Discrete Optimization最新文献

Exploiting symmetries in optimal quantum circuit design 利用最优量子电路设计中的对称性

IF 1.6 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2025-12-16 DOI: 10.1016/j.disopt.2025.100925

Frank de Meijer , Dion Gijswijt , Renata Sotirov

A physical limitation in quantum circuit design is the fact that gates in a quantum system can only act on qubits that are physically adjacent in the architecture. To overcome this problem, SWAP gates need to be inserted to make the circuit physically realizable. The nearest neighbour compliance problem (NNCP) asks for an optimal embedding of qubits in a given architecture such that the total number of SWAP gates to be inserted is minimized. In this paper we study the NNCP on general quantum architectures. Building upon an existing linear programming formulation, we show how the model can be reduced by exploiting the symmetries of the graph underlying the formulation. The resulting model is equivalent to a generalized network flow problem and follows from an in-depth analysis of the automorphism group of specific Cayley graphs. As a byproduct of our approach, we show that the NNCP is polynomial time solvable for several classes of symmetric quantum architectures. Numerical tests on various architectures indicate that the reduction in the number of variables and constraints is on average at least 90%. In particular, NNCP instances on the star architecture can be solved for quantum circuits up to 100 qubits and more than 1000 quantum gates within a very short computation time. These results are far beyond the computational capacity when solving the instances without the exploitation of symmetries.

量子电路设计的一个物理限制是，量子系统中的门只能作用于结构中物理相邻的量子位。为了克服这个问题，需要插入SWAP门以使电路在物理上可实现。最近邻遵从性问题（NNCP）要求在给定的体系结构中最优嵌入量子位，从而使要插入的SWAP门的总数最小化。本文研究了一般量子结构上的NNCP。在现有线性规划公式的基础上，我们展示了如何通过利用公式底层图的对称性来简化模型。所得到的模型等价于一个广义的网络流问题，它是对特定Cayley图的自同构群进行深入分析后得到的。作为我们方法的副产品，我们证明了NNCP对于几种对称量子体系结构是多项式时间可解的。在各种体系结构上的数值测试表明，变量和约束的数量平均减少了至少90%。特别是，星型架构上的NNCP实例可以在很短的计算时间内解决多达100个量子比特和超过1000个量子门的量子电路。这些结果远远超出了在不利用对称性的情况下求解实例的计算能力。

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引用次数: 0

An efficient solution approach to capacitated bilevel time minimizing transportation problem 一种有效的双层可容时间最小化运输问题的解决方法

IF 1.6 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2025-11-01 DOI: 10.1016/j.disopt.2025.100915

Nanlan Zhang , Fanrong Xie

Capacitated bilevel time minimizing transportation problem (CBTMTP) is a generalization of bilevel time minimizing transportation problem (BTMTP). Because of route shipping capacity realistic finiteness, CBTMTP is an optimization problem of crucial importance in logistics and emergency and project management. In the literature, no research report on CBTMTP is available due to its intractability, with exception of one BTMTP solving approach with defects of difficult computer implementation and difficult extension and inability used directly for efficiently solving CBTMTP. In this paper, by creating CBTMTP’s mathematical model along with auxiliary models and constructing network to make sufficient exploitation of CBTMTP’s network flow structure, two algorithms with one as exact optimum algorithm called CBTMTP-OA while another as heuristic algorithm called CBTMTP-HA are developed to solve CBTMTP efficiently. It is proved that CBTMTP-OA finds CBTMTP’s optimum solution by calling a polynomial time algorithm for a finite number of times, but CBTMTP-HA finds CBTMTP’s near optimum even exact optimum solution in a polynomial computation time. Computation study comprising distinct tests is conducted to verify the practical performance of CBTMTP-OA and CBTMTP-HA. It is revealed that CBTMTP-OA is capable of solving small and medium size instances efficiently, and inapplicable to solving large size instances because of too time consuming and memory overflow. But CBTMTP-HA is always capable of finding CBTMTP’s near optimum even exact optimum solution in high efficiency, and especially applicable to solving large size instances, with significant superiority to extant BTMTP solving approach. Both algorithms can serve as powerful tool for solving other relevant complicated optimization problems.

有能力双层时间最小化运输问题（CBTMTP）是双层时间最小化运输问题（BTMTP）的推广。由于航线运力的现实有限性，CBTMTP是物流应急管理和项目管理中一个至关重要的优化问题。在文献中，由于CBTMTP的顽固性，目前还没有关于其的研究报道，只有一种BTMTP求解方法存在计算机难以实现、难以扩展、无法直接用于有效求解CBTMTP的缺陷。本文通过建立CBTMTP的数学模型及其辅助模型，构建网络，充分利用CBTMTP的网络流结构，提出了两种算法，一种是精确优化算法CBTMTP- oa，另一种是启发式算法CBTMTP- ha，以有效地求解CBTMTP。证明了CBTMTP- oa通过有限次调用多项式时间算法找到CBTMTP的最优解，而CBTMTP- ha在多项式计算时间内找到CBTMTP的近最优甚至精确最优解。为验证cbttmp - oa和cbttmp - ha的实际性能，进行了包括不同测试的计算研究。结果表明，CBTMTP-OA能够有效地求解中小型实例，但由于耗时和内存溢出等原因，不适合求解大型实例。而CBTMTP- ha总是能够高效地找到CBTMTP的近最优甚至精确最优解，特别适用于求解大实例，与现有的BTMTP求解方法相比具有显著的优越性。这两种算法都可以作为解决其他相关复杂优化问题的有力工具。

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引用次数: 0

A preprocessing technique for quadratic unconstrained binary optimization 二次型无约束二值优化的预处理技术

IF 1.6 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2025-10-24 DOI: 10.1016/j.disopt.2025.100914

S. Gueye , P. Michelon

To solve combinatorial optimization problems more easily, it can be valuable to identify a set of necessary or sufficient conditions that an optimal solution of the problem must satisfy. For instance, weak and strong duality conditions in linear programming support the development of the well-known optimization algorithms for these problems. Similarly, the Karush–Kuhn–Tucker conditions give necessary and sufficient conditions for optimality in convex quadratic programming that underlie the development of optimization algorithms in this domain. Although such continuous conditions do not exist for integer programming, some necessary conditions can be derived from Karush–Kuhn–Tucker conditions for the Quadratic Unconstrained Binary Optimization (QUBO) problem. We present these conditions and show how they lead to a derivation of well-known criteria for fixing the values of single variables in the QUBO problem. From this, we show how to generalize these criteria to fix a product of any number of

p

(integer) literals, which also may be viewed as a generalization of the persistency notion, consisting of clauses of a Constraint Satisfaction Problem that the optimal solution must satisfy. We then couple our list of persistencies with state-of-the-art rules not covered by our approach. The resulting integrated set of conditions for fixing values of variables is tested in computational experiments on instances from standard databases available in the literature, showing that we can fix more variables and reduce problems more fully than previous approaches.

为了更容易地解决组合优化问题，确定问题的最优解必须满足的一组必要或充分条件是有价值的。例如，线性规划中的弱和强对偶条件支持了这些问题的著名优化算法的发展。类似地，Karush-Kuhn-Tucker条件给出了凸二次规划中最优性的充分必要条件，这是该领域优化算法发展的基础。虽然整数规划不存在这样的连续条件，但从二次型无约束二元优化问题的Karush-Kuhn-Tucker条件中可以导出一些必要条件。我们给出了这些条件，并展示了它们是如何推导出用于确定QUBO问题中单个变量值的著名标准的。由此，我们展示了如何推广这些准则来确定任意数量的p（整数）字量的乘积，这也可以看作是持久性概念的推广，由最优解必须满足的约束满足问题的子句组成。然后，我们将持久性列表与我们的方法未涵盖的最新规则结合起来。在文献中可用的标准数据库实例的计算实验中，对所得到的固定变量值的综合条件集进行了测试，表明我们可以比以前的方法更充分地固定更多的变量并减少问题。

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引用次数: 0

Minimizing maximum dissatisfaction in the allocation of indivisible items under a common preference graph 在共同偏好图下对不可分割物品分配的最大不满最小化

IF 1.6 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2025-10-03 DOI: 10.1016/j.disopt.2025.100913

Nina Chiarelli , Clément Dallard , Andreas Darmann , Stefan Lendl , Martin Milanič , Peter Muršič , Ulrich Pferschy

We consider the task of allocating indivisible items to agents, when the agents’ preferences over the items are identical. The preferences are captured by means of a directed acyclic graph, with vertices representing items and an edge

(a, b)

, meaning that each of the agents prefers item

a

over item

b

. The dissatisfaction of an agent is measured by the number of items that the agent does not receive and for which it also does not receive any more preferred item. The aim is to allocate the items to the agents in a fair way, i.e., to minimize the maximum dissatisfaction among the agents. We study the status of computational complexity of that problem and establish the following dichotomy: the problem is NP-hard for the case of at least three agents, even on fairly restricted graphs, but polynomially solvable for two agents. We also provide several polynomial-time results with respect to different underlying graph structures, such as graphs of width at most two and tree-like structures such as stars and matchings. These findings are complemented with fixed parameter tractability results related to path modules and independent set modules. Techniques employed in the paper include bottleneck assignment problem, greedy algorithm, dynamic programming, maximum network flow, and integer linear programming.

我们考虑了当智能体对物品的偏好相同时，将不可分割物品分配给智能体的任务。偏好是通过有向无环图的方式捕获的，顶点表示项目和一条边（a,b），这意味着每个代理更喜欢项目a而不是项目b。代理的不满意程度是通过代理没有接收到的项目数量来衡量的，并且它也没有接收到更多的首选项目。目标是以公平的方式将项目分配给代理，即最小化代理之间的最大不满。我们研究了该问题的计算复杂度状态，并建立了以下二分法：该问题对于至少三个智能体的情况是np困难的，即使在相当有限的图上也是如此，但对于两个智能体是多项式可解的。我们还提供了几个关于不同底层图结构的多项式时间结果，例如宽度最多为2的图和树状结构（如星形和匹配）的图。这些发现补充了与路径模块和独立集模块相关的固定参数可跟踪性结果。本文采用的技术包括瓶颈分配问题、贪心算法、动态规划、最大网络流量和整数线性规划。

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引用次数: 0

Computational aspects of lifted cover inequalities for knapsacks with few different weights 具有少量不同重量的背包的提升覆盖不等式的计算方面

IF 1.6 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2025-09-19 DOI: 10.1016/j.disopt.2025.100912

Christopher Hojny, Cédric Roy

Cutting planes are frequently used for solving integer programs. A common strategy is to derive cutting planes from building blocks or a substructure of the integer program. In this paper, we focus on knapsack constraints that arise from single row relaxations. Among the most popular classes derived from knapsack constraints are lifted minimal cover inequalities. The separation problem for these inequalities is NP-hard though, and one usually separates them heuristically, therefore not fully exploiting their potential.

For many benchmarking instances however, it turns out that many knapsack constraints only have few different coefficients. This motivates the concept of sparse knapsacks where the number of different coefficients is a small constant, independent of the number of variables present. For such knapsacks, we observe that there are only polynomially many different classes of structurally equivalent minimal covers. This opens the door to specialized techniques for using lifted minimal cover inequalities.

In this article we will discuss two such techniques, which are based on specialized sorting methods. On the one hand, we present new separation routines that separate equivalence classes of inequalities rather than individual inequalities. On the other hand, we derive compact extended formulations that express all lifted minimal cover inequalities by means of a polynomial number of constraints. These extended formulations are based on tailored sorting networks that express our separation algorithm by linear inequalities. We conclude the article by a numerical investigation of the different techniques for popular benchmarking instances.

切平面常用于求解整数程序。一种常见的策略是从构建块或整数程序的子结构中导出切割平面。在本文中，我们关注由单行松弛引起的背包约束。从背包约束中得到的最受欢迎的类是解除最小覆盖不等式。然而，这些不等式的分离问题是np困难的，人们通常会启发式地分离它们，因此不能充分利用它们的潜力。然而，对于许多基准测试实例，事实证明许多背包约束只有很少的不同系数。这激发了稀疏背包的概念，其中不同系数的数量是一个小常数，与存在的变量数量无关。对于这样的背包，我们观察到只有多项式多个不同类别的结构等效最小覆盖。这为使用提升的最小覆盖不等式的专门技术打开了大门。在本文中，我们将讨论两种这样的技术，它们基于专门的排序方法。一方面，我们提出了新的分离例程来分离不等式的等价类而不是单个不等式。另一方面，我们利用多项式个数的约束，导出了表示所有提升极小覆盖不等式的紧扩展公式。这些扩展的公式是基于定制的排序网络，通过线性不等式来表达我们的分离算法。我们通过对常用基准测试实例的不同技术进行数值研究来结束本文。

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引用次数: 0

On the circuit diameter conjecture for counterexamples to the Hirsch conjecture 赫希猜想反例的电路直径猜想

IF 1.6 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2025-09-17 DOI: 10.1016/j.disopt.2025.100910

Alexander E. Black , Steffen Borgwardt , Matthias Brugger

Circuit diameters of polyhedra are a fundamental tool for studying the complexity of circuit augmentation schemes for linear programming and for finding lower bounds on combinatorial diameters. The main open problem in this area is the circuit diameter conjecture, the analogue of the Hirsch conjecture in the circuit setting. A natural question is whether the well-known counterexamples to the Hirsch conjecture carry over. Previously, Stephen and Yusun showed that the Klee-Walkup counterexample to the unbounded Hirsch conjecture does not transfer to the circuit setting. Our main contribution is to show that the original counterexamples for other variants, using monotone walks or for bounded polytopes, also do not transfer. A challenge lies in the dependence of circuit diameters on the specific realization of a polyhedron. We discuss for which realizations, in addition to the original ones from the literature, our tools resolve this question.

Our results rely on new observations on structural properties of these counterexamples. To analyze the bounded case, we exploit the geometry of certain 2-faces of the polytopes underlying all known bounded Hirsch counterexamples in Santos’ work. For Todd’s monotone Hirsch counterexample, we study linear programs on spindles and prove sufficient conditions for short monotone circuit walks to exist. We then enumerate all linear programs over Todd’s polytope and find four new orientations that contradict the monotone Hirsch conjecture, while the remaining 7107 satisfy the bound. The conclusion then follows by applying these sufficient conditions to Todd’s counterexample.

多面体的电路直径是研究线性规划中电路增广方案的复杂性和求组合直径下界的基本工具。这一领域的主要开放问题是电路直径猜想，类似于电路设置中的赫希猜想。一个自然的问题是赫希猜想的著名反例是否适用。先前，Stephen和Yusun证明了无界Hirsch猜想的Klee-Walkup反例不能转移到电路设置中。我们的主要贡献是表明其他变体的原始反例，使用单调行走或有界多面体，也不会转移。一个挑战在于电路直径依赖于多面体的具体实现。我们讨论了除了文献中的原始实现之外，我们的工具解决了哪些实现。我们的结果依赖于对这些反例的结构特性的新观察。为了分析有界情况，我们利用了Santos工作中所有已知有界Hirsch反例的多面体的某些2面几何。对于Todd的单调Hirsch反例，我们研究了主轴上的线性规划，证明了短单调回路存在的充分条件。然后，我们枚举Todd的多面体上的所有线性规划，并找到四个与单调Hirsch猜想相矛盾的新方向，而其余的7107个满足界。然后通过将这些充分条件应用到Todd的反例中得出结论。

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引用次数: 0

An improved bound for the price of anarchy for related machine scheduling 相关机器调度的无政府状态代价的改进边界

IF 1.6 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2025-09-16 DOI: 10.1016/j.disopt.2025.100911

André Berger, Arman Rouhani, Marc Schröder

In this paper, we introduce an improved upper bound for the efficiency of Nash equilibria in utilitarian scheduling games on related machines. The machines have varying speeds and adhere to the shortest processing time first policy. The goal of each job is to minimize its completion time, while the social objective is to minimize the sum of completion times. Our main finding establishes an upper bound of

2 - 1 / (4 m - 2)

on the price of anarchy for the general case of

m

machines. We improve this bound to 3/2 for the case of two machines, and to

2 - 1 / (2 m)

for the general case of

m

machines when the machines have divisible speeds, i.e., if the speed of each machine is divisible by the speed of any slower machine.

本文引入了相关机器上功利调度对策纳什均衡效率的改进上界。机器有不同的速度，并坚持最短的处理时间优先的政策。每项工作的目标是使其完成时间最小化，而社会目标是使完成时间的总和最小化。我们的主要发现建立了m个机器的一般情况下无政府状态价格的上界为2−1/(4m−2)。对于两台机器的情况，我们将这个界限提高到3/2，对于m台机器的一般情况，当机器的速度可整除时，即，如果每台机器的速度可被任何较慢的机器的速度整除，则该界限提高到2 - 1/(2m)。

引用次数: 0

A criterion space search feasibility pump heuristic for solving maximum multiplicative programs 求解最大乘法规划的准则空间搜索可行性泵浦启发式

IF 1.6 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2025-08-01 DOI: 10.1016/j.disopt.2025.100903

Ashim Khanal, Hadi Charkhgard

We study a class of nonlinear optimization problems with diverse practical applications, particularly in cooperative game theory. These problems are referred to as Maximum Multiplicative Programs (MMPs), and can be conceived as instances of “Optimization Over the Frontier” in multi-objective optimization. To solve MMPs, we introduce a feasibility pump-based heuristic that is specifically designed to search the criterion space of their multi-objective optimization counterparts. Through a computational study, we show the efficacy of the proposed method.

我们研究了一类具有多种实际应用的非线性优化问题，特别是合作博弈问题。这些问题被称为最大乘法规划（MMPs），可以看作是多目标优化中的“边界优化”实例。为了解决MMPs问题，我们引入了一种基于可行性泵的启发式算法，专门用于搜索其多目标优化对应的准则空间。通过计算研究，我们证明了该方法的有效性。

引用次数: 0

Disjoint dominating and 2-dominating sets in graphs: Hardness and approximation results 图中的不相交支配集和2支配集：硬度和近似结果

IF 0.9 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2025-07-24 DOI: 10.1016/j.disopt.2025.100902

Soumyashree Rana , Sounaka Mishra , Bhawani Sankar Panda

<div><div>A set <span><math><mrow><mi>D</mi><mo>⊆</mo><mi>V</mi></mrow></math></span> of a graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> is a dominating set of <span><math><mi>G</mi></math></span> if each vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mo>∖</mo><mi>D</mi></mrow></math></span> is adjacent to at least one vertex in <span><math><mrow><mi>D</mi><mo>,</mo></mrow></math></span> whereas a set <span><math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⊆</mo><mi>V</mi></mrow></math></span> is a 2-dominating (double dominating) set of <span><math><mi>G</mi></math></span> if each vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mo>∖</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> is adjacent to at least two vertices in <span><math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>.</mo></mrow></math></span> A graph <span><math><mi>G</mi></math></span> is a <span><math><mrow><mi>D</mi><mspace></mspace><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>-graph if there exists a pair (<span><math><mrow><mi>D</mi><mo>,</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>) of dominating set and 2-dominating set of <span><math><mi>G</mi></math></span> which are disjoint. In this paper, we give approximation algorithms for the problem of determining a minimal spanning <span><math><mrow><mi>D</mi><mspace></mspace><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>-graph of minimum size (<span>Min-</span> <span><math><mrow><mi>D</mi><mspace></mspace><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>) with an approximation ratio of 3; a minimal spanning <span><math><mrow><mi>D</mi><mspace></mspace><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>-graph of maximum size (<span>Max-</span> <span><math><mrow><mi>D</mi><mspace></mspace><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>) with an approximation ratio of 3; and for the problem of adding minimum number of edges to a graph <span><math><mi>G</mi></math></span> to make it a <span><math><mrow><mi>D</mi><mspace></mspace><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>-graph (<span>Min-to-</span> <span><math><mrow><mi>D</mi><mspace></mspace><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>) with an <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mo>log</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> approximation ratio. The above three results answer the open problems mentioned in the paper, Miotk et al. (2020). Furthermore, we prove that <span>Min-</span> <span><math><mrow><mi>D</mi><mspace></mspace><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> and <s

当图G=（V，E）的每个顶点V∈V≠D与D中的至少一个顶点邻接，则集合D是G的控制集；当每个顶点V∈V∈D2与D2中的至少两个顶点邻接，则集合D2是G的2-控制（双控制）集。如果存在不相交的G的支配集和2-支配集对（D,D2），则图G是一个dd2图。本文给出了确定最小生成DD2图（Min- DD2）的近似算法，近似比为3；最小生成DD2图的最大尺寸（Max- DD2），近似比为3；以及为图G添加最小边数以使其成为具有O（logn）近似比的DD2图（Min-to- DD2）的问题。以上三个结果回答了Miotk et al.（2020）论文中提到的开放性问题。进一步证明了对于最大度为4的图，Min- DD2和Max- DD2是apx完全的。我们还表明，对于任何3正则图，Min- DD2和Max- DD2分别在1.8和1.5因子内近似。最后，我们给出了二部图的Max-Min-to- DD2的不可逼近性结果：除非P=NP，否则该问题不能在n16−_i内逼近任何_i >；0。

引用次数: 0

On the integrality gap of small Asymmetric Traveling Salesman Problems: A polyhedral and computational approach 非对称小旅行商问题的完整性缺口：一个多面体和计算方法

IF 0.9 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2025-07-04 DOI: 10.1016/j.disopt.2025.100901

Eleonora Vercesi , Janos Barta , Luca Maria Gambardella , Stefano Gualandi , Monaldo Mastrolilli

In this paper, we investigate the integrality gap of the Asymmetric Traveling Salesman Problem (ATSP) with respect to the linear relaxation given by the Asymmetric Subtour Elimination Problem (ASEP) for instances with

n

nodes, where

n

is small. In particular, we focus on the geometric properties and symmetries of the ASEP polytope (

P_{ASEP}^{n}

) and its vertices. The polytope’s symmetries are exploited to design a heuristic pivoting algorithm to search vertices where the integrality gap is maximized. Furthermore, a general procedure for the extension of vertices from

P_{ASEP}^{n}

to

P_{ASEP}^{n + 1}

is defined. The generated vertices improve the known lower bounds of the integrality gap for

16 \leq n \leq 22

and, provide small hard-to-solve ATSP instances.

本文研究了非对称旅行商问题（ATSP）在n个节点的情况下相对于非对称子游消除问题（ASEP）给出的线性松弛的完整性缺口，其中n很小。我们特别关注了ASEP多面体（PASEPn）及其顶点的几何性质和对称性。利用多面体的对称性，设计了一种启发式的旋转算法来搜索完整性间隙最大的顶点。此外，还定义了从PASEPn向PASEPn+1顶点扩展的一般过程。生成的顶点改善了已知的16≤n≤22的完整性间隙下界，并且提供了小的难以求解的ATSP实例。

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引用次数: 0