Consistency techniques for continuous constraints

We consider constraint satisfaction problems with variables in continuous, numerical domains. Contrary to most existing techniques, which focus on computing one single optimal solution, we address the problem of computing a compact representation of the space of all solutions admitted by the constraints. In particular, we show how globally consistent (also called decomposable) labelings of a constraint satisfaction problem can be computed.Our approach is based on approximating regions of feasible solutions by 2 ^k -trees, a representation commonly used in computer vision and image processing. We give simple and stable algorithms for computing labelings with arbitrary degrees of consistency. The algorithms can process constraints and solution spaces of arbitrary complexity, but with a fixed maximal resolution.Previous work has shown that when constraints are convex and binary, path-consistency is sufficient to ensure global consistency. We show that for continuous domains, this result can be generalized to ternary and in fact arbitrary n-ary constraints using the concept of (3,2)-relational consistency. This leads to polynomial-time algorithms for computing globally consistent labelings for a large class of constraint satisfaction problems with continuous variables.     

REASONING WITH TOPOLOGICAL AND DIRECTIONAL SPATIAL INFORMATION

Current research on qualitative spatial representation and reasoning mainly focuses on one single aspect of space. In real‐world applications, however, multiple spatial aspects are often involved simultaneously. This paper investigates problems arising in reasoning with combined topological and directional information. We use the RCC8 algebra and the rectangle algebra (RA) for expressing topological and directional information, respectively. We give examples to show that the bipath‐consistency algorithm Bipath‐Consistency is incomplete for solving even basic RCC8 and RA constraints. If topological constraints are taken from some maximal tractable subclasses of RCC8, and directional constraints are taken from a subalgebra, termed DIR49, of RA, then we show that Bipath‐Consistency is able to separate topological constraints from directional ones. This means, given a set of hybrid topological and directional constraints from the above subclasses of RCC8 and RA, we can transfer the joint satisfaction problem in polynomial time to two independent satisfaction problems in RCC8 and RA. For general RA constraints, we give a method to compute solutions that satisfy all topological constraints and approximately satisfy each RA constraint to any prescribed precision.     

Experimenting with a temporal constraint propagation algorithm

3-consistency algorithm for temporal constraint propagation over interval-based network, proposed by James Allen, is finding its use in many practical temporal reasoning systems. Apart from the polynomial behavior of this algorithm with respect to the number of nodes in the network, very little is known about its time complexity with respect to other properties of the initially given temporal constraints. In this article we have reported some of our results analyzing the complexity with respect to some structural parameters of the input constraint network. We have identified some regions, with respect to the structural parameters of the input network, where the algorithm takes much more time than it needs over other regions. Similar features have been observed in recent studies on NP-hard problems. Average case complexity of Allen's algorithm is also studied empirically, over a hundred thousand randomly generated networks, and the growth rate is observed to be of the order of quadratic with respect to the problem size (at least up to node 40, and expected to be lower above that). We have analyzed our data statistically to develop a model with which one can calculate the expected time to be consumed by the algorithm for a given input network.     

Classes of submodular constraints expressible by graph cuts

Submodular constraints play an important role both in theory and practice of valued constraint satisfaction problems (VCSPs). It has previously been shown, using results from the theory of combinatorial optimisation, that instances of VCSPs with submodular constraints can be minimised in polynomial time. However, the general algorithm is of order O(n ⁶) and hence rather impractical. In this paper, by using results from the theory of pseudo-Boolean optimisation, we identify several broad classes of submodular constraints over a Boolean domain which are expressible using binary submodular constraints, and hence can be minimised in cubic time. Furthermore, we describe how our results translate to certain optimisation problems arising in computer vision.     

Possibility theory in constraint satisfaction problems: Handling priority, preference and uncertainty

In classical Constraint Satisfaction Problems (CSPs) knowledge is embedded in a set of hard constraints, each one restricting the possible values of a set of variables. However constraints in real world problems are seldom hard, and CSP's are often idealizations that do not account for the preference among feasible solutions. Moreover some constraints may have priority over others. Lastly, constraints may involve uncertain parameters. This paper advocates the use of fuzzy sets and possibility theory as a realistic approach for the representation of these three aspects. Fuzzy constraints encompass both preference relations among possible instantiations and priorities among constraints. In a Fuzzy Constraint Satisfaction Problem (FCSP), a constraint is satisfied to a degree (rather than satisfied or not satisfied) and the acceptability of a potential solution becomes a gradual notion. Even if the FCSP is partially inconsistent, best instantiations are provided owing to the relaxation of some constraints. Fuzzy constraints are thus flexible. CSP notions of consistency and k-consistency can be extended to this framework and the classical algorithms used in CSP resolution (e.g., tree search and filtering) can be adapted without losing much of their efficiency. Most classical theoretical results remain applicable to FCSPs. In the paper, various types of constraints are modelled in the same framework. The handling of uncertain parameters is carried out in the same setting because possibility theory can account for both preference and uncertainty. The presence of uncertain parameters leads to ill-defined CSPs, where the set of constraints which defines the problem is not precisely known.     

相接行凸约束网络的快速识别算法

约束网络为计算机科学中的许多问题提供了一种有效的表示方法.一般而言,约束满足问题是NP完全的.然而,许多实际问题通常对约束的结构或形式施加了特殊的限制,从而能够高效地加以解决.迄今,为了识别易处理约束类,人们对特殊的约束或约束网络方面进行了许多研究.相接行凸(connected row-convex,简称CRC)约束网络是Deville等人提出的一类易处理问题.为了给该类问题寻求有效的快速识别算法,在CRC约束网络相关工作基础上,提出了CRC约束矩阵的标准型.在分析CRC约束矩阵的标准型性质的基础上,利用行凸(row-convex,简称RC)约束网络的判定,结合PQ树(由P节点和Q节点构成的树)的性质和矩阵的索引表示法,给出了CRC约束网络的快速识别算法.该算法的时间复杂度为O(n³d²),其中,n为约束网络涉及的变量数,d为各变量的定义域中最大定义域的大小.该时间复杂度达到该类问题的最佳时间复杂度,从而为实际的CRC约束满足问题的求解提供了可行的方法.     

Updating knowledge bases while maintaining their consistency

When updating a knowledge base, several problems may arise. One of the most important problems is that of integrity constraints satisfaction. The classic approach to this problem has been to develop methods forchecking whether a given update violates an integrity constraint. An alternative approach consists of trying to repair integrity constraints violations by performing additional updates thatmaintain knowledge base consistency. Another major problem in knowledge base updating is that ofview updating, which determines how an update request should be translated into an update of the underlying base facts. We propose a new method for updating knowledge bases while maintaining their consistency. Our method can be used for both integrity constraints maintenance and view updating. It can also be combined with any integrity checking method for view updating and integrity checking. The kind of updates handled by our method are: updates of base facts, view updates, updates of deductive rules, and updates of integrity constraints. Our method is based on events and transition rules, which explicity define the insertions and deletions induced by a knowledge base update. Using these rules, an extension of the SLDNF procedure allows us to obtain all possible minimal ways of updating a knowledge base without violating any integrity constraint.     

Solving finite domain constraint hierarchies by local consistency and tree search

We provide a reformulation of the constraint hierarchies (CHs) framework based on the notion of error indicators. Adapting the generalised view of local consistency in semiring-based constraint satisfaction problems, we define constraint hierarchy k-consistency (CH-k-C) and give a CH-2-C enforcement algorithm. We demonstrate how the CH-2-C algorithm can be seamlessly integrated into the ordinary branch-and-bound algorithm to make it a finite domain (FD) CH solver. Experimentation confirms the efficiency and robustness of our proposed solver prototype. Unlike other FD CH solvers, our proposed method works for both local and global comparators. In addition, our solver can support arbitrary error functions.     

A new tractable class of constraint satisfaction problems

In this paper we consider constraint satisfaction problems where the set of constraint relations is fixed. Feder and Vardi (1998) identified three families of constraint satisfaction problems containing all known polynomially solvable problems. We introduce a new class of problems called para-primal problems, incomparable with the families identified by Feder and Vardi (1998) and we prove that any constraint problem in this class is decidable in polynomial time. As an application of this result we prove a complete classification for the complexity of constraint satisfaction problems under the assumption that the basis contains all the permutation relations. In the proofs, we make an intensive use of algebraic results from clone theory about the structure of para-primal and homogeneous algebras. AMS subject classification 08A70     

New filtering algorithms for combinations of among constraints

Several combinatorial problems, such as car sequencing and rostering, feature sequence constraints, restricting the number of occurrences of certain values in every subsequence of a given length. We present three new filtering algorithms for the sequence constraint, including the first that establishes domain consistency in polynomial time. The filtering algorithms have complementary strengths: One borrows ideas from dynamic programming; another reformulates it as a regular constraint; the last is customized. The last two algorithms establish domain consistency, and the customized one does so in polynomial time. We provide experimental results that demonstrate the practical usefulness of each. We also show that the customized algorithm applies naturally to a generalized version of the sequence constraint that allows subsequences of varied lengths. The significant computational advantage of using a single generalized sequence constraint over a semantically equivalent collection of among or sequence constraints is demonstrated empirically.     

Tractable Decision for a Constraint Language Implies Tractable Search

A constraint satisfaction problem (CSP) instance has a set of variables, each of which can take values in some domain. It also has a set of constraints, each of which restricts the variables in its scope to take values limited by its constraint relation.The language of a constraint satisfaction problem instance is the set of different constraint relations used in its specification. For a given set of relations over some domain we define the problem CSP () to the set of CSP instances whose language is contained in .The decision problem for a set of CSP instances is, given an instance in the class, to decide whether a solution exists. The search problem is to find such a solution. Here we address the connection between the tractability of the decision and search problems. We prove that given a constraint language over a finite domain for which the decision problem for CSP () is tractable, the search problem is always tractable.We define a surjective language over a finite domain in a standard way. We also show that we can determine in polynomial time whether an instance over a surjective language with a tractable decision problem has fewer than k solutions, and that we can generate all of its solutions with polynomial delay.     

On the automatizability of resolution and related propositional proof systems

A propositional proof system is automatizable if there is an algorithm that, given a tautology, produces a proof in time polynomial in the size of its smallest proof. This notion can be weakened if we allow the algorithm to produce a proof in a stronger system within the same time bound. This new notion is called weak automatizability. Among other characterizations, we prove that a system is weakly automatizable exactly when a weak form of the satisfiability problem is solvable in polynomial time. After studying the robustness of the definition, we prove the equivalence between: (i) Resolution is weakly automatizable, (ii) Res(k) is weakly automatizable, and (iii) Res(k) has feasible interpolation, when k>1. In order to prove this result, we show that Res(2) has polynomial-size proofs of the reflection principle of Resolution, which is a version of consistency. We also show that Res(k), for every k>1, proves its consistency in polynomial size, while Resolution does not. In fact, we show that Resolution proofs of its own consistency require almost exponential size. This gives a better lower bound for the monotone interpolation of Res(2) and a separation from Resolution as a byproduct. Our techniques also give us a way to obtain a large class of examples that have small Resolution refutations but require relatively large width. This answers a question of Alekhnovich and Razborov related to whether Resolution is automatizable in quasipolynomial-time.     

On-line multiversion database concurrency control

This paper presents a new model for studying the concurrency vs. computation time tradeoffs involved in on-line multiversion database concurrency control. The basic problem that is studied in our model is the following: Given:a current database system state which includes information such as which transaction previously read a version from which other transaction; which transaction has written which versions into the database; and the ordering of versions previously written; anda set of read and write requests of requesting transactions. Question: Does there exist a new database system state in which the requesting transactions can be immediately put into execution (their read and write requests satisfied, or in the case of predeclared writeset transactions, write requests are guaranteed to be satisfied) while preserving consistency under a given set of additional constraints? (The amount of concurrency achieved is defined by the set of additional constraints). In this paper we derive “limits” of performance achievable by polynomial time concurrency control algorithms. Each limit is characterized by a minimal set of constraints that allow the on-line scheduling problem to be solved in polynomial time. If any one constraint in that minimal set is omitted, although it could increase the amount of concurrency, it would also have the dramatic negative effect of making the scheduling problem NP-complete; whereas if we do not omit any constraint in the minimal set, then the scheduling problem can be solved in polynomial time. With each of these limits, one can construct an efficient scheduling algorithm that achieves an optimal level of concurrency in polynomial computation time according to the constraints defined in the minimal set.     

Boosting Search with Variable Elimination in Constraint Optimization and Constraint Satisfaction Problems

There are two main solving schemas for constraint satisfaction and optimization problems: i) search, whose basic step is branching over the values of a variables, and ii) dynamic programming, whose basic step is variable elimination. Variable elimination is time and space exponential in a graph parameter called induced width, which renders the approach infeasible for many problem classes. However, by restricting variable elimination so that only low arity constraints are processed and recorded, it can be effectively combined with search, because the elimination of variables may reduce drastically the search tree size.In this paper we introduce BE-BB(k), a hybrid general algorithm that combines search and variable elimination. The parameter k controls the tradeoff between the two strategies. The algorithm is space exponential in k. Regarding time, we show that its complexity is bounded by k and a structural parameter from the constraint graph. We provide experimental evidence that the hybrid algorithm can outperform state-of-the-art algorithms in constraint satisfaction, Max-CSP and Weighted CSP. Especially in optimization tasks, the advantage of our approach over plain search can be overwhelming.     

Linear kernel for Rooted Triplet Inconsistency and other problems based on conflict packing technique

We develop a technique, that we call conflict packing, to obtain (and improve) polynomial kernels for several well-studied editing problems. We first illustrate our technique on Feedback Arc Set in Tournaments (k-FAST) yielding an alternative and simple proof of a linear kernel for this problem. The technique is then applied to obtain the first linear kernel for theDense Rooted Triplet Inconsistency (k-dense-RTI) problem. A linear kernel for Betweenness in Tournaments (k-BIT) is also proved. All these problems share common features. First, they can be expressed as modification problems on a dense set of constant-arity constraints. Also the consistency of the set of constraints can be characterized by means of a bounded size obstructions. The conflict packing technique basically consists of computing a maximal set of small obstructions allowing us either to bound the size of the input or to reduce the input.     

Representing and solving finite-domain constraint problems using systems of polynomials

In this paper we investigate the use of a system of multivariate polynomials to represent the restrictions imposed by a collection of constraints. One advantage of using polynomials to represent constraints is that it allows many different forms of constraints to be treated in a uniform way. Systems of polynomials have been widely studied, and a number of general techniques have been developed, including algorithms that generate an equivalent system with certain desirable properties, called a Gröbner Basis. General algorithms to compute a Gröbner Basis have doubly exponential complexity, but we observe that for the systems we use to describe constraint problems over finite domains a Gröbner Basis can be computed more efficiently than this. We also describe a family of algorithms, related to the calculation of Gröbner Bases, that can be used on any system of polynomials to compute an equivalent system in polynomial time. We show that these modified algorithms can simulate the effect of the local-consistency algorithms used in constraint programming and hence solve certain broad classes of constraint problems in polynomial time. Finally we discuss the use of adaptive consistency techniques for systems of polynomials.     

Localized event-based reasoning for multiagent domain1

This paper presents the GEM concurrency model and GEMPLAN, a multiagent planner based on this model. Unlike standard state-based AI representations, GEM is unique in its explicit emphasis on events and domain structure. In particular, a world domain is modeled as a set of regions composed of interrelated events. Event-based temporal-logic constraints are then associated with each region to delimit legal domain behavior. The GEMPLAN planner directly reflects this emphasis on domain structure and constraints. It can be viewed as a general-purpose constraint satisfaction facility which constructs a network of interrelated events (a “plan”) that is subdivided into regions (“subplans”), satisfies all applicable regional constraints, and also achieves some stated goal. GEMPLAN extends and generalizes previous planning architectures in the range of constraint forms it handles and in the flexibility of its constraint satisfaction search strategy. One critical aspect of our work has been an emphasis on localized reasoning—techniques that make explicit use of domain structure. For example, GEM localizes the applicability of domain constraints and imposes additional “locality constraints” on the basis of domain structure. Together, constraint localization and locality constraints provide semantic information that can be used to alleviate several aspects of the frame problem for multiagent domains. The GEMPLAN planner reflects the use of locality by subdividing its constraint satisfaction search space into regional planning search spaces. Utilizing constraint and property localization, GEMPLAN can pinpoint and rectify interactions among these regional search spaces, thus reducing the burden of “interaction analysis” ubiquitous to most planning systems. Because GEMPLAN is specifically geared towards parallel, multiagent domains, we believe that its natural application areas will include scheduling and other forms of organizational coordination.     

Generating customized yet factually consistent information: a constraint satisfaction approach

We present an information customization framework that leverages a hybrid of adaptive hypermedia and intelligent techniques, in particular constraint satisfaction methods, to generate customized and factually consistent information based on a user profile. Information customization is modeled as a constraint satisfaction problem, whereby a solution is derived by (a) satisfying user-model constraints to select a user-specific set of ‘information snippets’; and (b) establishing inter-snippet consistency to ensure that all snippets are compatible with each other. Our approach takes the unique step of establishing factually consistency – via the satisfaction of inter-snippet constraints – between heterogeneous information snippets. A customized information package is generated by systematically synthesizing the set of user-specific and factually consistent information snippets. The featured information customization framework incorporates variations of various search and constraint satisfaction methods. The work is applied in an E-Healthcare setting leading to the generation of customized healthcare information.     

Backtracking and random constraint satisfaction

The average running time used by backtracking on random constraint satisfaction problems is studied. This time is polynomial when the ratio of constraints to variables is large, and it is exponential when the ratio is small. When the number of variables goes to infinity, whether the average time is exponential or polynomial depends on the number of variables per constraint, the number of values per variable, and the probability that a random setting of variables satisfies a constraint. A method for computing the curve that separates polynomial from exponential time and several methods for approximating the curve are given. The version of backtracking studied finds all solutions to a problem, so the running time is exponential when the number of solutions per problem is exponential. For small values of the probability, the curve that separates exponential and polynomial average running time coincides with the curve that separates an exponential average number of solutions from a polynomial number. For larger probabilities the two curves diverge. Random problems similar to those that arise in understanding line drawings with shadows require a time that is mildly exponential when they are solved by simple backtracking. Slightly more sophisticated algorithms (such as constraint propagation combined with backtracking) should be able to solve these rapidly. This revised version was published online in June 2006 with corrections to the Cover Date.     

Tractable cases of the extended global cardinality constraint

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 D(x) í DD(x) \subseteq D 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.