Discrete Optimization最新文献

英文中文

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 Epub Date: 2025-10-27 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。

{"title":"An efficient solution approach to capacitated bilevel time minimizing transportation problem","authors":"Nanlan Zhang , Fanrong Xie","doi":"10.1016/j.disopt.2025.100915","DOIUrl":"10.1016/j.disopt.2025.100915","url":null,"abstract":"<div><div><em>Capacitated bilevel time minimizing transportation problem</em> (CBTMTP) is a generalization of <em>bilevel time minimizing transportation problem</em> (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.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"58 ","pages":"Article 100915"},"PeriodicalIF":1.6,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145417170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

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 : 2025-11-01 Epub 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。

{"title":"Computational aspects of lifted cover inequalities for knapsacks with few different weights","authors":"Christopher Hojny, Cédric Roy","doi":"10.1016/j.disopt.2025.100912","DOIUrl":"10.1016/j.disopt.2025.100912","url":null,"abstract":"<div><div>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.</div><div>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.</div><div>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.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"58 ","pages":"Article 100912"},"PeriodicalIF":1.6,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145099140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

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 : 2025-11-01 Epub 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

On degeneracy in the P-matroid oriented matroid complementarity problem 论面向p -拟阵互补问题的退化性

IF 0.9 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2025-08-01 Epub Date: 2025-06-11 DOI: 10.1016/j.disopt.2025.100891

Michaela Borzechowski , Simon Weber

Klaus showed that the Oriented Matroid Complementarity Problem (OMCP) can be solved by a reduction to the problem of sink-finding in a unique sink orientation (USO) if the input is promised to be given by a non-degenerate extension of a P-matroid. In this paper, we investigate the effect of degeneracy on this reduction. On the one hand, this understanding of degeneracies allows us to prove a linear lower bound on the number of vertex evaluations required for sink-finding in P-matroid USOs, the set of USOs obtainable through Klaus’ reduction. On the other hand, it allows us to adjust Klaus’ reduction to also work with degenerate instances. Furthermore, we introduce a total search version of the P-Matroid Oriented Matroid Complementarity Problem (P-OMCP). Given any extension of any oriented matroid

M

, by reduction to a total search version of USO sink-finding we can either solve the OMCP, or provide a polynomial-time verifiable certificate that

M

is not a P-matroid. This places the total search version of the P-OMCP in the complexity class Unique End of Potential Line (UEOPL).

Klaus证明了有向矩阵互补问题（OMCP）可以简化为在唯一集方向（USO）上寻找集的问题，如果输入是由p -矩阵的非退化扩展给出的话。在本文中，我们研究了简并对这种约简的影响。一方面，这种对退化的理解使我们能够证明p -矩阵USOs（通过Klaus约简可得到的USOs集合）中寻找sink所需顶点计算次数的线性下界。另一方面，它允许我们调整克劳斯的减少，也适用于退化的实例。进一步，我们引入了面向p -矩阵互补问题（P-OMCP）的一个全搜索版本。给定任意有向矩阵M的任意扩展，通过还原为USO sink-finding的总搜索版本，我们可以解出OMCP，或者提供一个多项式时间可验证的证明M不是p -矩阵。这将P-OMCP的总搜索版本放在复杂性类唯一潜在行结束（UEOPL）中。

{"title":"On degeneracy in the P-matroid oriented matroid complementarity problem","authors":"Michaela Borzechowski , Simon Weber","doi":"10.1016/j.disopt.2025.100891","DOIUrl":"10.1016/j.disopt.2025.100891","url":null,"abstract":"<div><div>Klaus showed that the <span>Oriented Matroid Complementarity Problem</span> (<span>OMCP</span>) can be solved by a reduction to the problem of sink-finding in a <em>unique sink orientation (USO)</em> if the input is promised to be given by a <em>non-degenerate</em> extension of a <em>P-matroid</em>. In this paper, we investigate the effect of degeneracy on this reduction. On the one hand, this understanding of degeneracies allows us to prove a linear lower bound on the number of vertex evaluations required for sink-finding in <em>P-matroid USOs</em>, the set of USOs obtainable through Klaus’ reduction. On the other hand, it allows us to adjust Klaus’ reduction to also work with degenerate instances. Furthermore, we introduce a total search version of the <span>P-Matroid Oriented Matroid Complementarity Problem</span> (<span>P-OMCP</span>). Given <em>any</em> extension of <em>any</em> oriented matroid <span><math><mi>M</mi></math></span>, by reduction to a total search version of USO sink-finding we can either solve the <span>OMCP</span>, or provide a polynomial-time verifiable certificate that <span><math><mi>M</mi></math></span> is <em>not</em> a P-matroid. This places the total search version of the <span>P-OMCP</span> in the complexity class <span>Unique End of Potential Line</span> (<span>UEOPL</span>).</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"57 ","pages":"Article 100891"},"PeriodicalIF":0.9,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144253723","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

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 Epub Date: 2025-07-31 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

Optimal partitions of the flat torus into parts of smaller diameter 平面环面最佳分割成较小直径的部分

IF 0.9 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2025-08-01 Epub Date: 2025-05-22 DOI: 10.1016/j.disopt.2025.100890

D.S. Protasov , A.D. Tolmachev , V.A. Voronov

We consider the problem of partitioning a two-dimensional flat torus

T^{2}

into

m

sets in order to minimize the maximum diameter of a part. For

m ⩽ 25

we give numerical estimates for the maximum diameter

d_{m} (T^{2})

at which the partition exists. Several approaches are proposed to obtain such estimates. In particular, we use the search for mesh partitions via the SAT solver, the global optimization approach for polygonal partitions, and the optimization of periodic hexagonal tilings. For

m = 3

, the exact estimate is proved using elementary topological reasoning.

为了使零件的最大直径最小，我们考虑将二维平面环面T2划分为m个集的问题。对于m≤25，我们给出了分区存在的最大直径dm（T2）的数值估计。提出了几种方法来获得这种估计。特别是，我们使用了通过SAT求解器搜索网格分区，多边形分区的全局优化方法，以及周期性六边形平铺的优化方法。对于m=3，利用初等拓扑推理证明了精确估计。

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 : 2025-08-01 Epub 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}

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。

{"title":"On the integrality gap of small Asymmetric Traveling Salesman Problems: A polyhedral and computational approach","authors":"Eleonora Vercesi , Janos Barta , Luca Maria Gambardella , Stefano Gualandi , Monaldo Mastrolilli","doi":"10.1016/j.disopt.2025.100901","DOIUrl":"10.1016/j.disopt.2025.100901","url":null,"abstract":"<div><div>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 <span><math><mi>n</mi></math></span> nodes, where <span><math><mi>n</mi></math></span> is small. In particular, we focus on the geometric properties and symmetries of the ASEP polytope (<span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>ASEP</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>) 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 <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>ASEP</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> to <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>ASEP</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></math></span> is defined. The generated vertices improve the known lower bounds of the integrality gap for <span><math><mrow><mn>16</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>22</mn></mrow></math></span> and, provide small hard-to-solve ATSP instances.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"57 ","pages":"Article 100901"},"PeriodicalIF":0.9,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144549222","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Approximation algorithms for the cluster editing problem with small clusters 小聚类聚类编辑问题的逼近算法

IF 0.9 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2025-08-01 Epub Date: 2025-06-13 DOI: 10.1016/j.disopt.2025.100900

Alexander Kononov , Victor Il’ev

Clustering is the task of dividing objects into groups (called clusters) so that objects in the same group are similar to each other. The Cluster Editing problem is one of the most natural ways to model clustering on graphs. In this problem, the similarity relation between objects is given by an undirected graph whose vertices correspond to the objects, edges connect couples of similar objects, and it is required to partition the set of vertices into disjoint subsets minimizing the number of edges between clusters and the number of missing edges within clusters. We present new approximation algorithms with better worst-case performance guarantees when cluster sizes are upper bounded by three or four vertices.

聚类是将对象分成组（称为集群）的任务，以便同一组中的对象彼此相似。聚类编辑问题是对图进行聚类建模的最自然的方法之一。在该问题中，目标之间的相似关系由一个无向图给出，该无向图的顶点对应于目标，边缘连接相似目标的对，并要求将顶点集划分为不相交的子集，以最小化聚类之间的边数和聚类内部的缺边数。我们提出了新的近似算法，当聚类大小的上界是三个或四个顶点时，它具有更好的最坏情况性能保证。

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 : 2025-08-01 Epub 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

Computation of lower tolerances of combinatorial bottleneck problems 组合瓶颈问题下容差的计算

IF 0.9 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization

Pub Date : 2025-05-01 Epub Date: 2025-04-10 DOI: 10.1016/j.disopt.2025.100887

Gerold Jäger , Marcel Turkensteen

This paper considers the computation of lower tolerances of combinatorial optimization problems with an objective of type bottleneck, in which the objective is to minimize the element with maximum cost of a feasible solution. A lower tolerance can be defined as the supremum decrease such that the objective value remains the same. We develop a computational approach for generic problems with objective of type bottleneck and two specific approaches for the Linear Bottleneck Assignment Problem and the Bottleneck Shortest Path Problem, which have a similar complexity as solution approaches for these two problems. Finally, we present some experimental results on random instances for these problems.

本文考虑的是目标为瓶颈类型的组合优化问题的下公差计算，其中目标是最小化可行解中成本最大的元素。下限公差可定义为目标值保持不变的至高减小值。我们为具有瓶颈类型目标的一般问题开发了一种计算方法，并为线性瓶颈分配问题和瓶颈最短路径问题开发了两种具体方法，这两种方法与这两个问题的求解方法具有相似的复杂性。最后，我们介绍了这些问题在随机实例上的一些实验结果。

引用次数: 0

首页上一页

下一页尾页

类型

全部化学•材料生命科学医学物理工程技术环境•农林材料科学地球科学法学管理学化学环境科学与生态学计算机科学教育学经济学农林科学人文科学生物学数学物理与天体物理心理学综合性期刊其他工业工程理学历史学农学文学信息工程

数据库

全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley

期刊

Discrete Optimization

全部 Acc. Chem. Res. ACS Applied Bio Materials ACS Appl. Electron. Mater. ACS Appl. Energy Mater. ACS Appl. Mater. Interfaces ACS Appl. Nano Mater. ACS Appl. Polym. Mater. ACS BIOMATER-SCI ENG ACS Catal. ACS Cent. Sci. ACS Chem. Biol. ACS Chemical Health & Safety ACS Chem. Neurosci. ACS Comb. Sci. ACS Earth Space Chem. ACS Energy Lett. ACS Infect. Dis. ACS Macro Lett. ACS Mater. Lett. ACS Med. Chem. Lett. ACS Nano ACS Omega ACS Photonics ACS Sens. ACS Sustainable Chem. Eng. ACS Synth. Biol. Anal. Chem. BIOCHEMISTRY-US Bioconjugate Chem. BIOMACROMOLECULES Chem. Res. Toxicol. Chem. Rev. Chem. Mater. CRYST GROWTH DES ENERG FUEL Environ. Sci. Technol. Environ. Sci. Technol. Lett. Eur. J. Inorg. Chem. IND ENG CHEM RES Inorg. Chem. J. Agric. Food. Chem. J. Chem. Eng. Data J. Chem. Educ. J. Chem. Inf. Model. J. Chem. Theory Comput. J. Med. Chem. J. Nat. Prod. J PROTEOME RES J. Am. Chem. Soc. LANGMUIR MACROMOLECULES Mol. Pharmaceutics Nano Lett. Org. Lett. ORG PROCESS RES DEV ORGANOMETALLICS J. Org. Chem. J. Phys. Chem. J. Phys. Chem. A J. Phys. Chem. B J. Phys. Chem. C J. Phys. Chem. Lett. Analyst Anal. Methods Biomater. Sci. Catal. Sci. Technol. Chem. Commun. Chem. Soc. Rev. CHEM EDUC RES PRACT CRYSTENGCOMM Dalton Trans. Energy Environ. Sci. ENVIRON SCI-NANO ENVIRON SCI-PROC IMP ENVIRON SCI-WAT RES Faraday Discuss. Food Funct. Green Chem. Inorg. Chem. Front. Integr. Biol. J. Anal. At. Spectrom. J. Mater. Chem. A J. Mater. Chem. B J. Mater. Chem. C Lab Chip Mater. Chem. Front. Mater. Horiz. MEDCHEMCOMM Metallomics Mol. Biosyst. Mol. Syst. Des. Eng. Nanoscale Nanoscale Horiz. Nat. Prod. Rep. New J. Chem. Org. Biomol. Chem. Org. Chem. Front. PHOTOCH PHOTOBIO SCI PCCP Polym. Chem.

﹀