Yat Chiu Law

Symmetry Breaking Constraints for Value Symmetries in Constraint Satisfaction

Constraint satisfaction problems (CSPs) sometimes contain both variable symmetries and value symmetries, causing adverse effects on CSP solvers based on tree search. As a remedy, symmetry breaking constraints are commonly used. While variable symmetry breaking constraints can be expressed easily and propagated efficiently using lexicographic ordering, value symmetry breaking constraints are often difficult to formulate. In this paper, we propose two methods of using symmetry breaking constraints to tackle value symmetries. First, we show theoretically when value symmetries in one CSP correspond to variable symmetries in another CSP of the same problem. We also show when variable symmetry breaking constraints in the two CSPs, combined using channeling constraints, are consistent. Such results allow us to tackle value symmetries efficiently using additional CSP variables and channeling constraints. Second, we introduce value precedence, a notion which can be used to break a common class of value symmetries, namely symmetries of indistinguishable values. While value precedence can be expressed using inefficient if-then constraints in existing CSP solvers, we propose efficient propagation algorithms for implementing global value precedence constraints. We also characterize several theoretical properties of the value precedence constraints. Extensive experiments are conducted to verify the feasibility and efficiency of the two proposals.

Breaking symmetry of interchangeable variables and values

A common type of symmetry is when both variables and values partition into interchangeable sets. Polynomial methods have been introduced to eliminate all symmetric solutions introduced by such interchangeability. Unfortunately, whilst eliminating all symmetric solutions is tractable in this case, pruning all symmetric values is NP-hard. We introduce a new global constraint called SIGLEX and its GAC propagator for pruning some (but not necessarily all) symmetric values. We also investigate how different postings of the SIGLEX constraints affect the pruning performance during constraint solving. Finally, we test these static symmetry breaking constraints experimentally for the first time.

Breaking value symmetries in matrix models using channeling constraints

Multi-aspect Assignment Problems (MAPs) can be naturally formulated into various matrix models of Constraint Satisfaction Problems (CSPs), which can contain both variable and value symmetries, using different viewpoints. While variable symmetry breaking constraints can be expressed relatively easily and executed efficiently by enforcing lexicographic ordering, value symmetry breaking constraints are difficult to formulate. We show when value symmetries in one viewpoint correspond to variable symmetries in another, and when symmetry breaking constraints in two viewpoints are consistent. Our results allow tackling value symmetries efficiently using additional viewpoints and channeling constraints. Experiments on the social golfer problem and a variant of the quasigroup existence problem confirm the benefits of our proposal against conventional methods.

Constraint programming on infinite data streams

Classical constraint satisfaction problems (CSPs) are commonly defined on finite domains. In real life, constrained variables can evolve over time. A variable can actually take an infinite sequence of values over discrete time points. In this paper, we propose constraint programming on infinite data streams, which provides a natural way to model constrained time-varying problems. In our framework, variable domains are specified by ω-regular languages. We introduce special stream operators as basis to form stream expressions and constraints. Stream CSPs have infinite search space. We propose a search procedure that can recognize and avoid infinite search over duplicate search space. The solution set of a stream CSP can be represented by a Buchi automaton allowing stream values to be non-periodic. Consistency notions are defined to reduce the search space early. We illustrate the feasibility of the framework by examples and experiments.

Automatic Generation of Redundant Models for Permutation Constraint Satisfaction Problems

If we have two representations of a problem as constraint satisfaction problem (CSP) models, it has been shown that combining the models using channeling constraints can increase constraint propagation in tree search CSP solvers. Handcrafting two CSP models for a problem, however, is often time-consuming. In this paper, we propose model induction, a process which generates a second CSP model from an existing model using channeling constraints, and study its theoretical properties. The generated induced model is in a different viewpoint, i.e., set of variables. It is mutually redundant to and can be combined with the input model, so that the combined model contains more redundant information, which is useful to increase constraint propagation. We also propose two methods of combining CSP models, namely model intersection and model channeling. The two methods allow combining two mutually redundant models in the same and different viewpoints respectively. We exploit the applications of model induction, intersection, and channeling and identify three new classes of combined models, which contain different amounts of redundant information. We construct combined models of permutation CSPs and show in extensive benchmark results that the combined models are more robust and efficient to solve than the single models.

Algebraic Properties of CSP Model Operators

The task at hand is to tackle Constraint Satisfaction Problems (CSPs) defined in the sense of Mackworth [4]. This paper aims to take a first step towards a CSP-based module systems for constraint programming languages and modeling tools. The call for such a system is two-fold. First, most existing constraint programming languages have some sort of module systems, but these systems are designed for the underlying languages. Thus these module systems facilitate the construction of large constraint programs in a particular language, but not of CSP models. Second, a module system designed for CSP models with clear and clean semantics should allow us to reason the properties of CSP models declaratively without actually solving the CSPs. As a first attempt, we introduce six operators for manipulating and transforming CSP models: namely intersection, union, channeling, induction, negation, and complementation. For each operator, we give its syntactic construction rule, define its set-theoretic meaning, and also examine its algebraic properties, all illustrated with examples where appropriate. Our results show that model intersection and union form abelian monoids respectively among others.

A parameterized local consistency for redundant modeling in weighted CSPs

The weighted constraint satisfaction problem (WCSP) framework is a soft constraint framework which can model many real life optimization or over-constrained problems.While there are many local consistency notions available to speed up WCSP solving, in this paper, we investigate how to effectively combine and channel mutually redundant WCSP models to increase constraint propagation. This successful technique for reducing search space in classical constraint satisfaction has been shown non-trivial when adapted for the WCSP framework. We propose a parameterized local consistency LB(m, φ), which can be instantiated with any local consistency φ for single models and applied to a combined model with m sub-models, and also provide a simple algorithm to enforce it. We instantiate LB(2, φ) with different state-of-the-art local consistencies AC*, FDAC*, and EDAC*, and demonstrate empirically the efficiency of the algorithm using different benchmark problems.

Speeding up weighted constraint satisfaction using redundant modeling

In classical constraint satisfaction, combining mutually redundant models using channeling constraints is effective in increasing constraint propagation and reducing search space for many problems. In this paper, we investigate how to benefit the same for weighted constraint satisfaction problems (WCSPs), a common soft constraint framework for modeling optimization and over-constrained problems. First, we show how to generate a redundant WCSP model from an existing WCSP using generalized model induction. We then uncover why naively combining two WCSPs by posting channeling constraints as hard constraints and relying on the standard NC* and AC* propagation algorithms does not work well. Based on these observations, we propose m -NC*c and m-AC*c and their associated algorithms for effectively enforcing node and arc consistencies on a combined model with m sub-models. The two notions are strictly stronger than NC* and AC* respectively. Experimental results confirm that applying the 2-NC*c and 2-AC*c algorithms on combined models reduces more search space and runtime than applying the state-of-the-art AC*, FDAC*, and EDAC* algorithms on single models.

Multiset variable representations and constraint propagation

Multisets generalize sets by allowing elements to have repetitions. In this paper, we study from a formal perspective representations of multiset variables, and the consistency and propagation of constraints involving multiset variables. These help us model problems more naturally and can, for example, prevent introducing unnecessary symmetries into a model. We identify a number of different representations for multiset variables, compare them in terms of effectiveness and efficiency, and propose inference rules to enforce bounds consistency for the representations. In addition, we propose to exploit the variety of a multiset—the number of distinct elements in it—to improve modeling expressiveness and further enhance constraint propagation. We derive a number of inference rules involving the varieties of multiset variables. The rules interact varieties with the traditional components of multiset variables (such as cardinalities) to obtain stronger propagation. We also demonstrate how to apply the rules to perform variety reasoning on some common multiset constraints. Experimental results show that performing variety reasoning on top of cardinality reasoning can effectively reduce more search space and achieve better runtime in solving some multiset CSPs.

Redundant modeling in permutation weighted constraint satisfaction problems

In classical constraint satisfaction, redundant modeling has been shown effective in increasing constraint propagation and reducing search space for many problem instances. In this paper, we investigate, for the first time, how to benefit the same from redundant modeling in weighted constraint satisfaction problems (WCSPs), a common soft constraint framework for modeling optimization and over-constrained problems. Our work focuses on a popular and special class of problems, namely, permutation problems. First, we show how to automatically generate a redundant permutation WCSP model from an existing permutation WCSP using generalized model induction. We then uncover why naively combining mutually redundant permutation WCSPs by posting channeling constraints as hard constraints and relying on the standard node consistency (NC*) and arc consistency (AC*) algorithms would miss pruning opportunities, which are available even in a single model. Based on these observations, we suggest two approaches to handle the combined WCSP models. In our first approach, we propose