Constraint-based Local Search Research Articles

Towards Sequence-to-Sequence Reinforcement Learning for Constraint Solving with Constraint-Based Local Search

This paper proposes a framework for solving constraint problems with reinforcement learning (RL) and sequence-tosequence recurrent neural networks. We approach constraint solving as a declarative machine learning problem, where for a variable-length input sequence a variable-length output sequence has to be predicted. Using randomly generated instances and the number of constraint violations as a reward function, a problem-specific RL agent is trained to solve the problem. The predicted solution candidate of the RL agent is verified and repaired by CBLS to ensure solutions, that satisfy the constraint model. We introduce the framework and its components and discuss early results and future applications.

A Constraint-Based Local Search for Offline and Online General Vehicle Routing

Vehicle routing is a class of combinatorial optimization problems arising in the industry of transportation and logistics. The goal of these problems is to compute an optimal route plan for a set of vehicles for serving transport requests of customers. There are many variants of the vehicle routing problems: routing for delivering goods, routing for demand responsive transport (taxi, school bus, …). Each problem might have different constraints, objectives. In this paper, we introduce a Constraint-Based Local Search (CBLS) framework for general offline and online vehicle routing problems. We extend existing neighborhood structures in the literature by proposing new neighborhoods to facilitate the resolution of different class of vehicle routing problems in a unified platform. A novel feature of the framework is the available APIs for online vehicle routing problems where requests arrive online during the execution of the computed route plan. Experimental results on three vehicle routing problems (the min-max capacitated vehicle routing problem, the multi-vehicle covering tour problem, and the online people-andparcel share-a-ride taxis problem) show the modelling flexibility, genericity, extensibility and efficiency of the proposed framework.

Estimating parallel runtimes for randomized algorithms in constraint solving

This paper presents a detailed analysis of the scalability and parallelization of Local Search algorithms for constraint-based and SAT (Boolean satisfiability) solvers. We propose a framework to estimate the parallel performance of a given algorithm by analyzing the runtime behavior of its sequential version. Indeed, by approximating the runtime distribution of the sequential process with statistical methods, the runtime behavior of the parallel process can be predicted by a model based on order statistics. We apply this approach to study the parallel performance of a constraint-based Local Search solver (Adaptive Search), two SAT Local Search solvers (namely Sparrow and CCASAT), and a propagation-based constraint solver (Gecode, with a random labeling heuristic). We compare the performance predicted by our model to actual parallel implementations of those methods using up to 384 processes. We show that the model is accurate and predicts performance close to the empirical data. Moreover, as we study different types of problems, we observe that the experimented solvers exhibit different behaviors and that their runtime distributions can be approximated by two types of distributions: exponential (shifted and non-shifted) and lognormal. Our results show that the proposed framework estimates the runtime of the parallel algorithm with an average discrepancy of 21 % w.r.t. the empirical data across all the experiments with the maximum allowed number of processors for each technique.

A constraint-based local search backend for MiniZinc

MiniZinc is a modelling language for combinatorial problems, which can then be solved by a solver provided in a backend. There are many backends, based on technologies such as constraint programming, integer programming, or Boolean satisfiability solving. However, to the best of our knowledge, there is currently no constraint-based local search (CBLS) backend. We discuss the challenges to develop such a backend and give an overview of the design of a CBLS backend for MiniZinc. Experimental results show that for some MiniZinc models, our CBLS backend, based on the OscaR/CBLS solver, is able to give good-quality results in competitive time.

Large-scale parallelism for constraint-based local search: the costas array case study

We present the parallel implementation of a constraint-based Local Search algorithm and investigate its performance on several hardware platforms with several hundreds or thousands of cores. We chose as the basis for these experiments the Adaptive Search method, an efficient sequential Local Search method for Constraint Satisfaction Problems (CSP). After preliminary experiments on some CSPLib benchmarks, we detail the modeling and solving of a hard combinatorial problem related to radar and sonar applications: the Costas Array Problem. Performance evaluation on some classical CSP benchmarks shows that speedups are very good for a few tens of cores, and good up to a few hundreds of cores. However for a hard combinatorial search problem such as the Costas Array Problem, performance evaluation of the sequential version shows results outperforming previous Local Search implementations, while the parallel version shows nearly linear speedups up to 8,192 cores. The proposed parallel scheme is simple and based on independent multi-walks with no communication between processes during search. We also investigated a cooperative multi-walk scheme where processes share simple information, but this scheme does not seem to improve performance.

How good are simplified models for protein structure prediction?

Protein structure prediction (PSP) has been one of the most challenging problems in computational biology for several decades. The challenge is largely due to the complexity of the all-atomic details and the unknown nature of the energy function. Researchers have therefore used simplified energy models that consider interaction potentials only between the amino acid monomers in contact on discrete lattices. The restricted nature of the lattices and the energy models poses a twofold concern regarding the assessment of the models. Can a native or a very close structure be obtained when structures are mapped to lattices? Can the contact based energy models on discrete lattices guide the search towards the native structures? In this paper, we use the protein chain lattice fitting (PCLF) problem to address the first concern; we developed a constraint-based local search algorithm for the PCLF problem for cubic and face-centered cubic lattices and found very close lattice fits for the native structures. For the second concern, we use a number of techniques to sample the conformation space and find correlations between energy functions and root mean square deviation (RMSD) distance of the lattice-based structures with the native structures. Our analysis reveals weakness of several contact based energy models used that are popular in PSP.

Dynamic Online Planning and Scheduling Using a Static Invariant-Based Evaluation Model

Sequential decision-making under uncertainty often uses an approach in which a plan is built over a given horizon ahead using a deterministic model, the first decisions in this plan are applied, new information is acquired, the plan is adapted or rebuilt from scratch over a sliding horizon, and so on. This paper introduces a generic local search library that can be used in this context to quickly build and rebuild good quality plans. This library is built upon the notion of invariant used in constraint-based local search. Invariants allow temporal constraints, resource constraints, and criteria to be very quickly evaluated from a variable assignment and re-evaluated from a small change in this assignment. The paper also shows how the library can be used to reason on dynamic problem instances using a unique static problem model, without dynamic memory allocation. The approach is illustrated on a problem of data download under uncertainty about the volume of data, coming from the space domain.

Underestimating the cost of a soft constraint is dangerous: revisiting the edit-distance based soft regular constraint

Many real-life problems are over-constrained, so that no solution satisfying all their constraints exists. Soft constraints, with costs denoting how much the constraints are violated, are used to solve these problems. We use the edit-distance based SoftRegular constraint as an example to show that a propagation algorithm that sometimes underestimates the cost may guide the search to incorrect (non-optimal) solutions to an over-constrained problem. To compute correctly the cost for the edit-distance based SoftRegular constraint, we present a quadratic-time propagation algorithm based on dynamic programming and a proof of its correctness. We also give an improved propagation algorithm using an idea of computing the edit distance between two strings, which may also be applied to other constraints with propagators based on dynamic programming. The asymptotic time complexity of our improved propagator is always at least as good as the one of our quadratic-time propagator, but significantly better when the edit distance is small. Our propagators achieve domain consistency on the problem variables and bounds consistency on the cost variable. Our method can also be adapted for the violation measure of the edit-distance based Regular constraint for constraint-based local search.

LS(Graph): a constraint-based local search for constraint optimization on trees and paths

Constrained optimum tree (COT) and constrained optimum path (COP) problems arise in many real-life applications and are ubiquitous in communication networks. They have been traditionally approached by dedicated algorithms, which are often hard to extend with side constraints and to apply widely. This paper proposes a constraint-based local search framework for COT/COP applications, bringing the compositionality, reuse, and extensibility at the core of constraint-based local search and constraint programming systems. The modeling contribution is the ability to express compositional models for various COT/COP applications at a high level of abstraction, while cleanly separating the model and the search procedure. The main technical contribution is a connected neighborhood based on rooted spanning trees to find high-quality solutions to COP problems. This framework is applied to some COT/COP problems, e.g., the quorumcast routing problem, the edge-disjoint paths problem, and the routing and wavelength assignment with delay side constraints problem. Computational results show the potential importance of the approach.

Performance analysis of parallel constraint-based local search

We present a parallel implementation of a constraint-based local search algorithm and investigate its performance results for hard combinatorial optimization problems on two different platforms up to several hundreds of cores. On a variety of classical CSPs benchmarks, speedups are very good for a few tens of cores, and good up to a hundred cores. More challenging problems derived from reallife applications (Costas array) shows even better speedups, nearly optimal up to 256 cores.

Constraint-Based Local Search for Inventory Control Under Stochastic Demand and Lead Time

In this paper, we address the general multiperiod production/inventory problem with nonstationary stochastic demand and supplier lead time under service-level constraints. A replenishment cycle policy is modeled. We propose two hybrid algorithms that blend constraint programming and local search for computing near-optimal policy parameters. Both algorithms rely on a coordinate descent local search strategy; what differs is the way this strategy interacts with the constraint programming solver. These two heuristics are first, compared for small instances against an existing optimal solution method. Second, they are tested and compared with each other in terms of solution quality and run time on a set of larger instances that are intractable for the exact approach. Our numerical experiments show the effectiveness of our methods.

Contingency plans for air traffic flow and capacity management using constraint programming

We present a constraint-based local search heuristic that contributes to solving the problem of generating contingency plans for air traffic flow and capacity management, which are to be used in the case of a catastrophic infrastructure failure within EUROCONTROL, the European Organisation for the Safety of Air Navigation. Experiments with the heuristic, implemented in Comet, on real-world flight plans for the entire European airspace show that it is feasible to automate the development of contingency plans, which is currently done by human experts. This is desirable as the development time goes down from two person months per year to a few CPU hours, and as it allows contingency plans to be generated with an increased frequency.

A heuristic to solve the synchronized log-truck scheduling problem

We present a synchronized routing and scheduling problem that arises in the forest industry, as a variation of the log-truck scheduling problem. It combines routing and scheduling of trucks with specific constraints related to the Canadian forestry context. This problem includes aspects such as pick-up and delivery, multiple products, inventory stock, multiple supply points and multiple demand points. We developed a decomposition approach to solve the weekly problem in two phases. In the first phase we use a MIP solver to solve a tactical model that determines the destinations of full truckloads from forest areas to woodmills. In the second phase, we make use of two different methods to route and schedule the daily transportation of logs: the first one consists in using a constraint-based local search approach while the second one is a hybrid approach involving a constraint programming based model and a constraint-based local search model. These approaches have been implemented using COMET2.0. The method, was tested on two industrial cases from forest companies in Canada.

Solving Steel Mill Slab Problems with constraint-based techniques: CP, LNS, and CBLS

The Steel Mill Slab Problem is an optimization benchmark that has been studied for a long time in the constraint-programming community but was only solved efficiently in the two last years. Gargani and Refalo solved the problem using Large Neighborhood Search and Van Hentenryck and Michel made use of constraint programming with an improved symmetry breaking scheme. In the first part of this paper, we build on those approaches, present improvements of those two techniques, and study how the problem can be tackled by Constraint-Based Local Search. As a result, the classical instances of CSPLib can now be solved in less than 50 ms. To improve our understanding of this problem, we also introduce a new set of harder instances, which highlight the strengths and the weaknesses of the various approaches. In a second part of the paper, we present a variation of the Steel Mill Slab Problem whose aim is to minimize the number of slabs. We show how this problem can be tackled with slight modifications of our proposed algorithms. In particular, the constraint-programming solution is enhanced by a global symmetric cardinality constraint, which, to our knowledge, has never been implemented and used before. All the proposed approaches to solve this problem have been modeled and evaluated using Comet.

Dynamic Demand-Capacity Balancing for Air Traffic Management Using Constraint-Based Local Search: First Results

Using constraint-based local search, we effectively model and efficiently solve the problem of balancing the traffic demands on portions of the European airspace while ensuring that their capacity constraints are satisfied. The traffic demand of a portion of airspace is the hourly number of flights planned to enter it, and its capacity is the upper bound on this number under which air-traffic controllers can work. Currently, the only form of demand-capacity balancing we allow is ground holding, that is the changing of the take-off times of not yet airborne flights. Experiments with projected European flight plans of the year 2030 show that already this first form of demand-capacity balancing is feasible without incurring too much total delay and that it can lead to a significantly better demand-capacity balance.

Parallel local search for solving Constraint Problems on the Cell Broadband Engine (Preliminary Results)

We explore the use of the Cell Broadband Engine (Cell/BE for short) for combinatorial optimization applications: we present a parallel version of a constraint-based local search algorithm that has been implemented on a multiprocessor BladeCenter machine with twin Cell/BE processors (total of 16 SPUs per blade). This algorithm was chosen because it fits very well the Cell/BE architecture and requires neither shared memory nor communication between processors, while retaining a compact memory footprint. We study the performance on several large optimization benchmarks and show that this achieves mostly linear time speedups, even sometimes super-linear. This is possible because the parallel implementation might explore simultaneously different parts of the search space and therefore converge faster towards the best sub-space and thus towards a solution. Besides getting speedups, the resulting times exhibit a much smaller variance, which benefits applications where a timely reply is critical.

Model-driven visualizations of constraint-based local search

Visualization is often invaluable to understand the behavior of optimization algorithms, identify their bottlenecks or pathological behaviors, and suggest remedial techniques. Yet developing visualizations is often a tedious activity requiring significant time and expertise. This paper presents a framework for the visualization of constraint-based local search (CBLS) algorithms. Given a high-level model and a declarative visualization specification, the CBLS visualizer systematically produces animations to visualize constraints and objectives, violations, and conflicts, as well as the temporal behavior of these measures. The visualization specification is declarative and typically composed of a triple (what,where,how) indicating what to display, where, and with which graphical objects. The visualizer architecture is compositional and extensible. It provides building blocks which can be assembled freely by the user and focuses almost exclusively on static aspects, the dynamic aspects being automated by the use of invariants. The paper highlights various functionalities of the visualizer and describes a blueprint for its implementation.

Revisiting constraint-directed search

In constraint-based local search the solutions are described declaratively by a conjunction of (often high-level) constraints. In this article we show that this opens up new ideas for constraint-directed search. For a constraint we introduce three neighbourhoods, where the penalty for that constraint alone is decreasing, increasing, or unchanged. We give specialised algorithms for common constraints that efficiently implement these neighbourhoods. Further, we give a general algorithm that implements these neighbourhoods from specifications of constraints in monadic existential second-order logic. Finally, we show how common constraint-directed local search algorithms are often easier to express using these neighbourhoods.