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Algorithms and complexity for the almost equal maximum flow problem 几乎相等的最大流量问题的算法和复杂性
IF 2.1 4区 计算机科学 Q4 COMPUTER SCIENCE, HARDWARE & ARCHITECTURE Pub Date : 2024-01-23 DOI: 10.1002/net.22209
Rebekka Haese, Till Heller, Sven O. Krumke
In the equal maximum flow problem (EMFP), we aim for a maximum flow where we require the same flow value on all arcs in some given subsets of the arc set, so called homologous arc sets. In this article, we study the closely related almost equal maximum flow problems (AEMFP) where the flow values on arcs of one homologous arc set differ at most by the valuation of a so called deviation function <mjx-container aria-label="normal upper Delta" ctxtmenu_counter="0" ctxtmenu_oldtabindex="1" jax="CHTML" role="application" sre-explorer- style="font-size: 103%; position: relative;" tabindex="0"><mjx-math aria-hidden="true" location="graphic/net22209-math-0001.png"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation="clearspeak:simple" data-semantic-font="normal" data-semantic- data-semantic-role="greekletter" data-semantic-speech="normal upper Delta" data-semantic-type="identifier"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden="true" display="inline" unselectable="on"><math altimg="urn:x-wiley:net:media:net22209:net22209-math-0001" display="inline" location="graphic/net22209-math-0001.png" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi data-semantic-="" data-semantic-annotation="clearspeak:simple" data-semantic-font="normal" data-semantic-role="greekletter" data-semantic-speech="normal upper Delta" data-semantic-type="identifier" mathvariant="normal">Δ</mi></mrow>$$ Delta $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>. We prove that the integer AEMFP is in general <mjx-container aria-label="script upper N script upper P" ctxtmenu_counter="1" ctxtmenu_oldtabindex="1" jax="CHTML" role="application" sre-explorer- style="font-size: 103%; position: relative;" tabindex="0"><mjx-math aria-hidden="true" location="graphic/net22209-math-0002.png"><mjx-mrow data-semantic-annotation="clearspeak:simple;clearspeak:unit" data-semantic-children="0,1" data-semantic-content="2" data-semantic- data-semantic-role="implicit" data-semantic-speech="script upper N script upper P" data-semantic-type="infixop"><mjx-mi data-semantic-annotation="clearspeak:simple" data-semantic-font="script" data-semantic- data-semantic-parent="3" data-semantic-role="latinletter" data-semantic-type="identifier"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added="true" data-semantic- data-semantic-operator="infixop,⁢" data-semantic-parent="3" data-semantic-role="multiplication" data-semantic-type="operator" style="margin-left: 0.056em; margin-right: 0.056em;"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation="clearspeak:simple" data-semantic-font="script" data-semantic- data-semantic-parent="3" data-semantic-role="latinletter" data-semantic-type="identifier"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-math><mjx-assistive-mml aria-hidden="true" display="inline" unselectable="on"><math altimg="urn:x-wiley:net:media:net22209:net22209-math-0002" display="inline" location="graphic/net22209-
在最大流量相等问题(EMFP)中,我们的目标是获得最大流量,其中我们要求弧集的某些给定子集(即所谓的同源弧集)中所有弧的流量值相同。在本文中,我们将研究与之密切相关的几乎相等的最大流量问题(AEMFP),其中一个同源弧集的弧上的流量值最多相差一个所谓的偏差函数 Δ$$ Delta $$$。我们证明了整数 AEMFP 一般是𝒩𝒫-完备的,并证明了在凸偏差函数的情况下,甚至寻找分数最大流的问题也是𝒩𝒫-完备的。这与 EMFP 形成了鲜明对比,后者在分数情况下是多项式时间可解的。此外,我们还提供了积分 AEMFP 的不可逼近性结果。对于(分数)凹 AEMFP,我们利用参数搜索方法,针对固定数量同源集的线性和凹片断多项式偏差函数情况,提出了一种强多项式算法。
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引用次数: 0
On the complexity of the upgrading version of the maximal covering location problem 论最大覆盖位置问题升级版的复杂性
IF 2.1 4区 计算机科学 Q4 COMPUTER SCIENCE, HARDWARE & ARCHITECTURE Pub Date : 2024-01-17 DOI: 10.1002/net.22207
Marta Baldomero-Naranjo, Jörg Kalcsics, Antonio M. Rodríguez-Chía
In this article, we study the complexity of the upgrading version of the maximal covering location problem with edge length modifications on networks. This problem is NP-hard on general networks. However, in some particular cases, we prove that this problem is solvable in polynomial time. The cases of star and path networks combined with different assumptions for the model parameters are analysed. In particular, we obtain that the problem on star networks is solvable in O(nlogn)�$$ Oleft(nlog nright) $$� time for uniform weights and NP-hard for non-uniform weights. On paths, the single facility problem is solvable in O(n3)�$$ Oleft({n}^3right) $$� time, while the p�$$ p $$�-facility problem is NP-hard even with uniform costs and upper bounds (maximal upgrading per edge), as well as, integer parameter values. Furthermore, a pseudo-polynomial algorithm is developed for the single facility problem on trees with integer parameters.
在本文中,我们研究了网络上最大覆盖位置问题升级版的复杂性,该问题的边长有修改。在一般网络上,这个问题是 NP 难的。然而,在某些特殊情况下,我们证明这个问题可以在多项式时间内求解。我们分析了星形网络和路径网络的情况,并结合了对模型参数的不同假设。特别是,我们得出星形网络上的问题对于均匀权重可在 O(nlogn)$$ Oleft(nlog nright) $$ 时间内求解,而对于非均匀权重则是 NP-hard。在路径上,单一设施问题可在 O(n3)$$ Oleft({n}^3right) $$ 时间内求解,而 p$$ p$ 设施问题即使有统一的成本和上限(每条边的最大升级)以及整数参数值,也是 NP-hard。此外,还针对具有整数参数的树上单一设施问题开发了一种伪多项式算法。
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引用次数: 0
Shortest path network interdiction with asymmetric uncertainty 具有不对称不确定性的最短路径网络拦截
IF 2.1 4区 计算机科学 Q4 COMPUTER SCIENCE, HARDWARE & ARCHITECTURE Pub Date : 2024-01-15 DOI: 10.1002/net.22208
She'ifa Z. Punla-Green, John E. Mitchell, Jared L. Gearhart, William E. Hart, Cynthia A. Phillips
This paper considers an extension of the shortest path network interdiction problem that incorporates robustness to account for parameter uncertainty. The shortest path interdiction problem is a game of two players with conflicting agendas and capabilities: an evader, who traverses the arcs of a network from a source node to a sink node using a path of shortest length, and an interdictor, who maximizes the length of the evader's shortest path by interdicting arcs on the network. It is usually assumed that the parameters defining the network are known exactly by both players. We consider the situation where the evader assumes the nominal parameter values while the interdictor uses robust optimization techniques to account for parameter uncertainty or sensor degradation. We formulate this problem as a nonlinear mixed-integer semi-infinite bilevel program and show that it can be converted into a mixed-integer linear program with a second order cone constraint. We use random geometric networks and transportation networks to perform computational studies and demonstrate the unique decision strategies that our variant produces. Solving the shortest path interdiction problem with asymmetric uncertainty protects the interdictor from investing in a strategy that hinges on key interdictions performing as promised. It also provides an alternate strategy that mitigates the risk of these worst-case possibilities.
本文探讨了最短路径网络阻截问题的扩展,其中包含了考虑参数不确定性的鲁棒性。最短路径拦截问题是一个由两个具有相互冲突的目的和能力的参与者组成的博弈:逃避者和拦截者,逃避者利用最短路径穿越从源节点到汇节点的网络弧线,而拦截者则通过拦截网络上的弧线使逃避者的最短路径长度最大化。通常假设定义网络的参数对双方来说都是完全已知的。我们考虑的情况是,规避者假设标称参数值,而拦截者使用鲁棒优化技术来考虑参数的不确定性或传感器的退化。我们将这一问题表述为非线性混合整数半无限双级程序,并证明它可以转换为带有二阶锥约束的混合整数线性程序。我们使用随机几何网络和交通网络进行计算研究,并展示了我们的变体所产生的独特决策策略。解决具有非对称不确定性的最短路径拦截问题可以保护拦截者,使其不必投资于依赖于关键拦截点是否如约执行的策略。它还提供了一种替代策略,可降低这些最坏情况可能性的风险。
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引用次数: 0
Special cases of the minimum spanning tree problem under explorable edge and vertex uncertainty 可探索边和顶点不确定性下最小生成树问题的特例
IF 2.1 4区 计算机科学 Q4 COMPUTER SCIENCE, HARDWARE & ARCHITECTURE Pub Date : 2024-01-11 DOI: 10.1002/net.22204
Corinna Mathwieser, Eranda Çela
This article studies the Minimum Spanning Tree Problem under Explorable Uncertainty as well as a related vertex uncertainty version of the problem. We particularly consider special instance types, including cactus graphs, for which we provide randomized algorithms. We introduce the problem of finding a minimum weight spanning star under uncertainty for which we show that no algorithm can achieve constant competitive ratio.
本文研究了可探索不确定性下的最小生成树问题以及该问题的相关顶点不确定性版本。我们特别考虑了包括仙人掌图在内的特殊实例类型,并提供了随机算法。我们介绍了在不确定条件下寻找最小权重生成星的问题,并证明没有算法能达到恒定的竞争比。
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引用次数: 0
How vulnerable is an undirected planar graph with respect to max flow 无向平面图在最大流量方面有多脆弱?
IF 2.1 4区 计算机科学 Q4 COMPUTER SCIENCE, HARDWARE & ARCHITECTURE Pub Date : 2024-01-03 DOI: 10.1002/net.22205
Lorenzo Balzotti, Paolo G. Franciosa
We study the problem of computing the vitality of edges and vertices with respect to the <mjx-container aria-label="Menu available. Press control and space , or space" ctxtmenu_counter="1" ctxtmenu_oldtabindex="1" jax="CHTML" role="application" sre-explorer- style="font-size: 103%; position: relative;" tabindex="0"><mjx-math aria-hidden="true" location="graphic/net22205-math-0001.png"><mjx-semantics><mjx-mrow data-semantic-annotation="clearspeak:simple;clearspeak:unit" data-semantic-children="0,1" data-semantic-content="2" data-semantic- data-semantic-role="implicit" data-semantic-speech="s t" data-semantic-type="infixop"><mjx-mi data-semantic-annotation="clearspeak:simple" data-semantic-font="italic" data-semantic- data-semantic-parent="3" data-semantic-role="latinletter" data-semantic-type="identifier"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added="true" data-semantic- data-semantic-operator="infixop,⁢" data-semantic-parent="3" data-semantic-role="multiplication" data-semantic-type="operator" style="margin-left: 0.056em; margin-right: 0.056em;"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation="clearspeak:simple" data-semantic-font="italic" data-semantic- data-semantic-parent="3" data-semantic-role="latinletter" data-semantic-type="identifier"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden="true" display="inline" unselectable="on"><math altimg="urn:x-wiley:net:media:net22205:net22205-math-0001" display="inline" location="graphic/net22205-math-0001.png" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow data-semantic-="" data-semantic-annotation="clearspeak:simple;clearspeak:unit" data-semantic-children="0,1" data-semantic-content="2" data-semantic-role="implicit" data-semantic-speech="s t" data-semantic-type="infixop"><mi data-semantic-="" data-semantic-annotation="clearspeak:simple" data-semantic-font="italic" data-semantic-parent="3" data-semantic-role="latinletter" data-semantic-type="identifier">s</mi><mo data-semantic-="" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-parent="3" data-semantic-role="multiplication" data-semantic-type="operator">⁢</mo><mi data-semantic-="" data-semantic-annotation="clearspeak:simple" data-semantic-font="italic" data-semantic-parent="3" data-semantic-role="latinletter" data-semantic-type="identifier">t</mi></mrow>$$ st $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-max flow in undirected planar graphs, where the vitality of an edge/vertex is the <mjx-container aria-label="Menu available. Press control and space , or space" ctxtmenu_counter="2" ctxtmenu_oldtabindex="1" jax="CHTML" role="application" sre-explorer- style="font-size: 103%; position: relative;" tabindex="0"><mjx-math aria-hidden="true" location="graphic/net22205-math-0002.png"><mjx-semantics><mjx-mrow data-semantic-annotation="clearspeak:simple;clearspeak:unit" data-semantic-children="0,1" data-semantic-content
我们研究的是计算无向平面图中边和顶点相对于 st$$ st $$$ 最大流量的活力问题,其中边/顶点的活力是指从图中删除该边/顶点时,该边/顶点的 st$$ st $$$ 最大流量的减少量。这样,我们就能确定图在 st$$ st $$-max 流量方面的脆弱性。我们给出了计算平面无向图中边和顶点活力的加法保证近似值的高效算法。我们证明,在一般情况下,高活力值可以在接近目前计算 st$$ st $$$-max 流量所需的时间 O(nloglogn)$$ Oleft(nmathrm{log}nright) $$ 的情况下被很好地近似。我们还给出了整数容量情况下的改进结果,有时甚至是最优结果。我们的所有算法都在 O(n)$$ O(n) $$ 空间内运行。
{"title":"How vulnerable is an undirected planar graph with respect to max flow","authors":"Lorenzo Balzotti, Paolo G. Franciosa","doi":"10.1002/net.22205","DOIUrl":"https://doi.org/10.1002/net.22205","url":null,"abstract":"We study the problem of computing the vitality of edges and vertices with respect to the &lt;mjx-container aria-label=\"Menu available. Press control and space , or space\" ctxtmenu_counter=\"1\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"&gt;&lt;mjx-math aria-hidden=\"true\" location=\"graphic/net22205-math-0001.png\"&gt;&lt;mjx-semantics&gt;&lt;mjx-mrow data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"0,1\" data-semantic-content=\"2\" data-semantic- data-semantic-role=\"implicit\" data-semantic-speech=\"s t\" data-semantic-type=\"infixop\"&gt;&lt;mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"&gt;&lt;mjx-c&gt;&lt;/mjx-c&gt;&lt;/mjx-mi&gt;&lt;mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,⁢\" data-semantic-parent=\"3\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"&gt;&lt;mjx-c&gt;&lt;/mjx-c&gt;&lt;/mjx-mo&gt;&lt;mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"&gt;&lt;mjx-c&gt;&lt;/mjx-c&gt;&lt;/mjx-mi&gt;&lt;/mjx-mrow&gt;&lt;/mjx-semantics&gt;&lt;/mjx-math&gt;&lt;mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"&gt;&lt;math altimg=\"urn:x-wiley:net:media:net22205:net22205-math-0001\" display=\"inline\" location=\"graphic/net22205-math-0001.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;semantics&gt;&lt;mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"0,1\" data-semantic-content=\"2\" data-semantic-role=\"implicit\" data-semantic-speech=\"s t\" data-semantic-type=\"infixop\"&gt;&lt;mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"&gt;s&lt;/mi&gt;&lt;mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,⁢\" data-semantic-parent=\"3\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"&gt;⁢&lt;/mo&gt;&lt;mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"&gt;t&lt;/mi&gt;&lt;/mrow&gt;$$ st $$&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/mjx-assistive-mml&gt;&lt;/mjx-container&gt;-max flow in undirected planar graphs, where the vitality of an edge/vertex is the &lt;mjx-container aria-label=\"Menu available. Press control and space , or space\" ctxtmenu_counter=\"2\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"&gt;&lt;mjx-math aria-hidden=\"true\" location=\"graphic/net22205-math-0002.png\"&gt;&lt;mjx-semantics&gt;&lt;mjx-mrow data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"0,1\" data-semantic-content","PeriodicalId":54734,"journal":{"name":"Networks","volume":"14 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139094100","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Math-based reinforcement learning for the adaptive budgeted influence maximization problem 自适应预算影响最大化问题的数学强化学习
IF 2.1 4区 计算机科学 Q4 COMPUTER SCIENCE, HARDWARE & ARCHITECTURE Pub Date : 2023-12-26 DOI: 10.1002/net.22206
Edoardo Fadda, Evelina Di Corso, Davide Brusco, Vlad Stefan Aelenei, Alexandru Balan Rares
In social networks, the influence maximization problem requires selecting an initial set of nodes to influence so that the spread of influence can reach its maximum under certain diffusion models. Usually, the problem is formulated in a two-stage un-budgeted fashion: The decision maker selects a given number of nodes to influence and observes the results. In the adaptive version of the problem, it is possible to select the nodes at each time step of a given time interval. This allows the decision-maker to exploit the observation of the propagation and to make better decisions. This paper considers the adaptive budgeted influence maximization problem, that is, the adaptive problem in which the decision maker has a finite budget to influence the nodes, and each node requires a cost to be influenced. We present two solution techniques: The first is an approximated value iteration leveraging mixed integer linear problems while the second exploits new concepts from graph neural networks. Extensive numerical experiments demonstrate the effectiveness of the proposed approaches.
在社交网络中,影响力最大化问题要求选择一组初始节点来施加影响,从而使影响力的传播在某些扩散模型下达到最大。通常,该问题以两阶段无预算方式提出:决策者选择一定数量的节点施加影响,然后观察结果。在该问题的自适应版本中,可以在给定时间间隔的每个时间步选择节点。这样,决策者就可以利用对传播的观察,做出更好的决策。本文考虑的是自适应预算影响最大化问题,即决策者有有限预算来影响节点,而每个节点都需要付出一定代价才能被影响的自适应问题。我们提出了两种解决方法:第一种是利用混合整数线性问题的近似值迭代,第二种是利用图神经网络的新概念。大量的数值实验证明了所提方法的有效性。
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引用次数: 0
Generalizing Horn's conditions for preemptive scheduling on identical parallel machines via network flow techniques 通过网络流技术推广相同并行机上抢占式调度的霍恩条件
IF 2.1 4区 计算机科学 Q4 COMPUTER SCIENCE, HARDWARE & ARCHITECTURE Pub Date : 2023-12-20 DOI: 10.1002/net.22202
Akiyoshi Shioura, Vitaly A. Strusevich, Natalia V. Shakhlevich
We study the use of flow-based algorithmic and proof techniques applied to preemptive scheduling of jobs on parallel identical machines. For the classical problem in which the jobs have individual release dates and must be finished by a common deadline, we present and prove unified necessary and sufficient conditions for the existence of a feasible schedule by examining the structure of minimum cuts in a special network. We then study an enhanced model that allows the presence of additional resources, provided that some jobs at any time of their processing require one unit of a particular resource. We extend our argument developed for the classical case to this enhanced model. The generalized necessary and sufficient conditions for the existence of a feasible schedule are presented and proved using the max-flow/min-cut reasoning.
我们研究了基于流的算法和证明技术在并行相同机器上作业抢先调度中的应用。对于作业有各自的发布日期且必须在共同截止日期前完成的经典问题,我们通过研究特殊网络中最小切割的结构,提出并证明了可行调度存在的统一必要条件和充分条件。然后,我们研究了一个允许额外资源存在的增强模型,条件是某些作业在其处理过程中的任何时候都需要一个单位的特定资源。我们将针对经典模型的论证扩展到这一增强模型。我们提出了可行计划表存在的广义必要条件和充分条件,并使用最大流量/最小切分推理进行了证明。
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引用次数: 0
Learning to repeatedly solve routing problems 学会反复解决路由问题
IF 2.1 4区 计算机科学 Q4 COMPUTER SCIENCE, HARDWARE & ARCHITECTURE Pub Date : 2023-12-14 DOI: 10.1002/net.22200
Mouad Morabit, Guy Desaulniers, Andrea Lodi
In the last years, there has been a great interest in machine-learning-based heuristics for solving NP-hard combinatorial optimization problems. The developed methods have shown potential on many optimization problems. In this paper, we present a learned heuristic for the reoptimization of a problem after a minor change in its data. We focus on the case of the capacited vehicle routing problem with static clients (i.e., same client locations) and changed demands. Given the edges of an original solution, the goal is to predict and fix the ones that have a high chance of remaining in an optimal solution after a change of client demands. This partial prediction of the solution reduces the complexity of the problem and speeds up its resolution, while yielding a good quality solution. The proposed approach resulted in solutions with an optimality gap ranging from 0% to 1.7% on different benchmark instances within a reasonable computing time.
在过去的几年里,人们对基于机器学习的启发式算法解决NP-hard组合优化问题产生了极大的兴趣。所开发的方法在许多优化问题上显示出潜力。在本文中,我们提出了一种学习启发式方法,用于在数据发生微小变化后对问题进行再优化。我们关注的是静态客户端(即相同的客户端位置)和需求变化情况下的可容车辆路径问题。给定原始解决方案的边缘,目标是在客户需求发生变化后,预测并修复最有可能保持在最优解决方案中的边缘。这种对解决方案的部分预测降低了问题的复杂性,加快了问题的解决速度,同时产生了高质量的解决方案。在合理的计算时间内,在不同的基准测试实例上,所提出的方法产生的解决方案的最优性差距从0%到1.7%不等。
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引用次数: 0
Asymptotic bounds for clustering problems in random graphs 随机图中聚类问题的渐近边界
IF 2.1 4区 计算机科学 Q4 COMPUTER SCIENCE, HARDWARE & ARCHITECTURE Pub Date : 2023-12-13 DOI: 10.1002/net.22203
Eugene Lykhovyd, Sergiy Butenko, Pavlo Krokhmal
Graph clustering is an important problem in network analysis. This problem can be approached by first finding a large cluster subgraph (i.e., a subgraph in which every connected component is a complete graph), perhaps in a relaxed form (connected components may have missing edges), and then assigning each of the remaining vertices to one of the connected components of the cluster subgraph according to some optimization criteria. The more vertices can be included in the initial cluster subgraph (also referred to as independent union of clusters), the more “clusterable” the graph is. This paper proposes a framework for establishing asymptotic bounds on the cardinality of independent unions of clusters in Erdős-Rényi random graphs G(n,p)�$$ Gleft(n,pright) $$� with constant p�$$ p $$�, referred to as uniform random graphs. In particular, sufficient conditions ensuring O(logn)�$$ Oleft(log nright) $$� (where n�$$ n $$� is the number of nodes) upper bounds with probability 1 are developed and shown to be applicable for the maximum independent union of cliques as well as some clique relaxations. In addition, it is shown that every graph must have an independent union of cliques of cardinality at least Ω(logn)�$$ Omega left(log nright) $$�. Since this bound is asymptotically tight on uniform random graphs, this suggests that these graphs can be viewed as a “least clusterable” class of graphs.
图聚类是网络分析中的一个重要问题。这个问题可以通过首先找到一个大的聚类子图(即,其中每个连接组件都是完整图的子图)来解决,可能是在一个宽松的形式(连接组件可能有缺失的边),然后根据一些优化标准将每个剩余的顶点分配给聚类子图的一个连接组件。初始聚类子图(也称为簇的独立并集)中包含的顶点越多,该图就越“可聚类”。本文提出了一个框架,用于建立具有常数p $$ p $$的Erdős-Rényi随机图G(n,p) $$ Gleft(n,pright) $$(称为一致随机图)中簇的独立并集基数的渐近界。特别是,给出了保证概率为1的O(logn) $$ Oleft(log nright) $$(其中n $$ n $$为节点数)上界的充分条件,并证明了该条件适用于团的最大独立联合以及一些团松弛。此外,还证明了每个图必须具有至少Ω(logn) $$ Omega left(log nright) $$的基数团的独立并集。由于这个界在均匀随机图上是渐近紧的,这表明这些图可以被看作是“最小可聚类”的图类。
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引用次数: 0
The complexity landscape of disaster-aware network extension problems 灾害感知网络扩展问题的复杂性状况
IF 2.1 4区 计算机科学 Q4 COMPUTER SCIENCE, HARDWARE & ARCHITECTURE Pub Date : 2023-12-12 DOI: 10.1002/net.22199
Balázs Vass, Beáta Éva Nagy, Balázs Brányi, János Tapolcai
This article deals with the complexity of problems related to finding cost-efficient, disaster-aware cable routes. We overview various mathematical problems studied to augment a backbone network topology to make it more robust against regional failures. These problems either consider adding a single cable, multiple cables, or even nodes too. They adapt simplistic or more sophisticated regional failure models. Their objective is to identify the network's weak points or minimize the investment cost concerning the risk of a network outage. We investigate the tradeoffs in mathematical modeling for the same real-world scenario, where more sophisticated models face more computationally challenging problems. We have seen how efficiently computational geometry algorithms can be used to solve simplified problems even for sufficiently large networks. In this article, we aim to understand why different mathematical models formulated for the same real-world scenario can or cannot be solved efficiently. In particular, we show simplistic mathematical models that formulate NP-hard problems.
本文讨论与寻找具有成本效益和灾害意识的电缆路由相关的复杂问题。我们概述了各种数学问题的研究,以增加骨干网络的拓扑结构,使其对区域故障更健壮。这些问题可以考虑添加单个电缆、多个电缆,甚至还可以添加节点。它们采用了简单或更复杂的区域失效模型。他们的目标是确定网络的弱点或最小化与网络中断风险有关的投资成本。我们研究了相同的现实世界场景的数学建模中的权衡,其中更复杂的模型面临更多的计算挑战问题。我们已经看到,即使对于足够大的网络,计算几何算法也可以有效地用于解决简化问题。在本文中,我们的目标是理解为什么为相同的现实世界场景制定的不同数学模型可以或不能有效地解决。特别是,我们展示了制定np困难问题的简单数学模型。
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