Propagating Conjunctions of AllDifferent Constraints

We study propagation algorithms for the conjunction of two AllDifferent constraints. Solutions of an AllDifferent constraint can be seen as perfect matchings on the variable/value bipartite graph. Therefore, we investigate the problem of finding simultaneous bipartite matchings. We present an extension of the famous Hall theorem which characterizes when simultaneous bipartite matchings exists. Unfortunately, finding such matchings is NP-hard in general. However, we prove a surprising result that finding a simultaneous matching on a convex bipartite graph takes just polynomial time. Based on this theoretical result, we provide the first polynomial time bound consistency algorithm for the conjunction of two AllDifferent constraints. We identify a pathological problem on which this propagator is exponentially faster compared to existing propagators. Our experiments show that this new propagator can offer significant benefits over existing methods.Comment: AAAI 2010, Proceedings of the Twenty-Fourth AAAI Conference on Artificial Intelligenc

Conjunctions of Among Constraints

Many existing global constraints can be encoded as a conjunction of among constraints. An among constraint holds if the number of the variables in its scope whose value belongs to a prespecified set, which we call its range, is within some given bounds. It is known that domain filtering algorithms can benefit from reasoning about the interaction of among constraints so that values can be filtered out taking into consideration several among constraints simultaneously. The present pa- per embarks into a systematic investigation on the circumstances under which it is possible to obtain efficient and complete domain filtering algorithms for conjunctions of among constraints. We start by observing that restrictions on both the scope and the range of the among constraints are necessary to obtain meaningful results. Then, we derive a domain flow-based filtering algorithm and present several applications. In particular, it is shown that the algorithm unifies and generalizes several previous existing results.Comment: 15 pages plus appendi

Estimating the Number of Solutions of Cardinality Constraints through range and roots Decompositions

International audienceThis paper introduces a systematic approach for estimating the number of solutions of cardinality constraints. A main difficulty of solutions counting on a specific constraint lies in the fact that it is, in general, at least as hard as developing the constraint and its propaga-tors, as it has been shown on alldifferent and gcc constraints. This paper introduces a probabilistic model to systematically estimate the number of solutions on a large family of cardinality constraints including alldifferent, nvalue, atmost, etc. Our approach is based on their decomposition into range and roots, and exhibits a general pattern to derive such estimates based on the edge density of the associated variable-value graph. Our theoretical result is finally implemented within the maxSD search heuristic, that aims at exploring first the area where there are likely more solutions

The AllDifferent Constraint with Precedences

We propose ALLDIFFPREC, a new global constraint that combines together an ALLDIFFERENT constraint with precedence constraints that strictly order given pairs of variables. We identify a number of applications for this global constraint including instruction scheduling and symmetry breaking. We give an efficient propagation algorithm that enforces bounds consistency on this global constraint. We show how to implement this propagator using a decomposition that extends the bounds consistency enforcing decomposition proposed for the ALLDIFFERENT constraint. Finally, we prove that enforcing domain consistency on this global constraint is NP-hard in general

Consistency techniques for linear global cost functions in weighted constraint satisfaction.

在加權約束滿足問題中使用多元價值函數需要強大的一致相容性技術，而在多元價值函數中維護一致相容性並不是一項簡單的工作。能在多項式時間內找出多元價值函數的最少價值，而且不被投影及擴展操作所破壞，是讓該多元價值函數實用的主要條件。但是，有很多有用的多元價值函數尚未有多項式時間的算法找出其最少價值，因而未能在加權約束滿足問題中實用地使用它們。我們定義了一類可被建構為整數線性規劃的多元價值函數，並稱它們為多項式線性投影安全(PLPS)價值函數。該類價值函數的最少價值能由解答整數線性規劃中找出，而這個特性並不會被投影及擴展操作所影響。線性鬆馳能讓我們找出一個最少價值的接近值，並避免了解答整數線性規劃的NP-難困難性。該最少價值的接近值能作為最少價值的下限以供維護鬆馳一致相容性概念。在實踐中我們示範了使用PLPS價值函數的組合的好處。我們定義了整數多項式線性投影安全(IPLPS)價值函數作為PLPS價值函數的一個子類，並讓我們表示組合該類價值函數的好處。在一個加權約束滿足問題的一致相容性α中，我們表示了在IPLPS價值函數的組合中維護鬆馳α比在單獨的IPLPS價值函數中維護α強大。這結果可用在能在多項式時間中找出最少價值，但不能在多項式時間中找出它們的組合的最少價值的IPLPS價值函數中。基於流量投影安全(flow-based projection-safe)及可多項式分解(polynomially decomposable)價值函數的一個重要的子類屬於這一類的IPLPS價值函數。在實驗中我們展示了我們的方法的可行性和效率。無論在時間或搜索空間的改進上，與現有的方法相比，在使用PLPS價值函數的組合和 IPLPS價值函數的組合時我們觀察到一個數量級的改進。The solving of Weighted CSP (WCSP) with global cost functions relies on powerful consistency techniques, but enforcing these consistencies on global cost functions is not a trivial task. Lee and Leung suggest that a global cost function can be used practically if we can find its minimum cost and perform projections/extensions on it in polynomial time, and at the same time projections and extensions should not destroy those conditions. However, there are many useful cost functions with no known polynomial time algorithms to compute the minimum costs yet.We propose a special class of global cost functions which can be modeled as integer linear programs, called polynomially linear projection-safe (PLPS) cost functions. We show that their minimum cost can be computed by integer programming and this property is unaffected by projections/extensions. By linear relaxation we can avoid the possible NP-hard time taken to solve the integer programs, as the approximation of their actual minimum costs can be obtained to serve as a good lower bound in enforcing the relaxed forms of common consistencies.We show the benets of using the conjunctions of PLPS cost functions empir-ically in terms of runtime. We introduce integral polynomially linear projection-safe (IPLPS) cost functions as a subclass of PLPS cost functions whose allow us to characterize the benets of using the conjunctions of them. Given a standard WCSP consistency α, we give theorems showing that maintaining relaxed α on a conjunction of IPLPS cost functions is stronger than maintaining α on the individual cost functions. A useful application of our method is on some IPLPS global cost functions, whose minimum cost computations are tractable and yet those for their conjunctions are not. We show that an important subclass of flow-based projection-safe and polynomially decomposable cost functions falls into this category.Experiments are conducted to demonstrate the feasibility and efciency of our framework. We observe orders of magnitude in runtime and search space improvements by using the conjunctions of PLPS and IPLPS cost functions with relaxed consistencies when compared with the existing approaches.Detailed summary in vernacular field only.Detailed summary in vernacular field only.Detailed summary in vernacular field only.Detailed summary in vernacular field only.Shum, Yu Wai.Thesis (M.Phil.)--Chinese University of Hong Kong, 2012.Includes bibliographical references (leaves 87-92).Abstracts also in Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Weighted Constraint Satisfaction Problems --- p.2Chapter 1.2 --- Motivation and Goal --- p.2Chapter 1.3 --- Outline of the Thesis --- p.4Chapter 2 --- Related Work --- p.6Chapter 2.1 --- Soft Constraint Frameworks --- p.6Chapter 2.2 --- Integer Linear Programming --- p.8Chapter 2.3 --- Global Cost Functions in WCSP --- p.8Chapter 3 --- Background --- p.11Chapter 3.1 --- Weighted Constraint Satisfaction Problems --- p.11Chapter 3.1.1 --- Branch and Bound Search --- p.14Chapter 3.1.2 --- Local consistencies in WCSP --- p.15Chapter 3.1.3 --- Global Cost Functions --- p.30Chapter 3.2 --- Integer Linear Programming --- p.31Chapter 4 --- Polynomially Linear Projection-Safe Cost Functions --- p.33Chapter 4.1 --- Non-tractable Global Cost Functions in WCSPs --- p.34Chapter 4.2 --- Polynomially Linear Projection-Safe Cost Functions --- p.37Chapter 4.3 --- Relaxed Consistencies on Polynomially Linear Projection-Safe Cost Functions --- p.44Chapter 4.4 --- Conjoining Polynomially Linear Projection-Safe Cost Functions --- p.50Chapter 4.5 --- Modeling Global Cost Functions as Polynomially Linear Projection- Safe Cost Functions --- p.53Chapter 4.5.1 --- The SOFT SLIDINGSUM{U+1D48}{U+1D52}{U+1D9C} Cost Function --- p.53Chapter 4.5.2 --- The SOFT EGCC{U+1D5B}{U+1D43}{U+02B3} Cost Function --- p.54Chapter 4.5.3 --- The SOFT DISJUNCTIVE/CUMULATIVE Cost Function --- p.56Chapter 4.6 --- Implementation Issues --- p.59Chapter 4.7 --- Experimental Results --- p.60Chapter 4.7.1 --- Generalized Car Sequencing Problem --- p.62Chapter 4.7.2 --- Magic Series Problem --- p.63Chapter 4.7.3 --- Weighted Tardiness Scheduling Problem --- p.65Chapter 5 --- Integral Polynomially Linear Projection-Safe Cost Functions --- p.68Chapter 5.1 --- Integral Polynomially Linear Projection-Safe Cost Functions --- p.69Chapter 5.2 --- Conjoining Global Cost Functions as IPLPS --- p.72Chapter 5.3 --- Experimental Results --- p.76Chapter 5.3.1 --- Car Sequencing Problem --- p.77Chapter 5.3.2 --- Examination Timetabling Problem --- p.78Chapter 5.3.3 --- Fair Scheduling --- p.79Chapter 5.3.4 --- Comparing WCSP Approach with Integer Linear programming Approach --- p.81Chapter 6 --- Conclusions --- p.83Chapter 6.1 --- Contributions --- p.83Chapter 6.2 --- Future Work --- p.85Bibliography --- p.8

Global constraints as graph properties on structured network of elementary constraints of the same type

This report introduces a classification scheme for the global constraints. This classification is based on four basic ingredients from which one can generate almost all existing global constraints and come up with new interesting constraints. Global constraints are defined in a very concise way, in term of graph properties that have to hold, where the graph is a structured network of same elementary constraints. Since this classification is based on the internal structure of the global constraints it is also a strong hint for the pruning algorithms of the global constraints

Generalized Support and Formal Development of Constraint Propagators

Abstract The concept of support is pervasive in constraint programming. Traditionally, when a domain value ceases to have support, it may be removed because it takes part in no solutions. Arc-consistency algorithms such as AC2001 [8] make use of support in the form of a single domain value. GAC algorithms such as GAC-Schema We design a methodology for developing correct propagators using generalized support. A constraint is expressed as a family of support properties, which may be proven correct against the formal semantics of the constraint. Using CurryHoward isomorphism to interpret constructive proofs as programs, we show how to derive correct propagators from the constructive proofs of the support properties. The framework is carefully designed to allow efficient algorithms to be produced. Derived algorithms may make use of dynamic literal triggers or watched literal

Global Constraint Catalog, 2nd Edition (revision a)

This report presents a catalogue of global constraints where each constraint is explicitly described in terms of graph properties and/or automata and/or first order logical formulae with arithmetic. When available, it also presents some typical usage as well as some pointers to existing filtering algorithms