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Recent advances in vehicle routing exact algorithms

4OR-A QUARTERLY JOURNAL OF OPERATIONS RESEARCH 2007年第4期5卷 269-298页

作者： Baldacci, Roberto Toth, Paolo Vigo, Daniele Univ Bologna DEIS I-47023 Cesena Italy Univ Bologna DEIS I-40136 Bologna Italy

The capacitated vehicle routing problem (CVRP) is the problem in which a set of identical vehicles located at a central depot is to be optimally routed to supply customers with known demands subject to vehicle capacity constraints. This paper provides a review of the most recent developments that had a major impact in the current state-of-the-art of exact algorithms for the CVRP. The most important mathematical formulations for the problem together with various CVRP relaxations are reviewed. The paper also describes the recent exact methods for the CVRP and reports a comparison of their computational performances.

关键词： vehicle routing exact algorithms survey

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Improved exact algorithms for M_AX-S_AT 5

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5th Latin American Symposium on Theoretical Informatics (LATIN 2002)

作者： Chen, JE Kanj, IA Depaul Univ Sch CTI Chicago IL 60604 USA Texas A&M Univ Dept Comp Sci College Stn TX 77843 USA

ISBN: (纸本)3540434003

In this paper, we present improved exact and parameterized algorithms for the maximum satisfiability problem. In particular, we give an algorithm that computes a truth assignment for a boolean formula F satisfying the maximum number of clauses in time O(1.3247(m)\F\), where m is the number of clauses in F, and \F\ is the sum of the number of literals appearing in each clause in F. Moreover, given a parameter k, we give an O(1.3695(k) + \F\) parameterized algorithm that decides whether a truth assignment for F satisfying at least k clauses exists. Both algorithms improve the previous best algorithms by Bansal and Raman for the problem. (C) 2004 Elsevier B.V. All rights reserved.

关键词： maximum satisfiability exact algorithms parameterized algorithms

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exact algorithms for a joint order acceptance and scheduling problem

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INTERNATIONAL JOURNAL OF PRODUCTION ECONOMICS 2020年 223卷 107516-107516页

作者： Li, Xin Ventura, Jose A. Penn State Univ Harold & Inge Marcus Dept Ind & Mfg Engn University Pk PA 16802 USA

This paper considers the scenario where a manufacturer receives multiple orders that are characterized by the revenue, processing time, due date, and tardiness penalty per time unit. The manufacturer can be seen as a single-machine system that adopts a make-to-order strategy. Due to the capacity limitation and tardiness penalty, the manufacturer cannot accept all orders. It needs to determine the optimal set of accepted orders and the corresponding production schedule that maximizes the total revenue minus tardiness penalty. Three different mathematical formulations are presented to model the problem: sequential relation-based formulation (SBF), assignment formulation (ASF), and time-index formulation (TIF). The first two models use both continuous and binary variables while (TIF) only uses binary variables but requires order processing times to be integer. Three exact algorithms are proposed to solve the problem. The first algorithm, denoted by DPA, follows a pure dynamic programming (DP) approach. The second algorithm, denoted by DPIA-SR, solves the problem in multiple stages. In the first stage, it solves a relaxed version of (TIF) using DP;then, in the following stages, the relaxed constraint is gradually recovered, and the resulting models are solved using DP until an optimal solution to the original model is found. The results of one stage are used to eliminate unnecessary states in the DP algorithm for the next stage. The third algorithm, denoted by DPIA-LRSR, improves DPIA-SR by incorporating Lagrangian relaxation to achieve higher computational efficiency. The numerical experiment shows that, when using CPLEX to solve the models, (ASF) and (TIF) take much less CPU time than (SBF);the last two algorithms and their variations significantly outperform CPLEX regarding computational time;and DPIA-LRSR shows the highest efficiency.

关键词： Order acceptance Single machine scheduling Tardiness penalty exact algorithms Dynamic programming Lagrangian relaxation

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exact algorithms for solving the constrained parallel-machine scheduling problems with divisible processing times and penalties

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JOURNAL OF COMBINATORIAL OPTIMIZATION 2023年第4期45卷 98-98页

作者： Li, Jianping Xie, Runtao Lichen, Junran Hu, Guojun Pan, Pengxiang Yang, Ping Yunnan Univ Univ Town Sch Math & Stat East Outer Ring South Rd Kunming 650504 Peoples R China Beijing Univ Chem Technol Sch Math & Phys 15 North Third Ring East Rd Beijing 100029 Peoples R China

In this paper, we address the constrained parallel-machine scheduling problem with divisible processing times and penalties (the CPS-DTP problem), which is a further generalization of the parallel-machine scheduling problem with divisible processing times (the PS-DT problem). Concretely, given a set M of m identical machines and a set J of n independent jobs, each job has a processing time and a penalty, the processing times of these n jobs are divisible, and we implement these n jobs under the requirement that each job in J must be either continuously executed on one machine with its processing time, or rejected with its penalty that we must pay for. We may consider three versions of the CPS-DTP problem, respectively. (1) The constrained parallel-machine scheduling problem with divisible processing times and total penalties (the CPS-DTTP problem) is asked to find a subset A of J and a schedule T for jobs in A to satisfy the aforementioned requirement, the objective is to minimize the of such a schedule T for jobs in A plus the summation of penalties paid for jobs not in A;(2) The constrained parallel-machine scheduling problem with divisible processing times and maximum penalty (the CPS-DTMP problem) is asked to find a subset A of J and a schedule T for jobs in A to satisfy the aforementioned requirement, the objective is to minimize themakespan of such a schedule T for jobs in A plus maximum penalty paid for jobs not in A;(3) The constrained parallel-machine scheduling problem with divisible processing times and bounded penalty (the CPS-DTBP problem) is asked to find a subset A of J and a schedule T for jobs in A to satisfy the aforementioned requirement and the summation of penalties paid for jobs not in A is no more than a fixed bound, the objective is to minimize the makespan of such a schedule T for jobs in A. As our main contributions, we design three exact algorithms to solve the CPS-DTTP problem, the CPS-DTMP problem and the CPS-DTBP problem, and these thre

关键词： Combinatorial optimization Constrained parallel-machine scheduling Divisible processing times Penalties exact algorithms

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Uniqueness, intractability and exact algorithms: Reflections on level-κ phylogenetic networks

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Journal of Bioinformatics and Computational Biology 2009年第4期7卷 597-623页

作者： Van Iersel, Leo Kelk, Steven Mnich, Matthias Department of Mathematics and Computer Science Technische Universiteit Eindhoven 5600 MB Eindhoven P. O. Box 513 Netherlands Centrum voor Wiskunde en Informatica (CWI) 1090 GB Amsterdam P. O. Box 94079 Netherlands

Phylogenetic networks provide a way to describe and visualize evolutionary histories that have undergone so-called reticulate evolutionary events such as recombination, hybridization or horizontal gene transfer. The level κ of a network determines how non-treelike the evolution can be, with level-0 networks being trees. We study the problem of constructing level-κ phylogenetic networks from triplets, i.e. phylogenetic trees for three leaves (taxa). We give, for each κ, a level-κ network that is uniquely defined by its triplets. We demonstrate the applicability of this result by using it to prove that (1) for all κ ≥ 1 it is NP-hard to construct a level-κ network consistent with all input triplets, and (2) for all κ ≥ 0 it is NP-hard to construct a level-κ network consistent with a maximum number of input triplets, even when the input is dense. As a response to this intractability, we give an exact algorithm for constructing level-1 networks consistent with a maximum number of input triplets. © 2009 Imperial College Press.

关键词： exact algorithms NP-hardness Phylogenetic networks

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exact algorithms for Maximum Lifetime Data-Gathering Tree in Wireless Sensor Networks

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INFORMS JOURNAL ON COMPUTING 2022年第4期34卷 1987-2002页

作者： Casazza, Marco Ceselli, Alberto Univ Milan Dipartimento Informat I-20133 Milan Italy

We tackle an optimization problem arising in the design of sensor networks: given a set of sensors, only one being connected to a backbone, to establish connection routes from each of them to the sink. Under a shortest path routing protocol, the set of connections form a spanning tree. Energy is required to transmit and receive data, and sensors have limited battery capacity: as soon as one sensor runs out of battery, a portion of the network is disconnected. We, therefore, search for the spanning tree maximizing the time elapsed before such a disconnection occurs, and therefore, maintenance is required. We propose new mathematical formulations for the problem, proving and exploiting theoretical results on its combinatorial structure. On that basis, we design algorithms offering a priori guarantees of global optimality. We undertake an extensive experimental campaign, showing our algorithms to outperform previous ones from the literature by orders of magnitude. We also identify which instance features have higher impact on network lifetime.

关键词： sensor networks energy conservation data aggregation spanning tree exact algorithms

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Resolution-exact algorithms for Link Robots 11th

Resolution-Exact Algorithms for Link Robots

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11th Workshop on Algorithmic Foundations of Robotics (WAFR)

作者： Luo, Zhongdi Chiang, Yi-Jen Lien, Jyh-Ming Yap, Chee NYU Dept Comp Sci New York NY USA NYU Dept Comp Sci & Engn Brooklyn NY USA George Mason Univ Dept Comp Sci Fairfax VA 22030 USA

ISBN: (纸本)9783319165950;9783319165943

Motion planning is a major topic in robotics. Divergent paths have been taken by practical roboticists and theoretical motion planners. Our goal is to produce algorithms that are practical and have strong theoretical guarantees. Recently, we have proposed a subdivision approach based on soft predicates Wang, C., Chiang, Y.-J., Yap, C.: On soft predicates in subdivision motion planning. In: 29th ACM Symposium on Computational Geometry (SoCG'13), pp. 349-358 (2013). To appear CGTA, Special Issue for SoCG' 13 [20], but with a new notion of correctness called resolution-exactness. Unlike mos ques for planar link robots. The technical contributions of this paper are the design of soft predicates for link robots, a novel "T/R splitting method" for subdivision, and feature-based search strategies. The T/R idea is to give primacy to the translational (T) components, and perform splitting of rotational components (R) only at the leaves of a subdivision tree. We implemented our algorithm for a 2-link robot with 4 degrees of freedom (DOFs). Our implementation achieves real-time performance on a variety of nontrivial scenarios. For comparison, our method outperforms sampling-based methods significantly. We extend our 2-link planner to thick link robots with little impact on performance. Note that there are no known exact algorithms for thick link robots.

关键词： exact algorithms Subdivision algorithms Motion planning Soft predicates Resolution-exact algorithms Link robots

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exact algorithms for the 0-1 Time-Bomb Knapsack Problem

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COMPUTERS & OPERATIONS RESEARCH 2022年 145卷 1页

作者： Monaci, Michele Pike-Burke, Ciara Santini, Alberto Univ Bologna Dept Elect Elect & Informat Engn Bologna Italy Imperial Coll London Dept Math London England Univ Pompeu Fabra Dept Econ & Business Barcelona Spain

We consider a stochastic version of the 0-1 Knapsack Problem in which, in addition to profit and weight, each item is associated with a probability of exploding and destroying all the contents of the knapsack. The objective is to maximise the expected profit of the selected items. The resulting problem, denoted as 0-1 Time-Bomb Knapsack Problem (01-TB-KP), has applications in logistics and cloud computing scheduling. We introduce a nonlinear mathematical formulation of the problem, study its computational complexity, and propose techniques to derive upper and lower bounds using convex optimisation and integer linear programming. We present three exact approaches based on enumeration, branch and bound, and dynamic programming, and computationally evaluate their performance on a large set of benchmark instances. The computational analysis shows that the proposed methods outperform the direct application of nonlinear solvers on the mathematical model, and provide high quality solutions in a limited amount of time.

关键词： Knapsack Problem Stochastic optimisation exact algorithms Computational experiments

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Branch & Memorize exact algorithms for sequencing problems: Efficient embedding of memorization into search trees

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COMPUTERS & OPERATIONS RESEARCH 2021年 128卷 105171-105171页

作者： Shang, Lei T'Kindt, Vincent Della Croce, Federico Univ Tours LIFAT EA 6300 ERL CNRS ROOT 7002 Tours France Politecn Torino DIGEP Turin Italy CNR IEIIT Turin Italy

Memorization, as an algorithm design technique, enables to speed up algorithms at the price of increased space usage. In this work, we focus on search tree algorithms applied to sequencing problems. In these algorithms, on lower branching levels, isomorphic sub-problems may appear exponentially many times and the use of memorization is twofold: on the one hand it avoids repetitive solutions, as they correspond to identical sub-problems;on the other hand, it allows to check for dominance conditions among permutations of the same subset of elements. The idea of memorization appeared for a long time, however, to the best of the authors? knowledge, it was only seldom applied in the number of branching algorithms proposed in the literature. In this paper, we propose a unifying framework for implementing memorization in exact branching algorithms dedicated to sequencing problems. Our proposal leads to the paradigm of Branch & Memorize and its implementation to three classical single machine problems is validated by an extensive computational experimentation that shows that the mentioned paradigm consistently improves existing exact branching algorithms. These results emphasize the idea of more systematically embedding memorization in branching algorithms. ? 2020 Elsevier Ltd. All rights reserved.

关键词： Scheduling exact algorithms Memorization Branch and memorize Sequencing

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On exact algorithms for Treewidth

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ACM TRANSACTIONS ON algorithms 2012年第1期9卷 12-12页

作者： Bodlaender, Hans L. Fomin, Fedor V. Koster, Arie M. C. A. Kratsch, Dieter Thilikos, Dimitrios M. Univ Utrecht Dept Informat & Comp Sci NL-3508 TB Utrecht Netherlands Univ Bergen Dept Informat N-5020 Bergen Norway Univ Aachen Aachen Germany Univ Metz LITA F-507045 Metz 01 France Univ Athens GR-10679 Athens Greece

We give experimental and theoretical results on the problem of computing the treewidth of a graph by exact exponential-time algorithms using exponential space or using only polynomial space. We first report on an implementation of a dynamic programming algorithm for computing the treewidth of a graph with running time O*(2(n)). This algorithm is based on the old dynamic programming method introduced by Held and Karp for the TRAVELING SALESMAN problem. We use some optimizations that do not affect the worst case running time but improve on the running time on actual instances and can be seen to be practical for small instances. We also consider the problem of computing TREEWIDTH under the restriction that the space used is only polynomial and give a simple O*(4(n)) algorithm that requires polynomial space. We also show that with a more complicated algorithm using balanced separators, TREEWIDTH can be computed in O*(2.9512(n)) time and polynomial space.

关键词： algorithms Design Theory Experimentation Graph algorithms exact algorithms treewidth dynamic programming separators

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