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
1985525233 2949393806 2007385254 1565834133 2164279585 2123103890 2343345052 1524973174 122825473 1818659651 1550483759 2067633747 2151239055 2130182605 1594857028 2164745460 2952621790 2953012544 2118406528 2065424480
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 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 . We initiate a study of when the value of mathematical relaxations such as linear and semi-definite programs for constraint satisfaction problems (CSPs) is approximately preserved when restricting the instance to a sub-instance induced by a small random subsample of the variables. Let C be a family of CSPs such as 3SAT, Max-Cut, etc., and let Π be a mathematical program that is a relaxation for C, in the sense that for every instance P ∈ C, Π(P) is a number in [0, 1] upper bounding the maximum fraction of satisfiable constraints of P. Loosely speaking, we say that subsampling holds for C and Π if for every sufficiently dense instance P ∈ C and every e > 0, if we let P' be the instance obtained by restricting P to a sufficiently large constant number of variables, then Π(P') ∈ (1 ± e)Π(P). We say that weak subsampling holds if the above guarantee is replaced with Π(P') = 1 − θ(γ) whenever Π(P) = 1 − γ, where θ hides only absolute constants. We obtain both positive and negative results, showing that: 1. Subsampling holds for the BasicLP and BasicSDP programs. BasicSDP is a variant of the semi-definite program considered by Raghavendra (2008), who showed it gives an optimal approximation factor for every constraint-satisfaction problem under the unique games conjecture. BasicLP is the linear programming analog of BasicSDP. 2. For tighter versions of BasicSDP obtained by adding additional constraints from the Lasserre hierarchy, weak subsampling holds for CSPs of unique games type. 3. There are non-unique CSPs for which even weak subsampling fails for the above tighter semi-definite programs. Also there are unique CSPs for which (even weak) subsampling fails for the Sherali-Adams linear programming hierarchy. As a corollary of our weak subsampling for strong semi-definite programs, we obtain a polynomial-time algorithm to certify that random geometric graphs (of the type considered by Feige and Schechtman, 2002) of max-cut value 1 − γ have a cut value at most 1 − γ 10. More generally, our results give an approach to obtaining average-case algorithms for CSPs using semi-definite programming hierarchies. This report outlines a more general formulation of sequential patterns, which uni es the generalized patterns proposed by Srikant and Agarwal [SA96] and episode discovery approach taken by [MTV97]. We show that just by varying the values of timing constraint parameters and counting methods, our formulation can be made identical to either one of these. Furthermore, our approach de nes several other counting methods which could be suitable for various applications. The algorithm used to discover these universal sequential patterns is based on a modi cation of the GSP algorithm proposed in [SA96]. Some of these modi cations are made to take care of the newly introduced timing constraints and pattern restrictions, whereas some modi cations are made for performance reasons. In the end, we present an application, which illustrates the de ciencies of current approaches that can be overcome by the proposed universal formulation. 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. 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. This paper concerns the worst-case complexity of Gauss-Seidel method for solving a positive semi-definite linear system; or equivalently, that of cyclic coordinate descent (C-CD) for minimizing a convex quadratic function. The known provable complexity of C-CD can be @math times slower than gradient descent (GD) and @math times slower than randomized coordinate descent (R-CD). However, these gaps seem rather puzzling since so far they have not been observed in practice; in fact, C-CD usually converges much faster than GD and sometimes comparable to R-CD. Thus some researchers believe the gaps are due to the weakness of the proof, but not that of the C-CD algorithm itself. In this paper we show that the gaps indeed exist. We prove that there exists an example for which C-CD takes at least @math or @math operations, where @math is the condition number, @math is a well-studied quantity that determines the convergence rate of R-CD, and @math hides the dependency on @math . This implies that C-CD can indeed be @math times slower than GD, and @math times slower than R-CD in the worst case. Our result establishes one of the few examples in continuous optimization that demonstrates a large gap between a deterministic algorithm and its randomized counterpart. Based on the example, we establish several almost tight complexity bounds of C-CD for quadratic problems. One difficulty with the analysis of the constructed example is that the spectral radius of a non-symmetric iteration matrix does not necessarily constitutes a lower bound for the convergence rate. 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. Gutierrez and Meseguer show how to enforce consistency in BnB-ADOPT+ for distributed constraint optimization, but they consider unconditional deletions only. However, during search, more values can be pruned conditionally according to variable instantiations that define subproblems. Enforcing consistency in these subproblems can cause further search space reduction. We introduce efficient methods to maintain soft arc consistencies in every subproblem during search, a non trivial task due to asynchronicity and induced overheads. Experimental results show substantial benefits on three different benchmarks. We are grateful to the anonymous referees for their constructive comments. The work of Lei and Lee was generously supported by grants CUHK413808, CUHK413710 and CUHK413713 from the Research Grants Council of Hong Kong SAR. The work of Gutierrez and Meseguer was partially supported by the Spanish project TIN2009-13591-C02-02 and Generalitat de Catalunya 2009-SGR-1434. The work of Gutierrez, Lee and Meseguer was also jointly supported by the CSIC RGC Joint Research Scheme grants S-HK003 12 and 2011HK0017. The work of Mak was performed while he was at CUHK. We prove a structure theorem for the feasible solutions of the Arora-Rao-Vazirani SDP relaxation on low threshold rank graphs and on small-set expanders. We show that if G is a graph of bounded threshold rank or a small-set expander, then an optimal solution of the Arora-Rao-Vazirani relaxation (or of any stronger version of it) can be almost entirely covered by a small number of balls of bounded radius. Then, we show that, if k is the number of balls, a solution of this form can be rounded with an approximation factor of O(sqrt log k ) in the case of the Arora-Rao-Vazirani relaxation, and with a constant-factor approximation in the case of the k-th round of the Sherali-Adams hierarchy starting at the Arora-Rao-Vazirani relaxation. The structure theorem and the rounding scheme combine to prove the following result, where G=(V,E) is a graph of expansion (G), is the k-th smallest eigenvalue of the normalized Laplacian of G, and (G) = disjoint S_1,...,S_k 1 > log^ 2.5 k phi(G) or k (G) >> log k sqrt log n loglog n (G), then the Arora-Rao-Vazirani relaxation can be rounded in polynomial time with an approximation ratio O(sqrt log k ). Stronger approximation guarantees are achievable in time exponential in k via relaxations in the Lasserre hierarchy. Guruswami and Sinop [GS13] and Arora, Ge and Sinop [AGS13] prove that 1+eps approximation is achievable in time 2^ O(k) poly(n) if either > (G) poly(eps), or if SSE_ n k > sqrt log k log n (G) poly(eps), where SSE_s is the minimal expansion of sets of size at most s. 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. 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. Given an arc-weighted directed graph G = (V, A, spl lscr ) and a pair of nodes s, t, we seek to find an s-t walk of length at most B that maximizes some given function f of the set of nodes visited by the walk. The simplest case is when we seek to maximize the number of nodes visited: this is called the orienteering problem. Our main result is a quasi-polynomial time algorithm that yields an O(log OPT) approximation for this problem when f is a given submodular set function. We then extend it to the case when a node v is counted as visited only if the walk reaches v in its time window [R(v), D(v)]. We apply the algorithm to obtain several new results. First, we obtain an O(log OPT) approximation for a generalization of the orienteering problem in which the profit for visiting each node may vary arbitrarily with time. This captures the time window problem considered earlier for which, even in undirected graphs, the best approximation ratio known [Bansal, (2004)] is O(log sup 2 OPT). The second application is an O(log sup 2 k) approximation for the k-TSP problem in directed graphs (satisfying asymmetric triangle inequality). This is the first non-trivial approximation algorithm for this problem. The third application is an O(log sup 2 k) approximation (in quasi-poly time) for the group Steiner problem in undirected graphs where k is the number of groups. This improves earlier ratios (Garg, ) by a logarithmic factor and almost matches the inapproximability threshold on trees (Halperin and Krauthgamer, 2003). This connection to group Steiner trees also enables us to prove that the problem we consider is hard to approximate to a ratio better than spl Omega (log sup 1- spl epsi OPT), even in undirected graphs. Even though our algorithm runs in quasi-poly time, we believe that the implications for the approximability of several basic optimization problems are interesting. Abstract This paper extends network-based methods of constraint satisfaction to include continuous variables, thus providing a framework for processing temporal constraints. In this framework, called temporal constraint satisfaction problem (TCSP), variables represent time points and temporal information is represented by a set of unary and binary constraints, each specifying a set of permitted intervals. The unique feature of this framework lies in permitting the processing of metric information, namely, assessments of time differences between events. We present algorithms for performing the following reasoning tasks: finding all feasible times that a given event can occur, finding all possible relationships between two given events, and generating one or more scenarios consistent with the information provided. We distinguish between simple temporal problems (STPs) and general temporal problems, the former admitting at most one interval constraint on any pair of time points. We show that the STP, which subsumes the major part of Vilain and Kautz's point algebra, can be solved in polynomial time. For general TCSPs, we present a decomposition scheme that performs the three reasoning tasks considered, and introduce a variety of techniques for improving its efficiency. We also study the applicability of path consistency algorithms as preprocessing of temporal problems, demonstrate their termination and bound their complexities. 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. The research in qualitative reasoning and in spatial CSP is always investigated in the backdrop of its temporal counterpart - qualitative temporal reasoning and TCSP. Unlike the case of interval algebra (IA), the composition table of RCC, IA's so-called spatial counterpart, is in general neither complete nor extensional, the compositional consistency can be still a valid reasoning mechanism. Even in such a restricted situation, many of the known properties of IA have not been investigated for validity in the context of RCC. We address, in this paper two such properties-convexity and minimality. The importance of minimality cannot be underestimated as in a minimal network every label is feasible and hence determining all the consistent scenarios can be accomplished very efficiently. It is known that path consistency does not yield a minimal network for tractable classes of RCC-8. We represent RCC-8 relations as a partially ordered set and exploit the properties of partial ordering to derive very interesting theoretical results. We show here that there exists a convex class of relations of RCC-8 for which path consistency yields a minimal network. Our results are very important as it gives a sufficient condition for minimality and useful to generate all consistent scenarios whenever compositional consistency is a valid reasoning mechanism 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. An important question in the study of constraint satisfaction problems (CSP) is understanding how the graph or hypergraph describing the incidence structure of the constraints influences the complexity of the problem. For binary CSP instances (i.e., where each constraint involves only two variables), the situation is well understood: the complexity of the problem essentially depends on the treewidth of the graph of the constraints. However, this is not the correct answer if constraints with unbounded number of variables are allowed, and in particular, for CSP instances arising from query evaluation problems in database theory. Formally, if H is a class of hypergraphs, then let CSP(H) be CSP restricted to instances whose hypergraph is in H. Our goal is to characterize those classes of hypergraphs for which CSP(H) is polynomial-time solvable or fixed-parameter tractable, parameterized by the number of variables. Note that in the applications related to database query evaluation, we usually assume that the number of variables is much smaller than the size of the instance, thus parameterization by the number of variables is a meaningful question. The most general known property of H that makes CSP(H) polynomial-time solvable is bounded fractional hypertree width. Here we introduce a new hypergraph measure called submodular width, and show that bounded submodular width of H implies that CSP(H) is fixed-parameter tractable. In a matching hardness result, we show that if H has unbounded submodular width, then CSP(H) is not fixed-parameter tractable, unless the Exponential Time Hypothesis fails. 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 propose novel cooperative spectrum sensing (CSS) algorithms for cognitive radio (CR) networks based on machine learning techniques which are used for pattern classification. In this regard, unsupervised (e.g., K-means clustering and Gaussian mixture model (GMM)) and supervised (e.g., support vector machine (SVM) and weighted K-nearest-neighbor (KNN)) learning-based classification techniques are implemented for CSS. For a radio channel, the vector of the energy levels estimated at CR devices is treated as a feature vector and fed into a classifier to decide whether the channel is available or not. The classifier categorizes each feature vector into either of the two classes, namely, the "channel available class" and the "channel unavailable class". Prior to the online classification, the classifier needs to go through a training phase. For classification, the K-means clustering algorithm partitions the training feature vectors into K clusters, where each cluster corresponds to a combined state of primary users (PUs) and then the classifier determines the class the test energy vector belongs to. The GMM obtains a mixture of Gaussian density functions that well describes the training feature vectors. In the case of the SVM, the support vectors (i.e., a subset of training vectors which fully specify the decision function) are obtained by maximizing the margin between the separating hyperplane and the training feature vectors. Furthermore, the weighted KNN classification technique is proposed for CSS for which the weight of each feature vector is calculated by evaluating the area under the receiver operating characteristic (ROC) curve of that feature vector. The performance of each classification technique is quantified in terms of the average training time, the sample classification delay, and the ROC curve. Our comparative results clearly reveal that the proposed algorithms outperform the existing state-of-the-art CSS techniques.