Propagation via lazy clause generation

Finite domain propagation solvers effectively represent the possible values of variables by a set of choices which can be naturally modelled as Boolean variables. In this paper we describe how to mimic a finite domain propagation engine, by mapping propagators into clauses in a SAT solver. This immediately results in strong nogoods for finite domain propagation. But a naive static translation is impractical except in limited cases. We show how to convert propagators to lazy clause generators for a SAT solver. The resulting system introduces flexibility in modelling since variables are modelled dually in the propagation engine and the SAT solver, and we explore various approaches to the dual modelling. We show that the resulting system solves many finite domain problems significantly faster than other techniques. This paper is an extension of results first published in [29, 30].     

Constraint Propagation and Decomposition Techniques for Highly Disjunctive and Highly Cumulative Project Scheduling Problems

In recent years, constraint satisfaction techniques have been successfully applied to disjunctive scheduling problems, i.e., scheduling problems where each resource can execute at most one activity at a time. Less significant and less generally applicable results have been obtained in the area of cumulative scheduling. Multiple constraint propagation algorithms have been developed for cumulative resources but they tend to be less uniformly effective than their disjunctive counterparts. Different problems in the cumulative scheduling class seem to have different characteristics that make them either easy or hard to solve with a given technique. The aim of this paper is to investigate one particular dimension along which problems differ. Within the cumulative scheduling class, we distinguish between highly disjunctive and highly cumulative problems: a problem is highly disjunctive when many pairs of activities cannot execute in parallel, e.g., because many activities require more than half of the capacity of a resource; on the contrary, a problem is highly cumulative if many activities can effectively execute in parallel. New constraint propagation and problem decomposition techniques are introduced with this distinction in mind. This includes an O(n²) edge-finding algorithm for cumulative resources (where n is the number of activities requiring the same resource) and a problem decomposition scheme which applies well to highly disjunctive project scheduling problems. Experimental results confirm that the impact of these techniques varies from highly disjunctive to highly cumulative problems. In the end, we also propose a refined version of the edge-finding algorithm for cumulative resources which, despite its worst case complexity in O(n³) , performs very well on highly cumulative instances.     

Arc-B-consistency of the Inter-distance Constraint

We study the “inter-distance constraint,” also known as the global minimum distance constraint, that ensures that the distance between any pair of variables is at least equal to a given value. When this value is 1, the inter-distance constraint reduces to the all-different constraint. We introduce an algorithm to propagate this constraint and we show that, when variables domains are intervals, our algorithm achieves arc-B-consistency. It provides tighter bounds than generic scheduling constraint propagation algorithms (like edge-finding) that could be used to capture this constraint. The worst case complexity of the algorithm is cubic but it behaves well in practice and it drastically reduces the search space. Experiments on special Job-Shop problems and on an Air-Traffic problem known as the “Runway Sequencing” problem.     

Extending constraint logic programming for temporal reasoning

In recent years, several constraint‐based temporal reasoning frameworks have been proposed. They consider temporal points or intervals as domain elements linked by temporal constraints. Temporal reasoning in these systems is based on constraint propagation. In this paper, we argue that a language based on constraint propagation can be a suitable tool for expressing and reasoning about temporal problems. We concentrate on Constraint Logic Programming (CLP) which is a powerful programming paradigm combining the advantages of Logic Programming and the efficiency of constraint solving. However, CLP presents some limitations in dealing with temporal reasoning. First, it uses an “arc consistency” propagation algorithm which is embedded in the inference engine, cannot be changed by the user, and is too weak in many temporal frameworks. Second, CLP is not able to deal with qualitative temporal constraints. We present a general meta CLP architecture which maintains the advantages of CLP, but overcomes these two main limitations. Each architectural level is a finite domain constraint solver(CLP(FD)) that reasons about constraints of the underlying level. Based on this conceptual architecture, we extend the CLP(FD)language and we specialize the extension proposed on Vilain and Kautz’sPoint Algebra, on Allen’s Interval Algebra and on the STP framework by Dechter, Meiri and Pearl. In particular, we show that we can cope effectively with disjunctive constraints even in an interval‐based framework. This revised version was published online in June 2006 with corrections to the Cover Date.     

Extending SMT solvers with support for finite domain <Literal>alldifferent</Literal> constraint

In this paper we consider integration of SMT solvers with the filtering algorithms for the finite domain alldifferent constraint. Such integration makes SMT solvers suitable for solving constraint satisfaction problems with the alldifferent constraint involved. First, we present a novel algorithm for explaining inconsistencies and propagations in the alldifferent constraint. We compare it to Katsirelos’ algorithm and flow-based algorithms that are commonly used for that purpose. Then we describe our DPLL(T)-compliant SMT theory solver for constraint satisfaction problems that include alldifferent constraints. We also provide an experimental evaluation of our approach.     

Explaining circuit propagation

The circuit constraint is used to constrain a graph represented by a successor for each node, such that the resulting edges form a circuit. Circuit and its variants are important for various kinds of tour-finding, path-finding and graph problems. In this paper we examine how to integrate the circuit constraint, and its variants, into a lazy clause generation solver. To do so we must extend the constraint to explain its propagation. We consider various propagation algorithms for circuit and examine how best to explain each of them. We compare the effectiveness of each propagation algorithm once we use explanation, since adding explanation changes the trade-off between propagation complexity and power. Simpler propagators, although less powerful, may produce more reusable explanations. Even though the most powerful propagator considered for circuit and variants creates huge explanations, we find that explanation is highly advantageous for solving problems involving this kind of constraint.     

Solving RCPSP/max by lazy clause generation

We present a generic exact method for minimizing the project duration of the resource-constrained project scheduling problem with generalized precedence relations (Rcpsp/max). This is a very general scheduling model with applications areas such as project management and production planning. Our method uses lazy clause generation, i.e., a hybrid of finite domain and Boolean satisfiability solving, in order to apply no-good learning and conflict-driven search to the solution generation. Our experiments show the benefit of lazy clause generation for finding an optimal solution and proving its optimality in comparison to other state-of-the-art exact and non-exact methods. In comparison to other methods, our method is able to find better solutions faster on the Rcpsp/max benchmarks. Indeed, our method closes 573 open problem instances and generates better solutions in most of the remaining instances. Surprisingly, although ours is an exact method, it outperforms the published non-exact methods on these benchmarks in terms of the quality of solutions.     

Don’t care in SMT: building flexible yet efficient abstraction/refinement solvers

This paper describes a method for combining “off-the-shelf” SAT and constraint solvers for building an efficient Satisfiability Modulo Theories (SMT) solver for a wide range of theories. Our method follows the abstraction/refinement approach to simplify the implementation of custom SMT solvers. The expected performance penalty by not using an interweaved combination of SAT and theory solvers is reduced by generalising a Boolean solution of an SMT problem first via assigning don’t care to as many variables as possible. We then use the generalised solution to determine a thereby smaller constraint set to be handed over to the constraint solver for a background theory. We show that for many benchmarks and real-world problems, this optimisation results in considerably smaller and less complex constraint problems. The presented approach is particularly useful for assembling a practically viable SMT solver quickly, when neither a suitable SMT solver nor a corresponding incremental theory solver is available. We have implemented our approach in the ABsolver framework and applied the resulting solver successfully to an industrial case-study: the verification problems arising in verifying an electronic car steering control system impose non-linear arithmetic constraints, which do not fall into the domain of any other available solver.     

The Design of the Zinc Modelling Language

Zinc is a new modelling language developed as part of the G12 project. It has four important characteristics. First, Zinc allows specification of models using a natural mathematical-like notation. To do so it supports overloaded functions and predicates and automatic coercion and provides arithmetic, finite domain and set constraints. Second, while Zinc is a relatively simple and small language, it can be readily extended to different application areas by means of powerful language constructs such as user-defined predicates and functions and constrained types. Third, Zinc provides sophisticated type and instantiation checking which allows early detection of errors in models. Finally, perhaps the main novelty in Zinc is that it is designed to support a modelling methodology in which the same conceptual model can be automatically mapped into different design models, thus allowing modellers to easily “plug and play” with different solving techniques and so choose the most appropriate for that problem. We describe in detail the various language features of Zinc and the many trade-offs we faced in its design.     

Limitations of restricted branching in clause learning

The techniques for making decisions, that is, branching, play a central role in complete methods for solving structured instances of constraint satisfaction problems (CSPs). In this work we consider branching heuristics in the context of propositional satisfiability (SAT), where CSPs are expressed as propositional formulas. In practice, there are cases when SAT solvers based on the Davis-Putnam-Logemann-Loveland procedure (DPLL) benefit from limiting the set of variables the solver is allowed to branch on to so called input variables which provide a strong unit propagation backdoor set to any SAT instance. Theoretically, however, restricting branching to input variables implies a super-polynomial increase in the length of the optimal proofs for DPLL (without clause learning), and thus input-restricted DPLL cannot polynomially simulate DPLL. In this paper we settle the case of DPLL with clause learning. Surprisingly, even with unlimited restarts, input-restricted clause learning DPLL cannot simulate DPLL (even without clause learning). The opposite also holds, and hence DPLL and input-restricted clause learning DPLL are polynomially incomparable. Additionally, we analyze the effect of input-restricted branching on clause learning solvers in practice with various structured real-world benchmarks. This is an extended version of a paper [27] presented at the 13th International Conference on Principles and Practice of Constraint Programming (CP 2007) in Providence, RI, USA. The first author gratefully acknowledges financial support from Helsinki Graduate School in Computer Science and Engineering, Academy of Finland (grants #211025 and #122399), Emil Aaltonen Foundation, Jenny and Antti Wihuri Foundation, Finnish Foundation for Technology Promotion TES, and Nokia Foundation. The second author gratefully acknowledges the financial support from Academy of Finland (grant #112016).     

A random generator of resource-constrained multi-project network problems

Many scheduling problems in project management, manufacturing, and elsewhere require the generation of activity networks to test proposed solution methods. Single-network generators have been used for the resource-constrained project scheduling problem (RCPSP). Since the first single-network generator was proposed in 1993, several advances have been reported in the literature. However, these generators create only one network or project at a time; they cannot generate multi-project problems to desired specifications. This paper presents the first multi-network problem generator. It is especially useful for investigating the resource-constrained multi-project scheduling problem (RCMPSP), where a controlled set of multi-project test problems is crucial for analyzing the performance of solution methods. In addition to the single-project characteristics handled by existing network generators—such as activity duration, resource types and usage, and network size, shape, and complexity—the proposed generator produces multi-project portfolios with controlled resource distributions and amounts of resource contention. To enable the generation of projects with desired levels of network complexity, we also develop several theoretical insights on the effects of network topology on the probability of successful network generation. Finally, we generate 12,320 test problems for a full-factorial experiment and use analysis of means to conclude that the generator produces “near-strongly random” problems. Fully “strongly random” problems require much greater computational expense.     

Resource Constraints for Preemptive Job-shop Scheduling

This paper presents an experimental study of constraint propagation algorithms for preemptive scheduling. We propose generalizations of non-preemptive constraint propagation techniques (based on timetables, on disjunctive constraints, and on edge-finding) to preemptive and mixed problems, i.e., problems in which some activities can be interrupted and some cannot. Another approach relies on incremental flow-based techniques. We theoretically compare these approaches and present an experimental comparison based on a branch and bound procedure for the preemptive variant of the job-shop scheduling problem. We show that both edge-finding and flow-based techniques allow the resolution of hard problem instances, including the preemptive variant of the famous FT10.     

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.     

Project scheduling under resource constraints: Application of the cumulative global constraint in a decision support framework

This paper concerns project scheduling under resource constraints. Traditionally, the objective is to find a unique solution that minimizes the project makespan, while respecting the precedence constraints and the resource constraints. This work focuses on developing a model and a decision support framework for industrial application of the cumulative global constraint. For a given project scheduling, the proposed approach allows the generation of different optimal solutions relative to the alternate availability of outsourcing and resources. The objective is to provide a decision-maker an assistance to construct, choose, and define the appropriate scheduling program taking into account the possible capacity resources. The industrial problem under consideration is modeled as a constraint satisfaction problem (CSP). It is implemented under the constraint programming language CHIP V5. The provided solutions determine values for the various variables associated to the tasks realized on each resource, as well as the curves with the profile of the total consumption of resources on time.     

Maintenance scheduling problems as benchmarks for constraint algorithms

The paper focuses on evaluating constraint satisfaction search algorithms on application based random problem instances. The application we use is a well-studied problem in the electric power industry: optimally scheduling preventive maintenance of power generating units within a power plant. We show how these scheduling problems can be cast as constraint satisfaction problems and used to define the structure of randomly generated non-binary CSPs. The random problem instances are then used to evaluate several previously studied algorithms. The paper also demonstrates how constraint satisfaction can be used for optimization tasks. To find an optimal maintenance schedule, a series of CSPs are solved with successively tighter cost-bound constraints. We introduce and experiment with an “iterative learning” algorithm which records additional constraints uncovered during search. The constraints recorded during the solution of one instance with a certain cost-bound are used again on subsequent instances having tighter cost-bounds. Our results show that on a class of randomly generated maintenance scheduling problems, iterative learning reduces the time required to find a good schedule. This revised version was published online in June 2006 with corrections to the Cover Date.     

Randomised restarted search in ILP

Recent statistical performance studies of search algorithms in difficult combinatorial problems have demonstrated the benefits of randomising and restarting the search procedure. Specifically, it has been found that if the search cost distribution of the non-restarted randomised search exhibits a slower-than-exponential decay (that is, a “heavy tail”), restarts can reduce the search cost expectation. We report on an empirical study of randomised restarted search in ILP. Our experiments conducted on a high-performance distributed computing platform provide an extensive statistical performance sample of five search algorithms operating on two principally different classes of ILP problems, one represented by an artificially generated graph problem and the other by three traditional classification benchmarks (mutagenicity, carcinogenicity, finite element mesh design). The sample allows us to (1) estimate the conditional expected value of the search cost (measured by the total number of clauses explored) given the minimum clause score required and a “cutoff” value (the number of clauses examined before the search is restarted), (2) estimate the conditional expected clause score given the cutoff value and the invested search cost, and (3) compare the performance of randomised restarted search strategies to a deterministic non-restarted search. Our findings indicate striking similarities across the five search algorithms and the four domains, in terms of the basic trends of both the statistics (1) and (2). Also, we observe that the cutoff value is critical for the performance of the search algorithm, and using its optimal value in a randomised restarted search may decrease the mean search cost (by several orders of magnitude) or increase the mean achieved score significantly with respect to that obtained with a deterministic non-restarted search. Editors: Rui Camacho     

Grouping Techniques for Scheduling Problems: Simpler and Faster

In this paper we describe a general grouping technique to devise faster and simpler approximation schemes for several scheduling problems. We illustrate the technique on two different scheduling problems: scheduling on unrelated parallel machines with costs and the job shop scheduling problem. The time complexity of the resulting approximation schemes is always linear in the number n of jobs, and the multiplicative constant hidden in the O(n) running time is reasonably small and independent of the error ε. Supported by Swiss National Science Foundation project 200020-109854, “Approximation Algorithms for Machine scheduling Through Theory and Experiments II”. A preliminary version of this paper appeared in the Proceedings of ESA’01.     

E(FG)<Superscript>2</Superscript>: a new fixed-grid shape optimization method based on the element-free galerkin mesh-free analysis

We propose a shape optimization method over a fixed grid. Nodes at the intersection with the fixed grid lines track the domain’s boundary. These “floating” boundary nodes are the only ones that can move/appear/disappear in the optimization process. The element-free Galerkin (EFG) method, used for the analysis problem, provides a simple way to create these nodes. The fixed grid (FG) defines integration cells for EFG method. We project the physical domain onto the FG and numerical integration is performed over partially cut cells. The integration procedure converges quadratically. The performance of the method is shown with examples from shape optimization of thermal systems involving large shape changes between iterations. The method is applicable, without change, to shape optimization problems in elasticity, etc. and appears to eliminate non-differentiability of the objective noticed in finite element method (FEM)-based fictitious domain shape optimization methods. We give arguments to support this statement. A mathematical proof is needed.     

具有优先关系的累积调度问题的约束传播算法

约束传播是约束规划成功应用的关键技术之一. 针对累积调度问题提出一种结合工作间优先关系和工作最早开始/最晚完成时间约束的约束传播算法, 给出了算法的理论依据. 引用资源受限项目调度问题库PSPLIB中的典型问题对算法进行了测试, 结果表明: 针对测试问题新的约束传播算法在总体约减效果上优于现有约束传播算法, 新算法与基于能量推理的约束传播算法可以互补, 两者结合推理效果更好.     

Packet transmission policies for battery operated wireless sensor networks

With the concept of “Cognitive Sense of China” and “Smart Planet” proposed, wireless sensor networking is considered to be one of the most important technologies of the new century. In wireless sensor networks, how to extend battery lifetime is a core problem. In this paper, we address the problem of designing battery-friendly packet transmission policies for wireless sensor networks. Our objective is to maximize the lifetime of batteries for wireless sensor nodes subject to certain delay constraints. We present three packet transmission schemes and evaluate them with respect to battery performance. The first scheme, based on combining multiple packets, utilizes battery charge recovery effect, which allows some charge to be recovered during long idle periods. The second scheme, based on a modified version of lazy packet scheduling, draws smoother and lower current and is battery efficient. The final scheme, based on a combination of the two previous schemes has superior battery performance at the expense of larger average packet delay. All three schemes are simulated for a wireless network framework with internet traffic, and the results were validated.