The service network design problem is commonly used to represent the tactical decisions encountered by a consolidation carrier operating a hub‐and‐spoke network: what transportation services to operate between hubs and how to route commodities from their origin to their destination through the network. In most settings, the capacity at hubs is not a limiting factor and can safely be ignored. However, in the context of city logistics networks, where space is limited and expensive, hub capacities typically have to be taken into account. The presence of hub capacity (and time) constraints implies that, contrary to traditional service network design problems, the existence of a feasible solution is no longer guaranteed. We present an exact dynamic discretization discovery algorithm for a variant of the service network design problem in which the number of vehicles that can be loaded and unloaded simultaneously at a hub is restricted. Novel techniques are introduced in the algorithm to handle the hub capacity constraints. A computational study using instances derived from real‐world data shows the potential of dynamic discretization discovery for this class of problems: integer program sizes are reduced by a factor of up to one thousand and small to mid size instances can be (optimally) solved in an acceptable amount of time.
{"title":"An exact algorithm for the service network design problem with hub capacity constraints","authors":"E. He, N. Boland, G. Nemhauser, M. Savelsbergh","doi":"10.1002/net.22128","DOIUrl":"https://doi.org/10.1002/net.22128","url":null,"abstract":"The service network design problem is commonly used to represent the tactical decisions encountered by a consolidation carrier operating a hub‐and‐spoke network: what transportation services to operate between hubs and how to route commodities from their origin to their destination through the network. In most settings, the capacity at hubs is not a limiting factor and can safely be ignored. However, in the context of city logistics networks, where space is limited and expensive, hub capacities typically have to be taken into account. The presence of hub capacity (and time) constraints implies that, contrary to traditional service network design problems, the existence of a feasible solution is no longer guaranteed. We present an exact dynamic discretization discovery algorithm for a variant of the service network design problem in which the number of vehicles that can be loaded and unloaded simultaneously at a hub is restricted. Novel techniques are introduced in the algorithm to handle the hub capacity constraints. A computational study using instances derived from real‐world data shows the potential of dynamic discretization discovery for this class of problems: integer program sizes are reduced by a factor of up to one thousand and small to mid size instances can be (optimally) solved in an acceptable amount of time.","PeriodicalId":54734,"journal":{"name":"Networks","volume":"80 1","pages":"572 - 596"},"PeriodicalIF":2.1,"publicationDate":"2022-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42481276","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}
The covering of a graph with (possibly disjoint) connected subgraphs is a fundamental problem in graph theory. In this paper, we study a version to cover a graph's vertices by connected subgraphs subject to lower and upper weight bounds, and propose a column generation approach to dynamically generate feasible and promising subgraphs. Our focus is on the solution of the pricing problem which turns out to be a variant of the NP‐hard Maximum Weight Connected Subgraph Problem. We compare different formulations to handle connectivity, and find that a single‐commodity flow formulation performs best. This is notable since the respective literature seems to have widely dismissed this formulation. We improve it to a new coarse‐to‐fine flow formulation that is theoretically and computationally superior, especially for large instances with many vertices of degree 2 like highway networks, where it provides a speed‐up factor of 5 over the non‐flow‐based formulations. We also propose a preprocessing method that exploits a median property of weight‐constrained subgraphs, a primal heuristic, and a local search heuristic. In an extensive computational study we evaluate the presented connectivity formulations on different classes of instances, and demonstrate the effectiveness of the proposed enhancements. Their speed‐ups essentially multiply to an overall factor of well over 10. Overall, our approach allows the reliable solution of instances with several hundreds of vertices in a few minutes. These findings are further corroborated in a comparison to existing districting models on a set of test instances from the literature.
{"title":"Vertex covering with capacitated trees","authors":"R. Borndörfer, Stephan Schwartz, William Surau","doi":"10.1002/net.22130","DOIUrl":"https://doi.org/10.1002/net.22130","url":null,"abstract":"The covering of a graph with (possibly disjoint) connected subgraphs is a fundamental problem in graph theory. In this paper, we study a version to cover a graph's vertices by connected subgraphs subject to lower and upper weight bounds, and propose a column generation approach to dynamically generate feasible and promising subgraphs. Our focus is on the solution of the pricing problem which turns out to be a variant of the NP‐hard Maximum Weight Connected Subgraph Problem. We compare different formulations to handle connectivity, and find that a single‐commodity flow formulation performs best. This is notable since the respective literature seems to have widely dismissed this formulation. We improve it to a new coarse‐to‐fine flow formulation that is theoretically and computationally superior, especially for large instances with many vertices of degree 2 like highway networks, where it provides a speed‐up factor of 5 over the non‐flow‐based formulations. We also propose a preprocessing method that exploits a median property of weight‐constrained subgraphs, a primal heuristic, and a local search heuristic. In an extensive computational study we evaluate the presented connectivity formulations on different classes of instances, and demonstrate the effectiveness of the proposed enhancements. Their speed‐ups essentially multiply to an overall factor of well over 10. Overall, our approach allows the reliable solution of instances with several hundreds of vertices in a few minutes. These findings are further corroborated in a comparison to existing districting models on a set of test instances from the literature.","PeriodicalId":54734,"journal":{"name":"Networks","volume":"81 1","pages":"253 - 277"},"PeriodicalIF":2.1,"publicationDate":"2022-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47850663","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}
In the Pickup‐and‐Delivery Traveling Salesman Problem with Handling Costs (PDTSPH), a single vehicle has to satisfy multiple customer requests, each defined by a pickup location and a delivery location. Cargo handling is performed at the rear end of the vehicle, in a Last‐In‐First‐Out (LIFO) order for PDTSPH. However, additional handling operations are permitted with a penalty if other loads that block the access to the delivery have to be unloaded and reloaded. The objective of PDTSPH is to minimize the total transportation and handling cost. In this paper, we present a new Mixed Integer Programming (MIP) model and a branch‐and‐cut algorithm to solve PDTSPH. We also present new integral separation procedures to effectively handle the exponential number of constraints in our MIP model. A family of inequalities are introduced to enhance the scalability of our implementation. The performance of our approach is compared with a compact formulation from the literature (Veenstra et al. [21]) in instances ranging from 9 to 21 customer requests. Computational results show our algorithm outperforming the compact formulation in 69% of instances with an average runtime improvement of 57%.
{"title":"A branch‐and‐cut algorithm for the pickup‐and‐delivery traveling salesman problem with handling costs","authors":"D. Krishnan, Tieming Liu","doi":"10.1002/net.22096","DOIUrl":"https://doi.org/10.1002/net.22096","url":null,"abstract":"In the Pickup‐and‐Delivery Traveling Salesman Problem with Handling Costs (PDTSPH), a single vehicle has to satisfy multiple customer requests, each defined by a pickup location and a delivery location. Cargo handling is performed at the rear end of the vehicle, in a Last‐In‐First‐Out (LIFO) order for PDTSPH. However, additional handling operations are permitted with a penalty if other loads that block the access to the delivery have to be unloaded and reloaded. The objective of PDTSPH is to minimize the total transportation and handling cost. In this paper, we present a new Mixed Integer Programming (MIP) model and a branch‐and‐cut algorithm to solve PDTSPH. We also present new integral separation procedures to effectively handle the exponential number of constraints in our MIP model. A family of inequalities are introduced to enhance the scalability of our implementation. The performance of our approach is compared with a compact formulation from the literature (Veenstra et al. [21]) in instances ranging from 9 to 21 customer requests. Computational results show our algorithm outperforming the compact formulation in 69% of instances with an average runtime improvement of 57%.","PeriodicalId":54734,"journal":{"name":"Networks","volume":"80 1","pages":"297 - 313"},"PeriodicalIF":2.1,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48752377","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}
This article introduces the consistent production routing problem in a setting with multiple plants and products. The problem consists in finding minimum‐cost production‐routing plans that also meet specific consistency requirements. In our context, consistency is defined as the degree to which some specified features of the solution remain invariant over time. We consider four forms of consistency, namely: driver, source, product, and plant consistency. For each of these consistency requirements, there is a target maximum value defining the decision‐maker's tolerance to deviations from a perfectly consistent solution. These targets are enforced as soft constraints whose violations need to be minimized when optimizing the integrated production and routing plan. We present a mathematical formulation for the problem and an exact branch‐and‐cut algorithm, enhanced with valid inequalities and specific branching priorities. We also propose a heuristic solution method based on iterated local search and several mathematical programming components. Experiments on a large benchmark set of newly introduced instances show that the enhancements substantially improve the performance of the exact algorithm and that the heuristic method performs robustly for production routing problems with different consistency requirements as well as for standard versions of the problem. We also analyze the cost‐consistency trade‐off of the solutions, confirming that it is possible to impose consistency without excessively increasing the cost. The results also reveal the impact of the first time period when optimizing and measuring the consistency features we study.
{"title":"The consistent production routing problem","authors":"Aldair Alvarez, J. Cordeau, R. Jans","doi":"10.1002/net.22112","DOIUrl":"https://doi.org/10.1002/net.22112","url":null,"abstract":"This article introduces the consistent production routing problem in a setting with multiple plants and products. The problem consists in finding minimum‐cost production‐routing plans that also meet specific consistency requirements. In our context, consistency is defined as the degree to which some specified features of the solution remain invariant over time. We consider four forms of consistency, namely: driver, source, product, and plant consistency. For each of these consistency requirements, there is a target maximum value defining the decision‐maker's tolerance to deviations from a perfectly consistent solution. These targets are enforced as soft constraints whose violations need to be minimized when optimizing the integrated production and routing plan. We present a mathematical formulation for the problem and an exact branch‐and‐cut algorithm, enhanced with valid inequalities and specific branching priorities. We also propose a heuristic solution method based on iterated local search and several mathematical programming components. Experiments on a large benchmark set of newly introduced instances show that the enhancements substantially improve the performance of the exact algorithm and that the heuristic method performs robustly for production routing problems with different consistency requirements as well as for standard versions of the problem. We also analyze the cost‐consistency trade‐off of the solutions, confirming that it is possible to impose consistency without excessively increasing the cost. The results also reveal the impact of the first time period when optimizing and measuring the consistency features we study.","PeriodicalId":54734,"journal":{"name":"Networks","volume":"80 1","pages":"356 - 381"},"PeriodicalIF":2.1,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47468107","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}
Raquel Bernardino, L. Gouveia, Ana Paias, Daniel Santos
In this article, we study the multi‐depot family traveling salesman problem (MDFTSP) and two clustered variants, the soft‐clustered MDFTSP (SC‐MDFTSP) and the hard‐clustered MDFTSP. We emphasize the relevance of this study by relating the problems with warehouse activities supported by scattered storage systems and by pointing out that clustered variants of routing problems have been scarcely addressed in the literature. For these three problems, we present several mixed integer linear programming formulations and develop appropriate branch‐&‐cut based algorithms which are tested with a newly generated data set including instances with up to 200 nodes and 40 depots. The results from the computational experiments allow us to identify the main differences between the three problems concerning modeling approaches as well as solution methods and put in evidence that these problems are challenging problems, in particular the SC‐MDFTSP.
{"title":"The multi‐depot family traveling salesman problem and clustered variants: Mathematical formulations and branch‐&‐cut based methods","authors":"Raquel Bernardino, L. Gouveia, Ana Paias, Daniel Santos","doi":"10.1002/net.22125","DOIUrl":"https://doi.org/10.1002/net.22125","url":null,"abstract":"In this article, we study the multi‐depot family traveling salesman problem (MDFTSP) and two clustered variants, the soft‐clustered MDFTSP (SC‐MDFTSP) and the hard‐clustered MDFTSP. We emphasize the relevance of this study by relating the problems with warehouse activities supported by scattered storage systems and by pointing out that clustered variants of routing problems have been scarcely addressed in the literature. For these three problems, we present several mixed integer linear programming formulations and develop appropriate branch‐&‐cut based algorithms which are tested with a newly generated data set including instances with up to 200 nodes and 40 depots. The results from the computational experiments allow us to identify the main differences between the three problems concerning modeling approaches as well as solution methods and put in evidence that these problems are challenging problems, in particular the SC‐MDFTSP.","PeriodicalId":54734,"journal":{"name":"Networks","volume":"80 1","pages":"502 - 571"},"PeriodicalIF":2.1,"publicationDate":"2022-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41733112","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}
EDWIN BAKER GAGER Edwin Baker Gager graduated from Yale College in 1877. He was admitted to the bar in I881 and became a member of the firm of Wooster & Torrance in Derby. On the appointment of Judge Torrance to the Superior Court the name of the firm was changed to Wooster, Williams & Gager and on the death of the senior partner to Williams & Gager. Judge Gager was appointed to the Superior Court in i9oi, and after seventeen years of service as a trial judge he became in 1918 an associate justice of the Supreme Court of Errors. From 1892 to 19o3 he was an instructor in law aid jurisprudence in the Yale School of Law, and in 1903 was appointed Professor of General Jurisprudence. As a lawyer he soon gained the confidence of the large business interests located in the Naugatuck Valley, and in the later years of his practice his counsel and advocacy were claimed chiefly by street railway, gas, and electric companies. He brought to the office of trial judge the poise of a man of affairs, the experience of an extensive practice, the learning of a widely read
{"title":"Editorial","authors":"B. Golden, D. Shier","doi":"10.1002/net.22126","DOIUrl":"https://doi.org/10.1002/net.22126","url":null,"abstract":"EDWIN BAKER GAGER Edwin Baker Gager graduated from Yale College in 1877. He was admitted to the bar in I881 and became a member of the firm of Wooster & Torrance in Derby. On the appointment of Judge Torrance to the Superior Court the name of the firm was changed to Wooster, Williams & Gager and on the death of the senior partner to Williams & Gager. Judge Gager was appointed to the Superior Court in i9oi, and after seventeen years of service as a trial judge he became in 1918 an associate justice of the Supreme Court of Errors. From 1892 to 19o3 he was an instructor in law aid jurisprudence in the Yale School of Law, and in 1903 was appointed Professor of General Jurisprudence. As a lawyer he soon gained the confidence of the large business interests located in the Naugatuck Valley, and in the later years of his practice his counsel and advocacy were claimed chiefly by street railway, gas, and electric companies. He brought to the office of trial judge the poise of a man of affairs, the experience of an extensive practice, the learning of a widely read","PeriodicalId":54734,"journal":{"name":"Networks","volume":" ","pages":""},"PeriodicalIF":2.1,"publicationDate":"2022-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42890156","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}
D. Ferone, P. Festa, Serena Fugaro, Tommaso Pastore
In this article, the Resource Constrained Clustered Shortest Path Tree Problem is defined. It generalizes the classic Resource Constrained Shortest Path Tree Problem since it is defined on an undirected, complete and weighted graph whose set of nodes is partitioned into clusters. The aim is then to find a shortest path tree respecting some resource consumption constraints and inducing a connected subgraph within each cluster. The main support and motivation for studying this problem are related, among the others, to the design of telecommunication networks, and to Disaster Operations Management. In this work, we present a path‐based formulation for the problem, addressing the case of local resource constraints, that is, resource constraints on single paths. For its resolution, a Branch&Price algorithm featuring a Column Generation approach with Multiple Pricing Scheme is devised. A comprehensive computational study is conducted, comparing the proposed method with the results achieved by the CPLEX solver, adopted to solve the mathematical model. The numerical results underline that the Branch&Price algorithm outperforms CPLEX, both in terms of solution cost and time.
{"title":"The resource constrained clustered shortest path tree problem: Mathematical formulation and Branch&Price solution algorithm","authors":"D. Ferone, P. Festa, Serena Fugaro, Tommaso Pastore","doi":"10.1002/net.22124","DOIUrl":"https://doi.org/10.1002/net.22124","url":null,"abstract":"In this article, the Resource Constrained Clustered Shortest Path Tree Problem is defined. It generalizes the classic Resource Constrained Shortest Path Tree Problem since it is defined on an undirected, complete and weighted graph whose set of nodes is partitioned into clusters. The aim is then to find a shortest path tree respecting some resource consumption constraints and inducing a connected subgraph within each cluster. The main support and motivation for studying this problem are related, among the others, to the design of telecommunication networks, and to Disaster Operations Management. In this work, we present a path‐based formulation for the problem, addressing the case of local resource constraints, that is, resource constraints on single paths. For its resolution, a Branch&Price algorithm featuring a Column Generation approach with Multiple Pricing Scheme is devised. A comprehensive computational study is conducted, comparing the proposed method with the results achieved by the CPLEX solver, adopted to solve the mathematical model. The numerical results underline that the Branch&Price algorithm outperforms CPLEX, both in terms of solution cost and time.","PeriodicalId":54734,"journal":{"name":"Networks","volume":"81 1","pages":"204 - 219"},"PeriodicalIF":2.1,"publicationDate":"2022-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48289592","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}
In a temporal graph, each edge appears and can be traversed at specific points in time. In such a graph, temporal reachability of one node from another is naturally captured by the existence of a temporal path where edges appear in chronological order. Inspired by the optimization of bus/metro/tramway schedules in a public transport network, we consider the problem of turning a collection of walks (called trips) in a directed graph into a temporal graph by assigning a starting time to each trip in order to maximize the reachability among pairs of nodes. Each trip represents the trajectory of a vehicle and its edges must be scheduled one right after another. Setting a starting time to the trip thus forces the appearance time of all its edges. We call such a starting time assignment a trip temporalization. We obtain several results about the complexity of maximizing reachability via trip temporalization. Among them, we show that maximizing reachability via trip temporalization is hard to approximate within a factor n/12$$ sqrt{n}/12 $$ in an n$$ n $$ ‐vertex digraph, even if we assume that for each pair of nodes, there exists a trip temporalization connecting them. On the positive side, we show that there must exist a trip temporalization connecting a constant fraction of all pairs if we additionally assume symmetry, that is, when the collection of trips to be scheduled is such that, for each trip, there is a symmetric trip visiting the same nodes in reverse order.
{"title":"Maximizing reachability in a temporal graph obtained by assigning starting times to a collection of walks","authors":"Filippo Brunelli, P. Crescenzi, L. Viennot","doi":"10.1002/net.22123","DOIUrl":"https://doi.org/10.1002/net.22123","url":null,"abstract":"In a temporal graph, each edge appears and can be traversed at specific points in time. In such a graph, temporal reachability of one node from another is naturally captured by the existence of a temporal path where edges appear in chronological order. Inspired by the optimization of bus/metro/tramway schedules in a public transport network, we consider the problem of turning a collection of walks (called trips) in a directed graph into a temporal graph by assigning a starting time to each trip in order to maximize the reachability among pairs of nodes. Each trip represents the trajectory of a vehicle and its edges must be scheduled one right after another. Setting a starting time to the trip thus forces the appearance time of all its edges. We call such a starting time assignment a trip temporalization. We obtain several results about the complexity of maximizing reachability via trip temporalization. Among them, we show that maximizing reachability via trip temporalization is hard to approximate within a factor n/12$$ sqrt{n}/12 $$ in an n$$ n $$ ‐vertex digraph, even if we assume that for each pair of nodes, there exists a trip temporalization connecting them. On the positive side, we show that there must exist a trip temporalization connecting a constant fraction of all pairs if we additionally assume symmetry, that is, when the collection of trips to be scheduled is such that, for each trip, there is a symmetric trip visiting the same nodes in reverse order.","PeriodicalId":54734,"journal":{"name":"Networks","volume":"81 1","pages":"177 - 203"},"PeriodicalIF":2.1,"publicationDate":"2022-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44155083","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}
In this article, we study robust transshipment under consistent flow constraints. We consider demand uncertainty represented by a finite set of scenarios and characterize a subset of arcs as so‐called fixed arcs. In each scenario, we require an integral flow that satisfies the respective flow balance constraints. In addition, on each fixed arc, we require equal flow for all scenarios. The objective is to minimize the maximum cost occurring among all scenarios. We show that the problem is strongly ‐complete on acyclic digraphs by a reduction from the ‐Sat problem. Furthermore, we prove that the problem is weakly ‐complete on series‐parallel digraphs by a reduction from a special case of the Partition problem. If in addition the number of scenarios is constant, we observe the pseudo‐polynomial‐time solvability of the problem. We provide poly‐nomial‐time algorithms for three special cases on series‐parallel digraphs. Finally, we present a polynomial‐time algorithm for pearl digraphs.
{"title":"Robust transshipment problem under consistent flow constraints","authors":"Christina Büsing, A. Koster, S. Schmitz","doi":"10.1002/net.22184","DOIUrl":"https://doi.org/10.1002/net.22184","url":null,"abstract":"In this article, we study robust transshipment under consistent flow constraints. We consider demand uncertainty represented by a finite set of scenarios and characterize a subset of arcs as so‐called fixed arcs. In each scenario, we require an integral flow that satisfies the respective flow balance constraints. In addition, on each fixed arc, we require equal flow for all scenarios. The objective is to minimize the maximum cost occurring among all scenarios. We show that the problem is strongly ‐complete on acyclic digraphs by a reduction from the ‐Sat problem. Furthermore, we prove that the problem is weakly ‐complete on series‐parallel digraphs by a reduction from a special case of the Partition problem. If in addition the number of scenarios is constant, we observe the pseudo‐polynomial‐time solvability of the problem. We provide poly‐nomial‐time algorithms for three special cases on series‐parallel digraphs. Finally, we present a polynomial‐time algorithm for pearl digraphs.","PeriodicalId":54734,"journal":{"name":"Networks","volume":"14 6","pages":""},"PeriodicalIF":2.1,"publicationDate":"2022-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50822310","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}
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