9 Search Results for "Régin, Jean-Charles"


Document
The Distance Constraint on Sequence Variables

Authors: Margaux Schmied, Augustin Delecluse, Jean-Charles Régin, and Pierre Schaus

Published in: LIPIcs, Volume 379, 32nd International Conference on Principles and Practice of Constraint Programming (CP 2026)


Abstract
Insertion sequence variables have recently been introduced as a computational domain for modeling routing and sequencing problems in constraint programming. Typically, search heuristics guide the insertion process of new nodes into a partial growing path, while constraints eliminate infeasible insertions. This paper investigates filtering for the (minimum) distance constraint over insertion sequence variables. This global constraint links a sequence to a distance variable based on a given distance matrix. So far, only a simple filtering algorithm has been proposed, which considers the partial path but ignores mandatory nodes. Our contribution is to introduce stronger lower bounds that also take mandatory nodes into account. These bounds further enable the derivation of additional filtering rules for node insertions. An experimental evaluation on the TourMustSee problem shows that the proposed filtering rules significantly reduce the search space compared to the existing filtering approach.

Cite as

Margaux Schmied, Augustin Delecluse, Jean-Charles Régin, and Pierre Schaus. The Distance Constraint on Sequence Variables. In 32nd International Conference on Principles and Practice of Constraint Programming (CP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 379, pp. 49:1-49:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)


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@InProceedings{schmied_et_al:LIPIcs.CP.2026.49,
  author =	{Schmied, Margaux and Delecluse, Augustin and R\'{e}gin, Jean-Charles and Schaus, Pierre},
  title =	{{The Distance Constraint on Sequence Variables}},
  booktitle =	{32nd International Conference on Principles and Practice of Constraint Programming (CP 2026)},
  pages =	{49:1--49:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-432-1},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{379},
  editor =	{Beldiceanu, Nicolas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2026.49},
  URN =		{urn:nbn:de:0030-drops-266828},
  doi =		{10.4230/LIPIcs.CP.2026.49},
  annote =	{Keywords: Constraint programming, sequence variable, global constraint, cost-based constraint, distance constraint, lower bound}
}
Document
Short Paper
Modeling and Solving a Composite Structure Design Problem with Constraint Programming (Short Paper)

Authors: Miguel Antoons, Augustin Delecluse, Samih Zein, and Pierre Schaus

Published in: LIPIcs, Volume 340, 31st International Conference on Principles and Practice of Constraint Programming (CP 2025)


Abstract
Composite structures are composed of plies (layers) of carbon fibers. For each ply, one must decide its orientation from the set of possible angles: -45°, 0°, 45°, and 90°. The stack of plies must follow strict constraints on the chosen orientations to achieve mechanical properties of the composite, such as sufficient buckling load. The design problem becomes more complex when determining the stack of plies for a complete surface material, that does not require the same number of plies in every region of the surface. Not only must the orientations be selected in each region, but it is also necessary to decide which plies are discontinued between adjacent regions. Thanks to its declarative nature, Constraint Programming (CP) offers an elegant modeling of the constraints, making it easy for designers to activate or deactivate them as needed. We propose a CP model, implemented in MiniZinc. The performance of this model on synthetic yet realistic instances when solved by different exact solvers, including Mixed Integer Programming (MIP) solvers, demonstrates the superiority of CP over MIP on our MiniZinc model, and over a commercial solution implemented by an industrial partner. It opens up the adoption of CP as an efficient building block of Computer-Aided Design tools for composite structures. By making the model and instances publicly available, we also hope to facilitate the inclusion of this problem in CP solver competitions and stimulate further research in this area.

Cite as

Miguel Antoons, Augustin Delecluse, Samih Zein, and Pierre Schaus. Modeling and Solving a Composite Structure Design Problem with Constraint Programming (Short Paper). In 31st International Conference on Principles and Practice of Constraint Programming (CP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 340, pp. 41:1-41:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)


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@InProceedings{antoons_et_al:LIPIcs.CP.2025.41,
  author =	{Antoons, Miguel and Delecluse, Augustin and Zein, Samih and Schaus, Pierre},
  title =	{{Modeling and Solving a Composite Structure Design Problem with Constraint Programming}},
  booktitle =	{31st International Conference on Principles and Practice of Constraint Programming (CP 2025)},
  pages =	{41:1--41:9},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-380-5},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{340},
  editor =	{de la Banda, Maria Garcia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2025.41},
  URN =		{urn:nbn:de:0030-drops-239022},
  doi =		{10.4230/LIPIcs.CP.2025.41},
  annote =	{Keywords: Constraint Programming, Composite Structures, Design Rules, MiniZinc}
}
Document
Understanding the Impact of Value Selection Heuristics in Scheduling Problems

Authors: Tim Luchterhand, Emmanuel Hebrard, and Sylvie Thiébaux

Published in: LIPIcs, Volume 340, 31st International Conference on Principles and Practice of Constraint Programming (CP 2025)


Abstract
It has been observed that value selection heuristics have less impact than other heuristic choices when solving hard combinatorial optimization (CO) problems. It is often thought that this is because more time is spent on unsatisfiable sub-problems where the value ordering is irrelevant. In this paper we investigate this belief in the scheduling domain and come up with a more detailed explanation. We find that, even though there are less relevant choices to be made on hard instances, each mistake tends to have a bigger impact, to a point where the potential gain from a value heuristic predominates. Moreover, we observe two interesting and relatively surprising phenomena when solving scheduling problems. First, the accuracy of a given value selection heuristic decreases with the optimality gap. Second, the computational penalty of a mistake increases with the accuracy of the heuristic. For the first observation, we argue that on hard problems, constraint propagation removes a large portion of choices that align with the intuition behind the heuristic. This means that the heuristic faces mostly difficult choices. For the second observation, we argue that simple heuristics tend to make more mistakes on intuitive choice points, and the computational cost for refuting these mistakes is smaller than for those made by a more accurate heuristic.

Cite as

Tim Luchterhand, Emmanuel Hebrard, and Sylvie Thiébaux. Understanding the Impact of Value Selection Heuristics in Scheduling Problems. In 31st International Conference on Principles and Practice of Constraint Programming (CP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 340, pp. 27:1-27:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)


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@InProceedings{luchterhand_et_al:LIPIcs.CP.2025.27,
  author =	{Luchterhand, Tim and Hebrard, Emmanuel and Thi\'{e}baux, Sylvie},
  title =	{{Understanding the Impact of Value Selection Heuristics in Scheduling Problems}},
  booktitle =	{31st International Conference on Principles and Practice of Constraint Programming (CP 2025)},
  pages =	{27:1--27:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-380-5},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{340},
  editor =	{de la Banda, Maria Garcia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2025.27},
  URN =		{urn:nbn:de:0030-drops-238885},
  doi =		{10.4230/LIPIcs.CP.2025.27},
  annote =	{Keywords: Scheduling, Branching Heuristics, Constraint Programming}
}
Document
Parallel MIP Solving with Dynamic Task Decomposition

Authors: Peng Lin, Shaowei Cai, Mengchuan Zou, and Shengqi Chen

Published in: LIPIcs, Volume 340, 31st International Conference on Principles and Practice of Constraint Programming (CP 2025)


Abstract
Mixed Integer Programming (MIP) is a foundational model in operations research. Although significant progress has been made in enhancing sequential MIP solvers through sophisticated techniques and heuristics, remarkable developments in computing resources have made parallel solving a promising direction for performance improvement. In this work, we propose a novel parallel MIP solving framework that employs dynamic task decomposition in a divide-and-conquer paradigm. Our framework incorporates a hardness estimate heuristic to identify challenging solving tasks and a reward decaying mechanism to reinforce the task decomposition decision. We apply our framework to two state-of-the-art open-source MIP solvers, SCIP and HiGHS, yielding efficient parallel solvers. Extensive experiments on the full MIPLIB benchmark, using up to 128 cores, demonstrate that our framework yields substantial performance improvements over modern divide-and-conquer parallel solvers. Moreover, our parallel solvers have established new best known solutions for 16 open MIPLIB instances.

Cite as

Peng Lin, Shaowei Cai, Mengchuan Zou, and Shengqi Chen. Parallel MIP Solving with Dynamic Task Decomposition. In 31st International Conference on Principles and Practice of Constraint Programming (CP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 340, pp. 26:1-26:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)


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@InProceedings{lin_et_al:LIPIcs.CP.2025.26,
  author =	{Lin, Peng and Cai, Shaowei and Zou, Mengchuan and Chen, Shengqi},
  title =	{{Parallel MIP Solving with Dynamic Task Decomposition}},
  booktitle =	{31st International Conference on Principles and Practice of Constraint Programming (CP 2025)},
  pages =	{26:1--26:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-380-5},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{340},
  editor =	{de la Banda, Maria Garcia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2025.26},
  URN =		{urn:nbn:de:0030-drops-238871},
  doi =		{10.4230/LIPIcs.CP.2025.26},
  annote =	{Keywords: Mixed Integer Programming, Parallel Computing, Complete Search, Task Decomposition}
}
Document
Unite and Lead: Finding Disjunctive Cliques for Scheduling Problems

Authors: Konstantin Sidorov, Imko Marijnissen, and Emir Demirović

Published in: LIPIcs, Volume 340, 31st International Conference on Principles and Practice of Constraint Programming (CP 2025)


Abstract
Constraint programming solvers have seen much success in scheduling problems owing to their efficient reasoning over constraints to solve complex problems in practice. Many algorithms have been proposed for propagating information from a single constraint. However, inferring and exchanging information across multiple constraints can provide deeper insight into the global structure of a problem. In this work, we propose to exchange information amongst constraints by inferring the disjointness of tasks in scheduling problems from many constraints. We do this by (i) augmenting existing propagators, such as the Cumulative and nogoods, to report when pairs of tasks are disjoint, and (ii) leveraging this information by introducing the SelectiveDisjunctive propagator which generates a lower bound on the earliest completion time of cliques of disjoint tasks to determine conflicts. This allows us to aggregate disjointness information spanning multiple constraints to gain a better global overview of the problem, as well as more precise local information. We also identify a problem structure where an LCG solver reasoning over Cumulative constraints separately, without any reformulations, requires an exponential amount of time to prove infeasibility, which we both justify theoretically and show empirically; on the other hand, our approach solves those instances in polynomial time. On particular known RCPSP and RCPSP/max benchmarks, our approach significantly reduces the number of conflicts required to prove optimality when resource contention is high. Additionally, we discover new lower bounds for 16 RCPSP/max instances (closing six of them) and four RCPSP instances (closing one), as well as new upper bounds for two RCPSP/max instances and four RCPSP instances. Furthermore, we empirically analyse our proposed approach to determine which features are beneficial for performance, showing that finding cliques is one of the main bottlenecks and that detecting disjointness during search can lead to improved bounds on certain instances, but it generally negatively impacts learning. This work paves the way for reasoning over the disjointness of tasks inferred from a variety of standard constraints to discover novel information sourced from multiple constraints during search.

Cite as

Konstantin Sidorov, Imko Marijnissen, and Emir Demirović. Unite and Lead: Finding Disjunctive Cliques for Scheduling Problems. In 31st International Conference on Principles and Practice of Constraint Programming (CP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 340, pp. 35:1-35:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)


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@InProceedings{sidorov_et_al:LIPIcs.CP.2025.35,
  author =	{Sidorov, Konstantin and Marijnissen, Imko and Demirovi\'{c}, Emir},
  title =	{{Unite and Lead: Finding Disjunctive Cliques for Scheduling Problems}},
  booktitle =	{31st International Conference on Principles and Practice of Constraint Programming (CP 2025)},
  pages =	{35:1--35:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-380-5},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{340},
  editor =	{de la Banda, Maria Garcia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2025.35},
  URN =		{urn:nbn:de:0030-drops-238969},
  doi =		{10.4230/LIPIcs.CP.2025.35},
  annote =	{Keywords: Constraint Programming, Lazy Clause Generation, Propagation, Scheduling, Cumulative, Disjunctive}
}
Document
Efficient Implementation of the Global Cardinality Constraint with Costs

Authors: Margaux Schmied and Jean-Charles Régin

Published in: LIPIcs, Volume 307, 30th International Conference on Principles and Practice of Constraint Programming (CP 2024)


Abstract
The success of Constraint Programming relies partly on the global constraints and implementation of the associated filtering algorithms. Recently, new ideas emerged to improve these implementations in practice, especially regarding the all different constraint. In this paper, we consider the cardinality constraint with costs. The cardinality constraint is a generalization of the all different constraint that specifies the number of times each value must be taken by a given set of variables in a solution. The version with costs introduces an assignment cost and bounds the total sum of assignment costs. The arc consistency filtering algorithm of this constraint is difficult to use in practice, as it systematically searches for many shortest paths. We propose a new approach that works with upper bounds on shortest paths based on landmarks. This approach can be seen as a preprocessing. It is fast and avoids, in practice, a large number of explicit computations of shortest paths.

Cite as

Margaux Schmied and Jean-Charles Régin. Efficient Implementation of the Global Cardinality Constraint with Costs. In 30th International Conference on Principles and Practice of Constraint Programming (CP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 307, pp. 27:1-27:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{schmied_et_al:LIPIcs.CP.2024.27,
  author =	{Schmied, Margaux and R\'{e}gin, Jean-Charles},
  title =	{{Efficient Implementation of the Global Cardinality Constraint with Costs}},
  booktitle =	{30th International Conference on Principles and Practice of Constraint Programming (CP 2024)},
  pages =	{27:1--27:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-336-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{307},
  editor =	{Shaw, Paul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2024.27},
  URN =		{urn:nbn:de:0030-drops-207126},
  doi =		{10.4230/LIPIcs.CP.2024.27},
  annote =	{Keywords: global constraint, filtering algorithm, cardinality constraints with costs, arc consistency}
}
Document
MDD Archive for Boosting the Pareto Constraint

Authors: Steve Malalel, Arnaud Malapert, Marie Pelleau, and Jean-Charles Régin

Published in: LIPIcs, Volume 280, 29th International Conference on Principles and Practice of Constraint Programming (CP 2023)


Abstract
Multi-objective problems are frequent in the real world. In general they involve several incomparable objectives and the goal is to find a set of Pareto optimal solutions, i.e. solutions that are incomparable two by two. In order to better deal with these problems in CP the global constraint Pareto was developed by Schaus and Hartert to handle the relations between the objective variables and the current set of Pareto optimal solutions, called the archive. This constraint handles three operations: adding a new solution to the archive, removing solutions from the archive that are dominated by a new solution, and reducing the bounds of the objective variables. The complexity of these operations depends on the size of the archive. In this paper, we propose to use a multi-valued Decision Diagram (MDD) to represent the archive of Pareto optimal solutions. MDDs are a compressed representation of solution sets, which allows us to obtain a compressed and therefore smaller archive. We introduce several algorithms to implement the above operations on compressed archives with a complexity depending on the size of the archive. We show experimentally on bin packing and multi-knapsack problems the validity of our approach.

Cite as

Steve Malalel, Arnaud Malapert, Marie Pelleau, and Jean-Charles Régin. MDD Archive for Boosting the Pareto Constraint. In 29th International Conference on Principles and Practice of Constraint Programming (CP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 280, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{malalel_et_al:LIPIcs.CP.2023.24,
  author =	{Malalel, Steve and Malapert, Arnaud and Pelleau, Marie and R\'{e}gin, Jean-Charles},
  title =	{{MDD Archive for Boosting the Pareto Constraint}},
  booktitle =	{29th International Conference on Principles and Practice of Constraint Programming (CP 2023)},
  pages =	{24:1--24:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-300-3},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{280},
  editor =	{Yap, Roland H. C.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2023.24},
  URN =		{urn:nbn:de:0030-drops-190610},
  doi =		{10.4230/LIPIcs.CP.2023.24},
  annote =	{Keywords: Constraint Programming, Global Constraint, MDD, Multi-Objective Problem, Pareto Constraint}
}
Document
A Linear Time Algorithm for the k-Cutset Constraint

Authors: Nicolas Isoart and Jean-Charles Régin

Published in: LIPIcs, Volume 210, 27th International Conference on Principles and Practice of Constraint Programming (CP 2021)


Abstract
In CP, the most efficient model solving the TSP is the Weighted Circuit Constraint (WCC) combined with the k-cutset constraint. The WCC is mainly based on the edges cost of a given graph whereas the k-cutset constraint is a structural constraint. Specifically, for each cutset in a graph, the k-cutset constraint imposes that the size of the cutset is greater than or equal to two. In addition, any solution contains an even number of elements from this cutset. Isoart and Régin introduced an algorithm for this constraint. Unfortunately, their approach leads to a time complexity growing with the size of the considered cutsets, i.e. with k. Thus, they introduced an algorithm with a quadratic complexity dealing with k lower or equal to three. In this paper, we introduce a linear time algorithm for any k based on a DFS checking the consistency of this constraint and performing its filtering. Experimental results show that the size of most of the k-cutsets is lower or equal than 3. In addition, since the time complexity is improved, our algorithm also improves the solving times.

Cite as

Nicolas Isoart and Jean-Charles Régin. A Linear Time Algorithm for the k-Cutset Constraint. In 27th International Conference on Principles and Practice of Constraint Programming (CP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 210, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{isoart_et_al:LIPIcs.CP.2021.29,
  author =	{Isoart, Nicolas and R\'{e}gin, Jean-Charles},
  title =	{{A Linear Time Algorithm for the k-Cutset Constraint}},
  booktitle =	{27th International Conference on Principles and Practice of Constraint Programming (CP 2021)},
  pages =	{29:1--29:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-211-2},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{210},
  editor =	{Michel, Laurent D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2021.29},
  URN =		{urn:nbn:de:0030-drops-153200},
  doi =		{10.4230/LIPIcs.CP.2021.29},
  annote =	{Keywords: k-Cutset, TSP, Linear algorithm, Constraint}
}
Document
A k-Opt Based Constraint for the TSP

Authors: Nicolas Isoart and Jean-Charles Régin

Published in: LIPIcs, Volume 210, 27th International Conference on Principles and Practice of Constraint Programming (CP 2021)


Abstract
The LKH algorithm based on k-opt is an extremely efficient algorithm solving the TSP. Given a non-optimal tour in a graph, the idea of k-opt is to iteratively swap k edges of this tour in order to find a shorter tour. However, the optimality of a tour cannot be proved with this method. In that case, exact solving methods such as CP can be used. The CP model is based on a graph variable with mandatory and optional edges. Through branch-and-bound and filtering algorithms, the set of mandatory edges will be modified. In this paper, we introduce a new constraint to the CP model named mandatory Hamiltonian path constraint searching for k-opt in the mandatory Hamiltonian paths. Experiments have shown that the mandatory Hamiltonian path constraint allows us to gain on average a factor of 3 on the solving time. In addition, we have been able to solve some instances that remain unsolved with the state of the art CP solver with a 1 week time out.

Cite as

Nicolas Isoart and Jean-Charles Régin. A k-Opt Based Constraint for the TSP. In 27th International Conference on Principles and Practice of Constraint Programming (CP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 210, pp. 30:1-30:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{isoart_et_al:LIPIcs.CP.2021.30,
  author =	{Isoart, Nicolas and R\'{e}gin, Jean-Charles},
  title =	{{A k-Opt Based Constraint for the TSP}},
  booktitle =	{27th International Conference on Principles and Practice of Constraint Programming (CP 2021)},
  pages =	{30:1--30:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-211-2},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{210},
  editor =	{Michel, Laurent D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2021.30},
  URN =		{urn:nbn:de:0030-drops-153212},
  doi =		{10.4230/LIPIcs.CP.2021.30},
  annote =	{Keywords: TSP, k-opt, 1-tree, Constraint}
}
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