A Cost-Regular Based Hybrid Column Generation Approach

Constraint Programming (CP) offers a rich modeling language of constraints embedding efficient algorithms to handle complex and heterogeneous combinatorial problems. To solve hard combinatorial optimization problems using CP alone or hybrid CP-ILP decomposition methods, costs also have to be taken into account within the propagation process. Optimization constraints, with their cost-based filtering algorithms, aim to apply inference based on optimality rather than feasibility. This paper introduces a new optimization constraint, cost-regular. Its filtering algorithm is based on the computation of shortest and longest paths in a layered directed graph. The support information is also used to guide the search for solutions. We believe this constraint to be particularly useful in modeling and solving Column Generation subproblems and evaluate its behaviour on complex Employee Timetabling Problems through a flexible CP-based column generation approach. Computational results on generated benchmark sets and on a complex real-world instance are given.A preliminary version of this paper appeared as [7]. This research was supported by the Mathematics of Information Technology and Complex Systems (MITACS) Internship program in association with Omega Optimisation Inc. (CA).     

Efficient handling of universally quantified inequalities

This paper introduces a new framework for solving quantified constraint satisfaction problems (QCSP) defined by universally quantified inequalities on continuous domains. This class of QCSPs has numerous applications in engineering and technology. We introduce a generic branch and prune algorithm to tackle these continuous CSPs with parametric constraints, where the pruning and the solution identification processes are dedicated to universally quantified inequalities. Special rules are proposed to handle the parameter domains of the constraints. The originality of our framework lies in the fact that it solves the QCSP as a non-quantified CSP where the quantifiers are handled locally, at the level of each constraint. Experiments show that our algorithm outperforms the state of the art methods based on constraint techniques. This paper is an extended version of a paper published at the SAC 2008 conference [15].     

Optimization-Oriented Global Constraints

In this paper, we propose a way of exploiting Operations Research techniques within global constraints for cost-based domain filtering. In Constraint Programming, constraint propagation is aimed at removing from variable domains combinations of values which are proven infeasible. Pruning derives from feasibility reasoning. When coping with optimization problems, pruning can be performed also on the basis of costs, i.e., optimality reasoning. Cost-based filtering removes combination of values which are proven sub-optimal. For this purpose, we encapsulate in global constraints optimization components representing suitable relaxations of the constraint itself. These components embed efficient Operations Research algorithms computing the optimal solution of the relaxed problem and a gradient function representing the estimated cost of each variable-value assignment. We exploit these pieces of information for pruning and for guiding the search. We have applied these techniques to a couple of ILOG Solver global constraints (a constraint of difference and a path constraint) and tested the approach on a variety of combinatorial optimization problems such as Timetabling, Travelling Salesman Problems and Scheduling Problems with sequence dependent setup times. Comparisons with pure Constraint Programming approaches and related literature clearly show the benefits of the proposed approach.     

Extension of <Emphasis Type="Italic">O</Emphasis>(<Emphasis Type="Italic">n</Emphasis> log <Emphasis Type="Italic">n</Emphasis>) Filtering Algorithms for the Unary Resource Constraint to Optional Activities

Scheduling is one of the most successful application areas of constraint programming mainly thanks to special global constraints designed to model resource restrictions. Among these global constraints, edge-finding and not-first/not-last are the most popular filtering algorithms for unary resources. In this paper we introduce new O(n log n) versions of these two filtering algorithms and one more O(n log n) filtering algorithm called detectable precedences. These algorithms use a special data structures Θ-tree and Θ-Λ-tree. These data structures are especially designed for “what-if” reasoning about a set of activities so we also propose to use them for handling so called optional activities, i.e. activities which may or may not appear on the resource. In particular, we propose new O(n log n) variants of filtering algorithms which are able to handle optional activities: overload checking, detectable precedences and not-first/not-last.     

An MDD-based generalized arc consistency algorithm for positive and negative table constraints and some global constraints

A table constraint is explicitly represented as its set of solutions or non-solutions. This ad hoc (or extensional) representation may require space exponential to the arity of the constraint, making enforcing GAC expensive. In this paper, we address the space and time inefficiencies simultaneously by presenting the mddc constraint. mddc is a global constraint that represents its (non-)solutions with a multi-valued decision diagram (MDD). The MDD-based representation has the advantage that it can be exponentially smaller than a table. The associated GAC algorithm (called mddc) has time complexity linear to the size of the MDD, and achieves full incrementality in constant time. In addition, we show how to convert a positive or negative table constraint into an mddc constraint in time linear to the size of the table. Our experiments on structured problems, car sequencing and still-life, show that mddc is also a fast GAC algorithm for some global constraints such as sequence and regular. We also show that mddc is faster than the state-of-the-art generic GAC algorithms in Gent et al. (2007), Lecoutre and Szymanek (2006), Lhomme and Régin (2005) for table constraint.     

The Complexity of Reasoning with Global Constraints

Constraint propagation is one of the techniques central to the success of constraint programming. To reduce search, fast algorithms associated with each constraint prune the domains of variables. With global (or non-binary) constraints, the cost of such propagation may be much greater than the quadratic cost for binary constraints. We therefore study the computational complexity of reasoning with global constraints. We first characterise a number of important questions related to constraint propagation. We show that such questions are intractable in general, and identify dependencies between the tractability and intractability of the different questions. We then demonstrate how the tools of computational complexity can be used in the design and analysis of specific global constraints. In particular, we illustrate how computational complexity can be used to determine when a lesser level of local consistency should be enforced, when constraints can be safely generalized, when decomposing constraints will reduce the amount of pruning, and when combining constraints is tractable.     

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.     

Root-finding by expansion with independent constraints

Elimination methods are highly effective for the solution of linear and nonlinear systems of equations, but reversal of the elimination principle can be beneficial as well: competent incorporation of additional independent constraints and variables or more generally immersion of the original computational problem into a larger task, defined by a larger number of independent constraints and variables can improve global convergence of iterative algorithms, that is their convergence from the start. A well known example is the dual linear and nonlinear programming, which enhances the power of optimization algorithms. We believe that this is just an ad hoc application of general Principle of Expansion with Independent Constraints; it should be explored systematically for devising iterative algorithms for the solution of equations and systems of equations and for optimization. At the end of this paper we comment on other applications and extensions of this principle.Presently we show it at work for the approximation of a single zero of a univariate polynomial p of a degree n. Empirical global convergence of the known algorithms for this task is much weaker than that of the algorithms for all n zeros, such as Weierstrass–Durand–Kerner’s root-finder, which reduces its root-finding task to Viète’s (Vieta’s) system of n polynomial equations with n unknowns. We adjust this root-finder to the approximation of a single zero of p, preserve its fast global convergence and decrease the number of arithmetic operations per iteration from quadratic to linear. Together with computing a zero of a polynomial p, the algorithm deflates this polynomial as by-product, and then could be reapplied to the quotient to approximate the next zero of p. Alternatively by using m processors that exchange no data, one can concurrently approximate up to m zeros of p. Our tests confirm the efficiency of the proposed algorithms.Technically our root-finding boils down to computations with structured matrices, polynomials and partial fraction decompositions. Our study of these links can be of independent interest; e.g., as by-product we express the inverse of a Sylvester matrix via its last column, thus extending the celebrated result of Gohberg and Sementsul (1972) [22] from Toeplitz to Sylvester matrix inverses.     

Solving Systems of Difference Constraints Incrementally

Difference constraints systems consisting of inequalities of the form x _i - x _j b _i,j occur in many applications, most notably those involving temporal reasoning. Often, it is necessary to maintain a solution to such a system as constraints are added, modified, and deleted. Existing algorithms handle modifications by solving the resulting system anew each time, which is inefficient. The best known algorithm to determine if a system of difference constraints is feasible (i.e., if it has a solution) and to compute a solution runs in Θ (mn) time, where n is the number of variables and m is the number of constraints. This paper presents a new efficient incremental algorithm for maintaining a solution to a system of difference constraints. As constraints are added, modified, or deleted, the algorithm determines if the new system is feasible and updates its solution. When the system becomes infeasible, the algorithm continues to process changes until it becomes feasible again, at which point a feasible solution will be produced. The algorithm processes the addition of a constraint in time O(m + n log n) and the removal of a constraint in constant time when the original system is feasible. More precisely, additions are processed in time O( || Δ || + |Δ| log|Δ| ) , where |Δ| is the number of variables whose values are changed to compute the new feasible solution, and || Δ || is the number of constraints involving the variables whose values are changed. When the original system is infeasible, the algorithm processes any change in O(m + n log n) amortized time. The new algorithm can also be used to check for the existence of negative cycles in dynamic graphs. Received September 25, 1997; revised November 16, 1997.     

Strong local consistency algorithms for table constraints

Table constraints are important in constraint programming as they are present in many real problems from areas such as configuration and databases. As a result, numerous specialized algorithms that achieve generalized arc consistency (GAC) on table constraints have been proposed. Since these algorithms achieve GAC, they operate on one constraint at a time. In this paper we propose new filtering algorithms for positive table constraints that achieve stronger local consistency properties than GAC by exploiting intersections between constraints. The first algorithm, called maxRPWC+, is a domain filtering algorithm that is based on the local consistency maxRPWC and extends the GAC algorithm of Lecoutre and Szymanek (2006). The second algorithm extends the state-of-the-art STR-based algorithms to stronger relation filtering consistencies, i.e., consistencies that can remove tuples from constraints’ relations. Experimental results from benchmark problems demonstrate that the proposed algorithms are quite competitive with standard GAC algorithms like STR2 in some classes of problems with intersecting table constraints, being orders of magnitude faster in some cases.     

Efficient Algorithms for Interface Timing Verification

This paper presents algorithms for computing separations between events that are constrained to obey prespecified relationships in their relative time of occurrence. The algorithms are useful for interface timing verification, where event separations are checked against timing requirements. The first algorithm computes separations when only linear and max constraints exist. The algorithm must converge to correct maximum separation values in a finite number of steps, or report an inconsistence of the constraints, irrespective of the existence of infinite constraint bounds or infinite event separations. It is conjectured to run in time, where V is the number of events, and E is the number of relationships between them. The other algorithms extend the first, and compute event separations in the NP-complete version of the problem where min constraints exist. Experiments demonstrate the algorithms are efficient in practice.     

一种带约束条件的关联规则频繁集挖掘

论文先提出顺序单调约束和反顺序单调约束的概念并对其所包含的数学性质进行了讨论,在此基础上将其运用于频繁集挖掘过程中,给出挖掘基于顺序反单调性约束的频繁集算法和挖掘基于顺序单调约束的频繁集算法。带约束条件的关联规则频繁集挖掘可减少生成无意义的规则;同时,在频繁集生成过程,利用约束条件对搜索空间进行修剪,可提高挖掘算法的效率。     

单调和反单调约束条件下关联规则的挖掘算法分析

本文充分利用了 Eclat算法的概念格理论和等价类划分方法,将约束条件融入基于垂直数据分布的关联规则挖掘算法中。提出了一种新的反单调和单调约束条件下关联规则的挖掘算法,分别为EclatA算法和EclatM算法。算法采用自底向上的搜索方法,在发现频繁项集的同时进行约束条件的检验。数据库的扫描次数较少,无需对候选项集进行剪枝,占用内存较小。实验证明:该算法的执行效率比已有算法有显著提高。     

Semi-supervised graph clustering: a kernel approach

Semi-supervised clustering algorithms aim to improve clustering results using limited supervision. The supervision is generally given as pairwise constraints; such constraints are natural for graphs, yet most semi-supervised clustering algorithms are designed for data represented as vectors. In this paper, we unify vector-based and graph-based approaches. We first show that a recently-proposed objective function for semi-supervised clustering based on Hidden Markov Random Fields, with squared Euclidean distance and a certain class of constraint penalty functions, can be expressed as a special case of the weighted kernel k-means objective (Dhillon et al., in Proceedings of the 10th International Conference on Knowledge Discovery and Data Mining, 2004a). A recent theoretical connection between weighted kernel k-means and several graph clustering objectives enables us to perform semi-supervised clustering of data given either as vectors or as a graph. For graph data, this result leads to algorithms for optimizing several new semi-supervised graph clustering objectives. For vector data, the kernel approach also enables us to find clusters with non-linear boundaries in the input data space. Furthermore, we show that recent work on spectral learning (Kamvar et al., in Proceedings of the 17th International Joint Conference on Artificial Intelligence, 2003) may be viewed as a special case of our formulation. We empirically show that our algorithm is able to outperform current state-of-the-art semi-supervised algorithms on both vector-based and graph-based data sets.     

Message delay and DisCSP search algorithms

Distributed constraint satisfaction problems (DisCSPs) are composed of agents, each holding its own variables, that are connected by constraints to variables of other agents. Due to the distributed nature of the problem, message delay can have unexpected effects on the behavior of distributed search algorithms on DisCSPs. This has been recently shown in experimental studies of asynchronous backtracking algorithms (Bejar et al., Artif. Intell., 161:117–148, 2005; Silaghi and Faltings, Artif. Intell., 161:25–54, 2005). To evaluate the impact of message delay on the run of DisCSP search algorithms, a model for distributed performance measures is presented. The model counts the number of non concurrent constraints checks, to arrive at a solution, as a non concurrent measure of distributed computation. A simpler version measures distributed computation cost by the non-concurrent number of steps of computation. An algorithm for computing these distributed measures of computational effort is described. The realization of the model for measuring performance of distributed search algorithms is a simulator which includes the cost of message delays. Two families of distributed search algorithms on DisCSPs are investigated. Algorithms that run a single search process, and multiple search processes algorithms. The two families of algorithms are described and associated with existing algorithms. The performance of three representative algorithms of these two families is measured on randomly generated instances of DisCSPs with delayed messages. The delay of messages is found to have a strong negative effect on single search process algorithms, whether synchronous or asynchronous. Multi search process algorithms, on the other hand, are affected very lightly by message delay.     

Satisfiability Testing of Boolean Combinations of Pseudo-Boolean Constraints using Local-search Techniques

Some search problems are most directly specified by conjunctions of (sets of) disjunctions of pseudo-Boolean (PB) constraints. We study a logic PL ^PB whose formulas are of such form, and design local-search methods to compute models of PL ^PB theories. In our approach we view a PL ^PB theory T as a data structure, a concise representation of a certain propositional conjunctive normal form (CNF) theory cl(T) logically equivalent to T. The key idea is an observation that parameters needed by local-search algorithms for CNF theories, such as walksat, can be estimated on the basis of T without the need to compute cl(T) explicitly. We compare our methods to a local-search algorithm wsat(oip). The experiments demonstrate that our approach performs better. In order for wsat(oip) to handle arbitrary PL ^PB theories, it is necessary to represent disjunctions of PB constraints by sets of PB constraints, which often increases the size of the theory dramatically. A better performance of our method underscores the importance of developing solvers that work directly on PL ^PB theories. This paper combines and extends results included in conference papers [14, 15].     

一种高效的FDE并行传播算法

约束规划(constraint programming, CP)是表示和求解组合问题的经典范式之一.扩展约束(extensional constraint)或称表约束(table constraint)是约束规划中最为常见的约束类型.绝大多数约束规划问题都可以用表约束表达.在问题求解时,相容性算法用于缩减搜索空间.目前,最为高效的表约束相容性算法是简单表约缩减(simple table reduction, STR)算法簇,如Compact-Table (CT)和STRbit算法.它们在搜索过程中维持广义弧相容(generalized arc consistency, GAC).此外,完全成对相容性(full pairwise consistency, fPWC)是一种比GAC剪枝能力更强的相容性.最为高效的维持fPWC算法是PW-CT算法.多年来,人们提出了多种表约束相容性算法来提高剪枝能力和执行效率.因子分解编码(factor-decomposition encoding, FDE)通过对平凡问题重新编码.它一定程度地扩大了问题模型,使在新的问题上维持相对较弱的GAC等价于在原问题...     

跳变约束下马尔可夫切换非线性系统滤波

针对系统状态演化多模不确定性和状态约束多样性,本文提出了跳变约束下马尔可夫切换非线性系统的交互式多假设估计方法.定义了包含跳变马尔可夫参数可能取值的假设集,根据最优贝叶斯滤波,推导出状态与假设的后验概率递推更新.基于统计线性回归线性化非线性函数,利用伪量测法,将线性化的约束扩维到真实量测中,给出了非线性系统滤波的近似解析最优解.最终给出所提算法的稀疏网格积分近似最优估计实现.在交叉道路机动目标跟踪仿真场景中,所提算法的滤波精度优于基于泰勒展开的交互式多模型算法,基于统计线性回归的交互式多模型算法,以及基于泰勒展开的非线性系统约束滤波算法.     

An affine scaling interior trust-region method combining with nonmonotone line search filter technique for linear inequality constrained minimization

This paper proposes an affine scaling interior trust-region method in association with nonmonotone line search filter technique for solving nonlinear optimization problems subject to linear inequality constraints. Based on a Newton step which is derived from the complementarity conditions of linear inequality constrained optimization, a trust-region subproblem subject only to an ellipsoidal constraint is defined by minimizing a quadratic model with an appropriate quadratic function and scaling matrix. The nonmonotone schemes combining with trust-region strategy and line search filter technique can bring about speeding up the convergence progress in the case of high nonlinear. A new backtracking relevance condition is given which assures global convergence without using the switching condition used in the traditional line search filter technique. The fast local convergence rate of the proposed algorithm is achieved which is not depending on any external restoration procedure. The preliminary numerical experiments are reported to show effectiveness of the proposed algorithm.