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## Über dieses Buch

This book constitutes the proceedings of the 15th International Conference on Integration of Artificial Intelligence and Operations Research Techniques in Constraint Programming for Combinatorial Optimization Problems, CPAIOR 2018, held in Delft, The Netherlands, in June 2018.

The 47 full papers presented together with 3 abstracts of invited talks and 3 abstracts of fast-track journal papers were carefully reviewed and selected from 111 submissions. The conference brings together interested researchers from constraint programming, artificial intelligence, and operations research to present new techniques or applications in the intersection of these fields and provides an opportunity for researchers in one area to learn about techniques in the others, and to show how the integration of techniques from different fields can lead to interesting results on large and complex problems.

## Inhaltsverzeichnis

### Time-Bounded Query Generator for Constraint Acquisition

QuAcq is a constraint acquisition algorithm that assists a non-expert user to model her problem as a constraint network. QuAcq generates queries as examples to be classified as positive or negative. One of the drawbacks of QuAcq is that generating queries can be time-consuming. In this paper we present Tq-gen, a time-bounded query generator. Tq-gen is able to generate a query in a bounded amount of time. We rewrite QuAcq to incorporate the Tq-gen generator. This leads to a new algorithm called T-quacq. We propose several strategies to make T-quacq efficient. Our experimental analysis shows that thanks to the use of Tq-gen, T-quacq dramatically improves the basic QuAcq in terms of time consumption, and sometimes also in terms of number of queries.

### Propagating lex, find and replace with Dashed Strings

Dashed strings have been recently proposed in Constraint Programming to represent the domain of string variables when solving combinatorial problems over strings. This approach showed promising performance on some classes of string problems, involving constraints like string equality and concatenation. However, there are a number of string constraints for which no propagator has yet been defined. In this paper, we show how to propagate lexicographic ordering (lex), find and replace with dashed strings. All of these are fundamental string operations: lex is the natural total order over strings, while find and replace are frequently used in string manipulation. We show that these propagators, that we implemented in G-Strings solver, allows us to be competitive with state-of-the-art approaches.

Roberto Amadini, Graeme Gange, Peter J. Stuckey

### Designing Fair, Efficient, and Interpretable Policies for Prioritizing Homeless Youth for Housing Resources

We consider the problem of designing fair, efficient, and interpretable policies for prioritizing heterogeneous homeless youth on a waiting list for scarce housing resources of different types. We focus on point-based policies that use features of the housing resources (e.g., permanent supportive housing, rapid rehousing) and the youth (e.g., age, history of substance use) to maximize the probability that the youth will have a safe and stable exit from the housing program. The policies can be used to prioritize waitlisted youth each time a housing resource is procured. Our framework provides the policy-maker the flexibility to select both their desired structure for the policy and their desired fairness requirements. Our approach can thus explicitly trade-off interpretability and efficiency while ensuring that fairness constraints are met. We propose a flexible data-driven mixed-integer optimization formulation for designing the policy, along with an approximate formulation which can be solved efficiently for broad classes of interpretable policies using Bender’s decomposition. We evaluate our framework using real-world data from the United States homeless youth housing system. We show that our framework results in policies that are more fair than the current policy in place and than classical interpretable machine learning approaches while achieving a similar (or higher) level of overall efficiency.

### An Efficient Relaxed Projection Method for Constrained Non-negative Matrix Factorization with Application to the Phase-Mapping Problem in Materials Science

In recent years, a number of methods for solving the constrained non-negative matrix factorization problem have been proposed. In this paper, we propose an efficient method for tackling the ever increasing size of real-world problems. To this end, we propose a general relaxation and several algorithms for enforcing constraints in a challenging application: the phase-mapping problem in materials science. Using experimental data we show that the proposed method significantly outperforms previous methods in terms of $$\ell _2$$-norm error and speed.

Junwen Bai, Sebastian Ament, Guillaume Perez, John Gregoire, Carla Gomes

### Dealing with Demand Uncertainty in Service Network and Load Plan Design

Less-than-Truckload (LTL) transportation carriers plan for their next operating season by deciding: (1) a load plan, which specifies how shipments are routed through the terminal network from origins to destinations, and (2) how many trailers to operate between each pair of terminals in the network. Most carriers also require that the load plan is such that shipments at an intermediate terminal and having the same ultimate destination are loaded onto trailers headed to a unique next terminal regardless of their origins. In practice, daily variations in demand are handled by relaxing this requirement and possibly loading shipments to an alternative next terminal. We introduce the p-alt model, which integrates routing and capacity decisions, and which allows p choices for the next terminal for shipments with a particular ultimate destination. We further introduce and computationally test three solution methods for the stochastic p-alt model, which shows that much can be gained from using the p-alt model and explicitly considering demand uncertainty.

Ahmad Baubaid, Natashia Boland, Martin Savelsbergh

### Energy-Aware Production Scheduling with Power-Saving Modes

This study addresses optimization of production processes where machines have high energy consumption. One efficient way to reduce the energy expenses in production is to turn a machine off when it is not being used or switch it into an energy-saving mode. If the production has several machines and production demand that varies in time, the energy saving can be substantial; the cost reduction can be achieved by an appropriate production schedule that could control the switching between the energy modes with respect to the required production volume. Therefore, inspired by real production processes of glass tempering and steel hardening, this paper addresses the scheduling of jobs with release times and deadlines on parallel machines. The objective is to find a schedule of the jobs and a switching between the power modes of the machines so that the total energy consumption is minimized. Moreover, to further generalize the scheduling problem to other production processes, we assume that the processing time of the jobs is mode-dependent, i.e., the processing time of a job depends on the mode in which a machine is operating. The study provides an efficient Branch-and-Price algorithm and compares two approaches (based on Integer Linear Programming and Constraint Programming) for solving the subproblem.

Ondřej Benedikt, Přemysl Šůcha, István Módos, Marek Vlk, Zdeněk Hanzálek

### EpisodeSupport: A Global Constraint for Mining Frequent Patterns in a Long Sequence of Events

The number of applications generating sequential data is exploding. This work studies the discovering of frequent patterns in a large sequence of events, possibly time-stamped. This problem is known as the Frequent Episode Mining (FEM). Similarly to the mining problems recently tackled by Constraint Programming (CP), FEM would also benefit from the modularity offered by CP to accommodate easily additional constraints on the patterns. These advantages do not offer a guarantee of efficiency. Therefore, we introduce two global constraints for solving FEM problems with or without time consideration. The time-stamped version can accommodate gap and span constraints on the matched sequences. Our experiments on real data sets of different levels of complexity show that the introduced constraints is competitive with the state-of-the-art methods in terms of execution time and memory consumption while offering the flexibility of adding constraints on the patterns.

Quentin Cappart, John O. R. Aoga, Pierre Schaus

### Off-Line and On-Line Optimization Under Uncertainty: A Case Study on Energy Management

Optimization problems under uncertainty arise in many application areas and their solution is very challenging. We propose here methods that merge off-line and on-line decision stages: we start with a two stage off-line approach coupled with an on-line heuristic. We improve this baseline in two directions: (1) by replacing the on-line heuristics with a simple anticipatory method; (2) by making the off-line component aware of the on-line heuristic. Our approach is grounded on a virtual power plant management system, where the load shifts can be planned off-line and the energy balance should be maintained on-line. The overall goal is to find the minimum cost energy flows at each point in time considering (partially shiftable) electric loads, renewable and non-renewable energy generators, and electric storages. We compare our models with an oracle operating under perfect information and we show that both our improved models achieve a high solution quality, while striking different trade-offs in terms of computation time and complexity of the off-line and on-line optimization techniques.

Allegra De Filippo, Michele Lombardi, Michela Milano

### Reasoning on Sequences in Constraint-Based Local Search Frameworks

This paper explains why global constraints for routing cannot be integrated into Constraint-Based Local Search (CBLS) frameworks. A technical reason for this is identified and defined as the multi-variable bottleneck. We solve this bottleneck by introducing a new type of variables: “sequence of integers”. We identify key requirements and defines a vocabulary for this variable type, through which it communicates with global constraints. Dedicated data structures are designed for efficiently representing sequences in this context. Benchmarks are presented to identify how to best parametrise those data structures and to compare our approach with other state-of-the-art local search frameworks: LocalSolver and GoogleCP. Our contribution is included in the CBLS engine of the open source OscaR framework.

Renaud De Landtsheer, Yoann Guyot, Gustavo Ospina, Fabian Germeau, Christophe Ponsard

### Constraint Programming for High School Timetabling: A Scheduling-Based Model with Hot Starts

High School Timetabling (HSTT) is a well-known and wide-spread problem. It consists of coordinating resources (e.g. teachers, rooms), times, and events (e.g. classes) with respect to a variety of constraints. In this paper, we study the applicability of constraint programming (CP) for high school timetabling. We formulate a novel CP model for HSTT using a scheduling-based point of view. We show that a drastic improvement in performance over the baseline CP model can be achieved by including solution-based phase saving, which directs the CP solver to first search in close proximity to the best solution found, and our hot start approach, where we use existing heuristic methods to produce a starting point for the CP solver. The experiments demonstrate that our approach outperforms the IP and maxSAT complete methods and provides competitive results when compared to dedicated heuristic solvers.

Emir Demirović, Peter J. Stuckey

### Epiphytic Trees: Relational Consistency Applied to Global Optimization Problems

Much effort has been spent to identify classes of CSPs in terms of the relationship between network structure and the amount of consistency that guarantees a backtrack-free solution. In this paper, we address Numerical Constrained global Optimization Problems (NCOPs) encoded as ternary networks, characterizing a class of such problems for which a combination of Generalized Arc-Consistency (GAC) and Relational Arc-Consistency (RAC) is sufficient to ensure a backtrack-free solution, called Epiphytic Trees. While GAC is a domain filtering technique, enforcing RAC creates new constraints in the network. Alternatively, we propose a branch and bound method to achieve a relaxed form of RAC, thus finding an approximation of the solution of NCOPs. We empirically show that Epiphytic Trees are relevant in practice. In addition, we extend this class to cover all ternary NCOPs, for which Strong Directional Relational k-Consistency ensures a backtrack-free solution.

Guilherme Alex Derenievicz, Fabiano Silva

### Learning Heuristics for the TSP by Policy Gradient

The aim of the study is to provide interesting insights on how efficient machine learning algorithms could be adapted to solve combinatorial optimization problems in conjunction with existing heuristic procedures. More specifically, we extend the neural combinatorial optimization framework to solve the traveling salesman problem (TSP). In this framework, the city coordinates are used as inputs and the neural network is trained using reinforcement learning to predict a distribution over city permutations. Our proposed framework differs from the one in [1] since we do not make use of the Long Short-Term Memory (LSTM) architecture and we opted to design our own critic to compute a baseline for the tour length which results in more efficient learning. More importantly, we further enhance the solution approach with the well-known 2-opt heuristic. The results show that the performance of the proposed framework alone is generally as good as high performance heuristics (OR-Tools). When the framework is equipped with a simple 2-opt procedure, it could outperform such heuristics and achieve close to optimal results on 2D Euclidean graphs. This demonstrates that our approach based on machine learning techniques could learn good heuristics which, once being enhanced with a simple local search, yield promising results.

Michel Deudon, Pierre Cournut, Alexandre Lacoste, Yossiri Adulyasak, Louis-Martin Rousseau

### Three-Dimensional Matching Instances Are Rich in Stable Matchings

Extensive studies have been carried out on the Stable Matching problem, but they mostly consider cases where the agents to match belong to either one or two sets. Little work has been done on the three-set extension, despite the many applications in which three-dimensional stable matching (3DSM) can be used. In this paper we study the Cyclic 3DSM problem, a variant of 3DSM where agents in each set only rank the agents from one other set, in a cyclical manner. The question of whether every Cyclic 3DSM instance admits a stable matching has remained open for many years. We give the exact number of stable matchings for the class of Cyclic 3DSM instances where all agents in the same set share the same master preference list. This number is exponential in the size of the instances. We also show through empirical experiments that this particular class contains the most constrained Cyclic 3DSM instances, the ones with the fewest stable matchings. This would suggest that not only do all Cyclic 3DSM instances have at least one stable matching, but they each have an exponential number of them.

Guillaume Escamocher, Barry O’Sullivan

### From Backdoor Key to Backdoor Completability: Improving a Known Measure of Hardness for the Satisfiable CSP

Many studies have been conducted on the complexity of Constraint Satisfaction Problem (CSP) classes. However, there exists little theoretical work on the hardness of individual CSP instances. In this context, the backdoor key fraction (BKF) [17] was introduced as a quantifier of problem hardness for individual satisfiable instances with regard to backtracking search. In our paper, after highlighting the weaknesses of the BKF, we propose a better characterization of the hardness of an individual satisfiable CSP instance based on the ratio between the size of the solution space and that of the search space. We formally show that our measure is negatively correlated with instance hardness. We also show through experiments that this measure evaluates more accurately the hardness of individual instances than the BKF.

Guillaume Escamocher, Mohamed Siala, Barry O’Sullivan

### Constrained-Based Differential Privacy: Releasing Optimal Power Flow Benchmarks Privately

Releasing Optimal Power Flow Benchmarks Privately

This paper considers the problem of releasing optimal power flow benchmarks that maintain the privacy of customers (loads) using the notion of Differential Privacy. It is motivated by the observation that traditional differential-privacy mechanisms are not accurate enough: The added noise fundamentally changes the nature of the underlying optimization and often leads to test cases with no solution. To remedy this limitation, the paper introduces the framework of Constraint-Based Differential Privacy (CBDP) that leverages the post- processing immunity of differential privacy to improve the accuracy of traditional mechanisms. More precisely, CBDP solves an optimization problem to satisfies the problem-specific constraints by redistributing the noise. The paper shows that CBDP enjoys desirable theoretical properties and produces orders of magnitude improvements on the largest set of test cases available.

Ferdinando Fioretto, Pascal Van Hentenryck

### Chasing First Queens by Integer Programming

The n-queens puzzle is a well-known combinatorial problem that requires to place n queens on an $$n\times n$$ chessboard so that no two queens can attack each other. Since the 19th century, this problem was studied by many mathematicians and computer scientists. While finding any solution to the n-queens puzzle is rather straightforward, it is very challenging to find the lexicographically first (or smallest) feasible solution. Solutions for this type are known in the literature for $$n\le 55$$, while for some larger chessboards only partial solutions are known. The present paper was motivated by the question of whether Integer Linear Programming (ILP) can be used to compute solutions for some open instances. We describe alternative ILP-based solution approaches, and show that they are indeed able to compute (sometimes in unexpectedly-short computing times) many new lexicographically optimal solutions for n ranging from 56 to 115.

Matteo Fischetti, Domenico Salvagnin

### Accelerating Counting-Based Search

Counting-based search, a branching heuristic used in constraint programming, relies on computing the proportion of solutions to a constraint in which a given variable-value assignment appears in order to build an integrated variable- and value-selection heuristic to solve constraint satisfaction problems. The information it collects has led to very effective search guidance in many contexts. However, depending on the constraint, computing such information can carry a high computational cost. This paper presents several contributions to accelerate counting-based search, with supporting empirical evidence that solutions can thus be obtained orders of magnitude faster.

Samuel Gagnon, Gilles Pesant

### Model Agnostic Solution of CSPs via Deep Learning: A Preliminary Study

Deep Neural Networks (DNNs) have been shaking the AI scene, for their ability to excel at Machine Learning tasks without relying on complex, hand-crafted, features. Here, we probe whether a DNN can learn how to construct solutions of a CSP, without any explicit symbolic information about the problem constraints. We train a DNN to extend a feasible solution by making a single, globally consistent, variable assignment. The training is done over intermediate steps of the construction of feasible solutions. From a scientific standpoint, we are interested in whether a DNN can learn the structure of a combinatorial problem, even when trained on (arbitrarily chosen) construction sequences of feasible solutions. In practice, the network could also be used to guide a search process, e.g. to take into account (soft) constraints that are implicit in past solutions or hard to capture in a traditional declarative model. This research line is still at an early stage, and a number of complex issues remain open. Nevertheless, we already have intriguing results on the classical Partial Latin Square and N-Queen completion problems.

Andrea Galassi, Michele Lombardi, Paola Mello, Michela Milano

### Boosting Efficiency for Computing the Pareto Frontier on Tree Structured Networks

Multi-objective optimization plays a key role in the study of real-world problems, as they often involve multiple criteria. In multi-objective optimization it is important to identify the so-called Pareto frontier, which characterizes the trade-offs between the objectives of different solutions. We show how a divide-and-conquer approach, combined with batched processing and pruning, significantly boosts the performance of an exact and approximation dynamic programming (DP) algorithm for computing the Pareto frontier on tree-structured networks, proposed in [18]. We also show how exploiting restarts and a new instance selection strategy boosts the performance and accuracy of a mixed integer programming (MIP) approach for approximating the Pareto frontier. We provide empirical results demonstrating that our DP and MIP approaches have complementary strengths and outperform previous algorithms in efficiency and accuracy. Our work is motivated by a problem in computational sustainability concerning the evaluation of trade-offs in ecosystem services due to the proliferation of hydropower dams throughout the Amazon basin. Our approaches are general and can be applied to computing the Pareto frontier of a variety of multi-objective problems on tree-structured networks.

Jonathan M. Gomes-Selman, Qinru Shi, Yexiang Xue, Roosevelt García-Villacorta, Alexander S. Flecker, Carla P. Gomes

### Bandits Help Simulated Annealing to Complete a Maximin Latin Hypercube Design

Simulated Annealing (SA) is commonly considered as an efficient method to construct Maximin Latin Hypercube Designs (LHDs) which are widely employed for Experimental Design. The Maximin LHD construction problem may be generalized to the Maximin LHD completion problem in an instance of which the measurements have already been taken at certain points.As the Maximin LHD completion is NP-complete, the choice of SA to treat it shows itself naturally. The SA performance varies greatly depending on the mutation used. The completion problem nature changes when the number of given points varies. We thus provide SA with a mechanism which selects an appropriate mutation. In our approach the choice of a mutation is seen as a bandit problem. It copes with changes in the environment, which evolves together with the thermal descent.The results obtained prove that the bandit-driven SA adapts itself on the fly to the completion problem nature. We believe that other parametrized problems, where SA can be employed, may also benefit from the use of a decision-making algorithm which selects the appropriate mutation.

Christian Hamelain, Kaourintin Le Guiban, Arpad Rimmel, Joanna Tomasik

### A Dynamic Discretization Discovery Algorithm for the Minimum Duration Time-Dependent Shortest Path Problem

We present an exact algorithm for the Minimum Duration Time-Dependent Shortest Path Problem with piecewise linear arc travel time functions. The algorithm iteratively refines a time-expanded network model, which allows for the computation of a lower and an upper bound, until - in a finite number of iterations - an optimal solution is obtained.

Edward He, Natashia Boland, George Nemhauser, Martin Savelsbergh

### Observations from Parallelising Three Maximum Common (Connected) Subgraph Algorithms

We discuss our experiences adapting three recent algorithms for maximum common (connected) subgraph problems to exploit multi-core parallelism. These algorithms do not easily lend themselves to parallel search, as the search trees are extremely irregular, making balanced work distribution hard, and runtimes are very sensitive to value-ordering heuristic behaviour. Nonetheless, our results show that each algorithm can be parallelised successfully, with the threaded algorithms we create being clearly better than the sequential ones. We then look in more detail at the results, and discuss how speedups should be measured for this kind of algorithm. Because of the difficulty in quantifying an average speedup when so-called anomalous speedups (superlinear and sublinear) are common, we propose a new measure called aggregate speedup.

Ruth Hoffmann, Ciaran McCreesh, Samba Ndojh Ndiaye, Patrick Prosser, Craig Reilly, Christine Solnon, James Trimble

### Horizontally Elastic Not-First/Not-Last Filtering Algorithm for Cumulative Resource Constraint

Fast and powerful propagators are the main key to the success of constraint programming on scheduling problems. It is, for example, the case with the cumulative constraint, which is used to model tasks sharing a resource of discrete capacity. In this paper, we propose a new not-first/not-last rule, which we call the horizontally elastic not-first/not-last, based on strong relaxation of the earliest completion time of a set of tasks. This computation is obtained when scheduling the tasks in a horizontally elastic way. We prove that the new rule is sound and is able to perform additional adjustments missed by the classic not-first/not-last rule. We use the new data structure called Profile to propose a $$\mathcal {O}(n^3)$$ filtering algorithm for a relaxed variant of the new rule where n is the number of tasks sharing the resource. We prove that the proposed algorithm still dominates the classic not-first/not-last algorithm. Experimental results on highly cumulative instances of resource constrained project scheduling problems (RCPSP) show that using this new algorithm can substantially improve the solving process of instances with an occasional and marginal increase of running time.

Roger Kameugne, Sévérine Betmbe Fetgo, Vincent Gingras, Yanick Ouellet, Claude-Guy Quimper

### Soft-Regular with a Prefix-Size Violation Measure

In this paper, we propose a variant of the global constraint soft-regular by introducing a new violation measure that relates a cost variable to the size of the longest prefix of the assigned variables, which is consistent with the constraint automaton. This measure allows us to guarantee that first decisions (assigned variables) respect the rules imposed by the automaton. We present a simple algorithm, based on a Multi-valued Decision Diagram (MDD), that enforces Generalized Arc Consistency (GAC). We provide an illustrative case study on nurse rostering, which shows the practical interest of our approach.

Minh Thanh Khong, Christophe Lecoutre, Pierre Schaus, Yves Deville

### Constraint and Mathematical Programming Models for Integrated Port Container Terminal Operations

This paper considers the integrated problem of quay crane assignment, quay crane scheduling, yard location assignment, and vehicle dispatching operations at a container terminal. The main objective is to minimize vessel turnover times and maximize the terminal throughput, which are key economic drivers in terminal operations. Due to their computational complexities, these problems are not optimized jointly in existing work. This paper revisits this limitation and proposes Mixed Integer Programming (MIP) and Constraint Programming (CP) models for the integrated problem, under some realistic assumptions. Experimental results show that the MIP formulation can only solve small instances, while the CP model finds optimal solutions in reasonable times for realistic instances derived from actual container terminal operations.

Damla Kizilay, Deniz Türsel Eliiyi, Pascal Van Hentenryck

### Heuristic Variants of A Search for 3D Flight Planning

A crucial component of a flight plan to be submitted for approval to a control authority in the pre-flight phase is the prescription of a sequence of airways and airway points in the sky that an aircraft has to follow to cover a given route. The generation of such a path in the 3D network that models the airways must respect a number of constraints. They generally state that if a set of points or airways is visited then another set of points or airways must be avoided or visited. Paths are then selected on the basis of cost considerations. The cost of traversing an airway depends, directly, on fuel consumption and on traversing time, and, indirectly, on weight and on weather conditions.Path finding algorithms based on A$$^*$$ search are commonly used in automatic planning. However, the constraints and the dependency structure of the costs invalidate the classic domination criterion in these algorithms. A common approach to tackle the increased computational effort is to decompose the problem heuristically into a sequence of horizontal and vertical route optimizations. Using techniques recently designed for the simplified 2D context, we address the 3D problem directly. We compare the direct approach with the decomposition approach. We enhance both approaches with ad hoc heuristics that exploit the expected appeal of routes to speed-up the solution process. We show that, on data resembling those arising in the context of European airspaces, the direct approach is computationally practical and leads to results of better quality than the decomposition approach.

Anders N. Knudsen, Marco Chiarandini, Kim S. Larsen

### Juniper: An Open-Source Nonlinear Branch-and-Bound Solver in Julia

Nonconvex mixed-integer nonlinear programs (MINLPs) represent a challenging class of optimization problems that often arise in engineering and scientific applications. Because of nonconvexities, these programs are typically solved with global optimization algorithms, which have limited scalability. However, nonlinear branch-and-bound has recently been shown to be an effective heuristic for quickly finding high-quality solutions to large-scale nonconvex MINLPs, such as those arising in infrastructure network optimization. This work proposes Juniper, a Julia-based open-source solver for nonlinear branch-and-bound. Leveraging the high-level Julia programming language makes it easy to modify Juniper’s algorithm and explore extensions, such as branching heuristics, feasibility pumps, and parallelization. Detailed numerical experiments demonstrate that the initial release of Juniper is comparable with other nonlinear branch-and-bound solvers, such as Bonmin, Minotaur, and Knitro, illustrating that Juniper provides a strong foundation for further exploration in utilizing nonlinear branch-and-bound algorithms as heuristics for nonconvex MINLPs.

Ole Kröger, Carleton Coffrin, Hassan Hijazi, Harsha Nagarajan

### Objective Landscapes for Constraint Programming

This paper presents the concept of objective landscape in the context of Constraint Programming. An objective landscape is a light-weight structure providing some information on the relation between decision variables and objective values, that can be quickly computed once and for all at the beginning of the resolution and is used to guide the search. It is particularly useful on decision variables with large domains and with a continuous semantics, which is typically the case for time or resource quantity variables in scheduling problems. This concept was recently implemented in the automatic search of CP Optimizer and resulted in an average speed-up of about 50% on scheduling problems with up to almost 2 orders of magnitude for some applications.

Philippe Laborie

### An Update on the Comparison of MIP, CP and Hybrid Approaches for Mixed Resource Allocation and Scheduling

We consider a well known resource allocation and scheduling problem for which different approaches like mixed-integer programming (MIP), constraint programming (CP), constraint integer programming (CIP), logic-based Benders decompositions (LBBD) and SAT-modulo theories (SMT) have been proposed and experimentally compared in the last decade. Thanks to the recent improvements in CP Optimizer, a commercial CP solver for solving generic scheduling problems, we show that a standalone tiny CP model can out-perform all previous approaches and close all the 335 instances of the benchmark. The article explains which components of the automatic search of CP Optimizer are responsible for this success. We finally propose an extension of the original benchmark with larger and more challenging instances.

Philippe Laborie

### Modelling and Solving the Senior Transportation Problem

This paper defines a novel transportation problem, the Senior Transportation Problem (STP), which is inspired by the elderly door-to-door transportation services provided by non-profit organizations. Building on the vehicle routing literature, we develop solution approaches including mixed integer programming (MIP), constraint programming (CP), two logic-based Benders decompositions (LBBD), and a construction heuristic. Empirical analyses on both randomly generated datasets and large real-life datasets are performed. CP achieved the best results, solving to optimality 89% of our real-life instances of up to 270 vehicles with 385 requests in under 600 s. The best LBBD model can only solve 17% of those instances to optimality. Further investigation of this somewhat surprising result indicates that, compared to the LBBD approaches, the pure CP model is able to find better solutions faster and then is able to use the bounds from these sub-optimal solutions to reduce the search space slightly more effectively than the decomposition models.

Chang Liu, Dionne M. Aleman, J. Christopher Beck

### Solver Independent Rotating Workforce Scheduling

The rotating workforce scheduling problem aims to schedule workers satisfying shift sequence constraints and ensuring enough shifts are covered on each day, where every worker completes the same schedule, just starting at different days in the schedule. We give two solver independent models for the rotating workforce scheduling problem and compare them using different solving technology, both constraint programming and mixed integer programming. We show that the best of these models outperforms the state-of-the-art for the rotating workforce scheduling problem, and that solver independent modeling allows us to use different solvers to achieve different aims: e.g., speed to solution or robustness of solving (particular for unsatisfiable problems). We give the first complete method able to solve all of the standard benchmarks for this problem.

Nysret Musliu, Andreas Schutt, Peter J. Stuckey

### Greedy Randomized Search for Scalable Compilation of Quantum Circuits

This paper investigates the performances of a greedy randomized algorithm to optimize the realization of nearest-neighbor compliant quantum circuits. Current technological limitations (decoherence effect) impose that the overall duration (makespan) of the quantum circuit realization be minimized. One core contribution of this paper is a lexicographic two-key ranking function for quantum gate selection: the first key acts as a global closure metric to minimize the solution makespan; the second one is a local metric acting as “tie-breaker” for avoiding cycling. Our algorithm has been tested on a set of quantum circuit benchmark instances of increasing sizes available from the recent literature. We demonstrate that our heuristic approach outperforms the solutions obtained in previous research against the same benchmark, both from the CPU efficiency and from the solution quality standpoint.

Angelo Oddi, Riccardo Rasconi

### A Comparison of Optimization Methods for Multi-objective Constrained Bin Packing Problems

Despite the existence of efficient solution methods for bin packing problems, in practice these seldom occur in such a pure form but feature instead various considerations such as pairwise conflicts or profits between items, or aiming for balanced loads amongst the bins. The Wedding Seating Problem is a combinatorial optimization problem incorporating elements of bin packing with conflicts, bin packing with profits, and load balancing. We use this representative problem to present and compare constraint programming, integer programming, and metaheuristic approaches.

Philippe Olivier, Andrea Lodi, Gilles Pesant

### A Checker and Filtering Algorithm for the Energetic Reasoning

Energetic reasoning is a strong filtering technique for the Cumulative constraint. However, the best algorithms process $$O(n^2)$$ time intervals to perform the satisfiability check which makes it too costly to use in practice. We present how to apply the energetic reasoning by processing only $$O(n \log n)$$ intervals. We show how to compute the energy in an interval in $$O(\log n)$$ time. This allows us to propose a $$O(n \log ^2 n)$$ checker and a filtering algorithm for the energetic reasoning with $$O(n^2 \log n)$$ average time complexity. Experiments show that these two algorithms outperform their state of the art counterparts.

Yanick Ouellet, Claude-Guy Quimper

### The WeightedCircuitsLmax Constraint

The travelling salesman problem is a well-known problem that can be generalized to the m-travelling salesmen problem with min-max objective. In this problem, each city must be visited by exactly one salesman, among m travelling salesmen. We want to minimize the longest circuit travelled by a salesman. This paper generalizes the Circuit and WeightedCircuit constraints and presents a new constraint that encodes m cycles all starting from the same city and whose lengths are bounded by a variable $$L_{max}$$. We propose two filtering algorithms, each based on a relaxation of the problem that uses the structure of the graph and the distances between each city. We show that this new constraint improves the solving time for the m travelling salesmen problem.

### A Local Search Framework for Compiling Relaxed Decision Diagrams

This paper presents a local search framework for constructing and improving relaxed decision diagrams (DDs). The framework consists of a set of elementary DD manipulation operations including a redirect operation introduced in this paper and a general algorithmic scheme. We show that the framework can be used to reproduce several standard DD compilation schemes and to create new compilation and improvement strategies. In computational experiments for the 0–1 knapsack problem, the multidimensional knapsack problem and the set covering problem we compare different compilation methods. It turns out that a new strategy based on the local search framework consistently yields better bounds, in many cases far better bounds, for limited-width DDs than previously published heuristic strategies.

Michael Römer, Andre A. Cire, Louis-Martin Rousseau

### Symmetry Breaking Inequalities from the Schreier-Sims Table

We propose a way to derive symmetry breaking inequalities for a mixed-integer programming (MIP) model from the Schreier-Sims table of its formulation group. We then show how to consider only the action of the formulation group onto a subset of the variables. Computational results show that this can lead to considerable speedups on some classes of models.

Domenico Salvagnin

### Frequency-Based Multi-agent Patrolling Model and Its Area Partitioning Solution Method for Balanced Workload

Multi-agent patrolling problem has received growing attention from many researchers due to its wide range of potential applications. In realistic environment, e.g., security patrolling, each location has different visitation requirement according to the required security level. Therefore, a patrolling system with non-uniform visiting frequency is preferable. The difference in visiting frequency generally causes imbalanced workload amongst agents leading to inefficiency. This paper, thus, aims at partitioning a given area to balance agents’ workload by considering that different visiting frequency and then generating route inside each sub-area. We formulate the problem of frequency-based multi-agent patrolling and propose its semi-optimal solution method, whose overall process consists of two steps – graph partitioning and sub-graph patrolling. Our work improve traditional k-means clustering algorithm by formulating a new objective function and combine it with simulated annealing – a useful tool for operations research. Experimental results illustrated the effectiveness and reasonable computational efficiency of our approach.

Vourchteang Sea, Ayumi Sugiyama, Toshiharu Sugawara

### Algorithms for Sparse k-Monotone Regression

The problem of constructing k-monotone regression is to find a vector $$z\in \mathbb {R}^n$$ with the lowest square error of approximation to a given vector $$y\in \mathbb {R}^n$$ (not necessary k-monotone) under condition of k-monotonicity of z. The problem can be rewritten in the form of a convex programming problem with linear constraints. The paper proposes two different approaches for finding a sparse k-monotone regression (Frank-Wolfe-type algorithm and k-monotone pool adjacent violators algorithm). A software package for this problem is developed and implemented in R. The proposed algorithms are compared using simulated data.

Sergei P. Sidorov, Alexey R. Faizliev, Alexander A. Gudkov, Sergei V. Mironov

### Revisiting the Self-adaptive Large Neighborhood Search

This paper revisits the Self-Adaptive Large Neighborhood Search introduced by Laborie and Godard. We propose a variation in the weight-update mechanism especially useful when the LNS operators available in the portfolio exhibit unequal running times. We also propose some generic relaxations working for a large family of problems in a black-box fashion. We evaluate our method on various problem types demonstrating that our approach converges faster toward a selection of efficient operators.

Charles Thomas, Pierre Schaus

### A Warning Propagation-Based Linear-Time-and-Space Algorithm for the Minimum Vertex Cover Problem on Giant Graphs

A vertex cover (VC) of a graph $$G$$ is a subset of vertices in $$G$$ such that at least one endpoint vertex of each edge in $$G$$ is in this subset. The minimum VC (MVC) problem is to identify a VC of minimum size (cardinality) and is known to be NP-hard. Although many local search algorithms have been developed to solve the MVC problem close-to-optimally, their applicability on giant graphs (with no less than 100,000 vertices) is limited. For such graphs, there are two reasons why it would be beneficial to have linear-time-and-space algorithms that produce small VCs. Such algorithms can: (a) serve as preprocessing steps to produce good starting states for local search algorithms and (b) also be useful for many applications that require finding small VCs quickly. In this paper, we develop a new linear-time-and-space algorithm, called MVC-WP, for solving the MVC problem on giant graphs based on the idea of warning propagation, which has so far only been used as a theoretical tool for studying properties of MVCs on infinite random graphs. We empirically show that it outperforms other known linear-time-and-space algorithms in terms of sizes of produced VCs.

Hong Xu, Kexuan Sun, Sven Koenig, T. K. Satish Kumar

### Symbolic Bucket Elimination for Piecewise Continuous Constrained Optimization

Bucket elimination and its approximation extensions have proved to be effective techniques for discrete optimization. This paper addresses the extension of bucket elimination to continuous constrained optimization by leveraging the recent innovation of the extended algebraic decision diagram (XADD). XADDs support symbolic arithmetic and optimization operations on piecewise linear or univariate quadratic functions that permit the solution of continuous constrained optimization problems with a symbolic form of bucket elimination. The proposed framework is an efficient alternative for solving optimization problems with low tree-width constraint graphs without using a big-M formulation for piecewise, indicator, or conditional constraints. We apply this framework to difficult constrained optimization problems including XOR’s of linear constraints and temporal constraint satisfaction problems with “repulsive” preferences, and show that this new approach significantly outperforms Gurobi. Our framework also enables symbolic parametric optimization where closed-form solutions cannot be computed with tools like Gurobi, where we demonstrate a final novel application to parametric optimization of learned Relu-based deep neural networks.

Zhijiang Ye, Buser Say, Scott Sanner

### Learning a Classification of Mixed-Integer Quadratic Programming Problems

Within state-of-the-art solvers such as IBM-CPLEX, the ability to solve both convex and nonconvex Mixed-Integer Quadratic Programming (MIQP) problems to proven optimality goes back few years, yet presents unclear aspects. We are interested in understanding whether for solving an MIQP it is favorable to linearize its quadratic part or not. Our approach exploits machine learning techniques to learn a classifier that predicts, for a given instance, the most suitable resolution method within CPLEX’s framework. We aim as well at gaining first methodological insights about the instances’ features leading this discrimination. We examine a new dataset and discuss different scenarios to integrate learning and optimization. By defining novel measures, we interpret and evaluate learning results from the optimization point of view.

Pierre Bonami, Andrea Lodi, Giulia Zarpellon

### Fleet Scheduling in Underground Mines Using Constraint Programming

The profitability of an underground mine is greatly affected by the scheduling of the mobile production fleet. Today, most mine operations are scheduled manually, which is a tedious and error-prone activity. In this contribution, we present and formalize the underground mine scheduling problem, and propose a CP-based model for solving it. The model is evaluated on instances generated from real data. The results are promising and show a potential for further extensions.

Max Åstrand, Mikael Johansson, Alessandro Zanarini

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