraint satisfaction problems (CSPs) and propositional satisfiability (SAT). We analyse four different mappings of SAT problems into CSPs, and two of CSPs into SAT problems. For each mapping, we compare the impact of achieving arc-consistency on the CSP with unit propagation on the SAT problem. We then extend these results to CSP algorithms that maintain (some level of) arc-consistency during search like FC and MAC, and to the Davis-Putnam procedure (which performs unit propagation at each search node). Because of differences in the branching structure of their search, a result showing the dominance of achieving arc-consistency on the CSP over unit propagation on the SAT problem does not necessarily translate to the dominance of MAC over the Davis-Putnam procedure. These results provide insight into the relationship between propositional satisfiability and constraint satisfaction. We study the consistency and domain consistency problem for extended global cardinality (EGC) constraints. An EGC constraint consists of a set X of variables, a set D of values, a domain @math for each variable x, and a "cardinality set" K(d) of non-negative integers for each value d. The problem is to instantiate each variable x with a value in D(x) such that for each value d, the number of variables instantiated with d belongs to the cardinality set K(d). It is known that this problem is NP-complete in general, but solvable in polynomial time if all cardinality sets are intervals. First we pinpoint connections between EGC constraints and general factors in graphs. This allows us to extend the known polynomial-time case to certain non-interval cardinality sets. Second we consider EGC constraints under restrictions in terms of the treewidth of the value graph (the bipartite graph representing variable-value pairs) and the cardinality-width (the largest integer occurring in the cardinality sets). We show that EGC constraints can be solved in polynomial time for instances of bounded treewidth, where the order of the polynomial depends on the treewidth. We show that (subject to the complexity theoretic assumption FPT???W[1]) this dependency cannot be avoided without imposing additional restrictions. If, however, also the cardinality-width is bounded, this dependency gets removed and EGC constraints can be solved in linear time. We develop a framework for addressing correctness and timeliness-of-propagation issues for reactive constraints - global constraints or user-defined constraints that are implemented through constraint propagation. The notion of propagation completeness is introduced to capture timeliness of constraint propagation. A generalized form of arc-consistency is formulated which unifies many local consistency conditions in the literature. We show that propagation complete implementations of reactive constraints achieve this arc-consistency when propagation quiesces. Finally, we use the framework to state and prove an impossibility result: that CHR cannot implement a common relation with a desirable degree of timely constraint propagation. I describe and study the 'support encoding' of binary constraint satisfaction problems (CSP's) into boolean satisfiability (SAT). This is based on work by Kasif in 1990, in which clauses encode support information, rather than the encoding of conflicts in the standard 'direct encoding'. This enables arc consistency in the original CSP to be established by propagation in the translated SAT instance, providing an optimal worst case time algorithm for arc consistency [14]. I prove that the DP algorithm in SAT applied to the suport encoding behaves exactly like the MAC algorithm in CSP's. Using the SAT solver Chaff, I show that the support encoding can be used effectively to solve hard instances of random binary CSP's, and more effectively than the direct encoding on hard instances. Finally, I show that the local search algorithm WalkSAT can perform many times better on the support encoding than on the direct encoding. We study in this paper the problem of global consistency for qualitative constraints networks (QCNs) of the Point Algebra (PA) and the Interval Algebra (IA). In particular, we consider the subclass @math corresponding to the set of relations of PA except the relations ,= , and the subclass @math corresponding to pointizable relations of IA one can express by means of relations of @math . We prove that path-consistency implies global consistency for QCNs defined on these subclasses. Moreover, we show that with the subclasses corresponding to convex relations, there are unique greatest subclasses of PA and IA containing singleton relations satisfying this property. We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDP-hierarchy based algorithm for constraint satisfaction problems with 2-variable constraints (2-CSP's). More concretely, we show for every 2-CSP instance I a rounding algorithm for r rounds of the Lasserre SDP hierarchy for I that obtains an integral solution that is at most worse than the relaxation's value (normalized to lie in [0,1]), as long as r > k ( ) ( ) ;, where k is the alphabet size of I, @math , and @math denotes the number of eigenvalues larger than @math in the normalized adjacency matrix of the constraint graph of @math . In the case that @math is a instance, the threshold @math is only a polynomial in @math , and is independent of the alphabet size. Also in this case, we can give a non-trivial bound on the number of rounds for instance. In particular our result yields an SDP-hierarchy based algorithm that matches the performance of the recent subexponential algorithm of Arora, Barak and Steurer (FOCS 2010) in the worst case, but runs faster on a natural family of instances, thus further restricting the set of possible hard instances for Khot's Unique Games Conjecture. Our algorithm actually requires less than the @math constraints specified by the @math level of the Lasserre hierarchy, and in some cases @math rounds of our program can be evaluated in time @math . For certain subclasses of NP, @math P, or #P characterized by local constraints, it is known that if there exist any problems within that subclass that are not polynomial time computable, then all the problems in the subclass are NP-complete, @math P-complete, or #P-complete. Such dichotomy results have been proved for characterizations such as constraint satisfaction problems and directed and undirected graph homomorphism problems, often with additional restrictions. Here we give a dichotomy result for the more expressive framework of Holant problems. For example, these additionally allow for the expression of matching problems, which have had pivotal roles in the development of complexity theory. As our main result we prove the dichotomy theorem that, for the class @math P, every set of symmetric Holant signatures of any arities that is not polynomial time computable is @math P-complete. The result exploits some special properties of the class @math P and characterizes four distinct tractable ... We propose a simple declarative language for specifying a wide range of counting and occurrence constraints. This specification language is executable since it immediately provides a polynomial propagation algorithm. To illustrate the capabilities of this language, we specify a dozen global constraints taken from the literature. We observe one of three outcomes: we achieve generalized arc-consistency; we do not achieve generalized arc-consistency, but achieving generalized arc-consistency is NP-hard; we do not achieve generalized arc-consistency, but specialized propagation algorithms can do so in polynomial time. Experiments demonstrate that this specification language is both efficient and effective in practice. This paper introduces the SEQ BIN meta-constraint with a polytime algorithm achieving generalized arc-consistency. SEQ BIN can be used for encoding counting constraints such as CHANGE, SMOOTH, or INCREASING NVALUE. For all of them the time and space complexity is linear in the sum of domain sizes, which improves or equals the best known results of the literature. Artificial intelligence tasks which can be formulated as constraint satisfaction problems, with which this paper is for the most part concerned, are usually by solved backtracking the examining the thrashing behavior that nearly always accompanies backtracking, identifying three of its causes and proposing remedies for them we are led to a class of algorithms whoch can profitably be used to eliminate local (node, arc and path) inconsistencies before any attempt is made to construct a complete solution. A more general paradigm for attacking these tasks is the altenation of constraint manipulation and case analysis producing an OR problem graph which may be searched in any of the usual ways. Many authors, particularly Montanari and Waltz, have contributed to the development of these ideas; a secondary aim of this paper is to trace that history. The primary aim is to provide an accessible, unified framework, within which to present the algorithms including a new path consistency algorithm, to discuss their relationships and the may applications, both realized and potential of network consistency algorithms. This paper introduces the Increasing_Nvalue constraint, which restricts the number of distinct values assigned to a sequence of variables so that each variable in the sequence is less than or equal to its successor. This constraint is a specialization of the Nvalue constraint, motivated by symmetry breaking. Propagating the Nvalue constraint is known as an NP-hard problem. However, we show that the chain of non strict inequalities on the variables makes the problem polynomial. We propose an algorithm achieving generalized arc-consistency in O(ΣDi) time, where ΣDi is the sum of domain sizes. This algorithm is an improvement of filtering algorithms obtained by the automaton-based or the Slide-based reformulations. We evaluate our constraint on a resource allocation problem. For the classical minimum spanning tree problem we introduce disjunctive constraints for pairs of edges which can not be both included in the spanning tree at the same time. These constraints are represented by a conflict graph whose vertices correspond to the edges of the original graph. Edges in the conflict graph connect conflicting edges of the original graph. It is shown that the problem becomes strongly @math -hard even if the connected components of the conflict graph consist only of paths of length two. On the other hand, for conflict graphs consisting of disjoint edges (i.e. paths of length one) the problem remains polynomially solvable. In the centralized context, global constraints have been essential for the advancement of constraint reasoning. In this paper we propose to include soft global constraints in distributed constraint optimization problems (DCOPs). Looking for efficiency, we study possible decompositions of global constraints, including the use of extra variables. We extend the distributed search algorithm BnB-ADOPT+ to support these representations of global constraints. In addition, we explore the relation of global constraints with soft local consistency in DCOPs, in particular for the generalized soft arc consistency (GAC) level. We include specific propagators for some well-known soft global constraints. Finally, we provide empirical results on several benchmarks. We revisit the SeqBin constraint. This meta-constraint subsumes a number of important global constraints like Change, Smooth and IncreasingNValue. We show that the previously proposed filtering algorithm for SeqBin has two drawbacks even under strong restrictions: it does not detect bounds disentailment and it is not idempotent. We identify the cause for these problems, and propose a new propagator that overcomes both issues. Our algorithm is based on a connection to the problem of finding a path of a given cost in a restricted @math -partite graph. Our propagator enforces domain consistency in O(nd^2) and, for special cases of SeqBin that include Change, Smooth and IncreasingNValue, in O(nd) time.