Minimum Cost Network Flow Problem Research Articles

Scalable Computation of Dynamic Flow Problems via Multimarginal Graph-Structured Optimal Transport

In this work, we develop a new framework for dynamic network flow problems based on optimal transport theory. We show that the dynamic multicommodity minimum-cost network flow problem can be formulated as a multimarginal optimal transport problem, where the cost function and the constraints on the marginals are associated with a graph structure. By exploiting these structures and building on recent advances in optimal transport theory, we develop an efficient method for such entropy-regularized optimal transport problems. In particular, the graph structure is utilized to efficiently compute the projections needed in the corresponding Sinkhorn iterations, and we arrive at a scheme that is both highly computationally efficient and easy to implement. To illustrate the performance of our algorithm, we compare it with a state-of-the-art linear programming (LP) solver. We achieve good approximations to the solution at least one order of magnitude faster than the LP solver. Finally, we showcase the methodology on a traffic routing problem with a large number of commodities. Funding: This work was supported by KTH Digital Futures, Knut och Alice Wallenbergs Stiftelse [Grants KAW 2018.0349, KAW 2021.0274, the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation], Vetenskapsrådet [Grant 2020-03454], and the National Science Foundation [Grants 1942523 and 2206576].

Scalable Computation of Dynamic Flow Problems via Multimarginal Graph-Structured Optimal Transport

In this work, we develop a new framework for dynamic network flow problems based on optimal transport theory. We show that the dynamic multicommodity minimum-cost network flow problem can be formulated as a multimarginal optimal transport problem, where the cost function and the constraints on the marginals are associated with a graph structure. By exploiting these structures and building on recent advances in optimal transport theory, we develop an efficient method for such entropy-regularized optimal transport problems. In particular, the graph structure is utilized to efficiently compute the projections needed in the corresponding Sinkhorn iterations, and we arrive at a scheme that is both highly computationally efficient and easy to implement. To illustrate the performance of our algorithm, we compare it with a state-of-the-art linear programming (LP) solver. We achieve good approximations to the solution at least one order of magnitude faster than the LP solver. Finally, we showcase the methodology on a traffic routing problem with a large number of commodities. Funding: This work was supported by KTH Digital Futures, Knut och Alice Wallenbergs Stiftelse [Grants KAW 2018.0349, KAW 2021.0274, the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation], Vetenskapsrådet [Grant 2020-03454], and the National Science Foundation [Grants 1942523 and 2206576].

Risk‐averse optimization and resilient network flows

AbstractWe propose an approach to constructing metrics of network resilience, where resilience is understood as the network's amenability to restoring its optimal or near‐optimal operations subsequent to unforeseen (stochastic) disruptions of its topology or operational parameters, and illustrated it on the examples of the resilient maximum network flow problem and the resilient minimum cost network problem. Specifically, the network flows in these problems are designed for resilience against unpredictable losses of network carrying capacity, and the mechanism of attaining a degree of resilience is through preallocation of resources toward (at least partial) restoration of the capacities of the arcs. The obtained formulations of resilient network flow problems possess a number of useful properties, for example, similarly to the standard network flow problems, the network flow is integral if the arc capacities, costs, and so forth, are integral. It is also shown that the proposed formulations of resilient network flow problems can be viewed as “network measures of risk”, similar in properties and behavior to convex measures of risk. Efficient decomposition algorithms have been proposed for both the resilient maximum network flow problem and the resilient minimum cost network flow problem, and a study of the network flow resilience as a function of network's structure has been conducted on networks with three types of topology: that of uniform random graphs, scale‐free graphs, and grid graphs.

Modeling Minimum Cost Network Flows With Port‐Hamiltonian Systems

AbstractWe give a short overview of advantages and drawbacks of the classical formulation of minimum cost network flow problems and solution techniques, to motivate a reformulation of classical static minimum cost network flow problems as optimal control problems constrained by port‐Hamiltonian systems (pHS). The first‐order optimality system for the port‐Hamiltonian system‐constrained optimal control problem is formally derived. Then we propose a gradient‐based algorithm to find optimal controls. The port‐Hamiltonian system formulation naturally conserves flow and supports a wide array of further modeling options as, for example, node reservoirs, flow dependent costs, leaking pipes (dissipation) and coupled sub‐networks (ports). They thus provide a versatile alternative to state‐of‐the art approaches towards dynamic network flow problems, which are often based on computationally costly time‐expanded networks. We argue that this opens the door for a plethora of modeling options and solution approaches for network flow problems.

Joint chance-constrained multi-objective multi-commodity minimum cost network flow problem with copula theory

This study focuses on a specific problem in network optimization, namely the minimum cost multi-commodity network flow (MCNF) problem. The problem is complicated by the presence of uncertain parameters, including various types of costs associated with each arc in the network. The study presents a multi-objective approach to solving this problem, where the coefficients of the capacity constraints are modelled as random variables with a normal distribution, and the dependence between them is modelled using an Archimedean copula. The capacity constraints are presented as joint chance constraints, and a multi-objective problem is formulated to deal with the uncertainty. This uncertain multi-objective problem is then converted into a certain multi-objective problem using fuzzy programming. The resulting certain multi-objective MCNF problem is converted to a certain single objective problem using second-order cone programming (SOCP), which is solved using either piecewise tangent approximation or piecewise linear approximation methods. The proposed approaches are tested using numerical examples and experimental tests to demonstrate their effectiveness in solving large-scale network problems efficiently. The results show that the proposed approaches are a useful tool for solving uncertain multi-objective MCNF problems in real-world applications.

A linear input dependence model for interdependent networks

We consider a linear relaxation of a generalized minimum-cost network flow problem with binary input dependencies. In this model the flows through certain arcs are bounded by linear (or more generally, piecewise linear concave) functions of the flows through other arcs. This formulation can be used to model interrelated systems in which the components of one system require the delivery of material from another system in order to function (for example, components of a subway system may require delivery of electrical power from a separate system). We propose and study randomized rounding schemes for how this model can be used to approximate solutions to a related mixed integer linear program for modeling binary input dependencies. The introduction of side constraints prevents this problem from being solved using the well-known network simplex algorithm, however by characterizing its basis structure we develop a generalization of network simplex algorithm that can be used for its computationally efficient solution.

Algorithm of Ecocompensation in Sloping Land Conversion Program Based on Heckman’s Two-Step Model

In this paper, we broadly generalize the assignment auction algorithm to solve linear minimum cost network flow problems. It is significant to establish a market-based compensation mechanism by way of conservation auctions based on peasant households’ willingness, which can promote the innovation of ecocompensation policies, green development, and balanced growth. Using the survey data collected from 453 households within 3 national pilot counties in ecologically fragile regions in northwest Liaoning for the Sloping Land Conversion Programme, measuring peasant households’ willingness to accept ecocompensation through sealed auctions, we built a database through cloud computing to realize information collation and query and applied the Heckman’s Two-Step Model to study the impact of risk preference, social capital, cognitive preference, land parcel characteristics, and family endowments on farmers’ willingness to participate in protection auctions and their bid prices. The results reveal that the average bid price of peasant households in the ecologically fragile region in northwest Liaoning for the Sloping Land Conversion Programme is annually 274.5 yuan per mu and that risk preference and social capital have positive impacts on peasant households’ willingness to participate in conservation auctions and on their bid prices, cognitive preference has a positive impact on peasant households’ bid prices in conservation auctions, and land plot characteristics have a negative impact on peasant households’ bid prices in conservation auctions. It is suggested that ecocompensation policies should be optimized with such methods as lowering peasant households’ perception of high risks, setting role models for them to follow, and strengthening their perception of the environment, income, and property rights.

A Bi-Objective Simulation-Based Optimization Approach for Optimizing Price, Warranty, and Spare Part Production Decisions Under Imperfect Repair

This paper proposes an efficient methodology based on the Monte-Carlo simulation-based bi-objective optimization, to determine base-warranty (BW) and extended warranty (EW) parameters based on the product lifecycle. The first objective, which is from the manufacturer’s perspective, maximizes the profit while the second objective minimizes the expected number of failures that occurred during the out-of-warranty period. The manufacturer can rectify failed products via minimal repair, imperfect repair and perfect repair. The optimization model has decision variables including the product price, BW length, EW length, EW price, product failure rate, imperfect repair level, and spare part production rate in each time interval. The structure of the model admits the design of a hybrid method based on the multi-objective optimization search algorithm, Monte-Carlo simulation and an exact Out-Of-Kilter (OOK) algorithm. The nondominated sorting genetic algorithm (NSGA-II) and the multi-objective particle swarm optimization (MOPSO) algorithm are used as search algorithms. The proposed approach consists of three stages, where in the first stage, product price, BW length, EW length, EW price, product failure rate, imperfect repair level are set by NSGA-II/MOPSO. In the second stage, the number of failed products is calculated by the Monte-Carlo simulation and in the third stage, we show that the spare part inventory control sub problem can be transformed to a minimum cost network flow problem which is optimized by the OOK algorithm to attain a unified solution. A Taguchi approach is used to find the optimum level of parameters. The performance of algorithms is compared based on three different metrics. Results on a real-world problem demonstrate that the NSGA-II-OOK algorithm is more effective than the MOPSO-OOK algorithm. Through a sensitivity analysis, we analyze how various levels of planning horizon can affect Pareto-set which indicates valuable managerial insight.

Knights Exchange Puzzle—Teaching the Efficiency of Modeling

Puzzles and games enhance the quality of teaching by creating an enjoyable, interactive, and playful atmosphere. The knight exchange is a famous, very old, and amusing game on the chessboard. This puzzle was used by the author to teach modeling in a mathematical programming course designed for graduate students. The aim was to teach the students the efficiency of the models. Accordingly, first, a binary programming formulation was developed. This formulation was, however, found to be inefficient, and tremendous time (i.e., more than four hours) and a large amount of processing memory were needed to solve the puzzle. The puzzle was subsequently formulated as a minimum cost network flow problem. The latter formulation outperformed the general binary formulation by solving the puzzle in less than a minute. The network formulation could also save the required processing memory. The results could help students to learn the value of modeling combinatorial optimization problems as network flows.

Video Game Level Repair via Mixed Integer Linear Programming

Recent advancements in procedural content generation via machine learning enable the generation of video-game levels that are aesthetically similar to human-authored examples. However, the generated levels are often unplayable without additional editing. We propose a “generate-then-repair” framework for automatic generation of playable levels adhering to specific styles. The framework constructs levels using a generative adversarial network (GAN) trained with human-authored examples and repairs them using a mixed-integer linear program (MIP) with playability constraints. A key component of the framework is computing minimum cost edits between the GAN generated level and the solution of the MIP solver, which we cast as a minimum cost network flow problem. Results show that the proposed framework generates a diverse range of playable levels, that capture the spatial relationships between objects exhibited in the human-authored levels.

Optimal Steering of Ensembles With Origin-Destination Constraints

We consider the optimal control problem of steering a collection of agents over a network. The group behavior of an ensemble is often modeled by a distribution, and thus the optimal control problem we study can be cast as a distribution steering problem. While most existing works for steering distributions require the agents in the ensemble to be indistinguishable, we consider the setting where agents have specified origin-destination constraints. This control problem also resembles a minimum cost network flow problem with a massive number of commodities. We propose a novel optimal transport based framework for this problem and derive an efficient algorithm for solving it. This framework extends multi-marginal optimal transport theory to settings with capacity and origin-destination constraints. The proposed method is illustrated on a numerical simulation for traffic planning.

Long-Term Action Dependence-Based Hierarchical Deep Association for Multi-Athlete Tracking in Sports Videos

Tracking multiple athletes in sports videos is a very challenging Multi-Object Tracking (MOT) task, as athletes generally share high similarity in appearance with large deformations. In this paper, unlike the existing hand-crafted solutions, we propose a novel and effective approach to this issue, which hierarchically associates detections of the same identity through discriminative and robust deep features. First, in detection association, we make use of athlete appearances and poses instead of traditional position cues to generate short tracklets for better initialization. Second, in tracklet association, a new deep architecture, namely Siamese Tracklet Affinity Networks (STAN), is presented, which is able to bi-directionally simulate the unseen dynamics of actions, comprehensively models the long-term action dependences, and sequentially estimates their affinity. Such hierarchical association is finally solved as a minimum-cost network flow problem. We extensively evaluate the proposed approach on the APIDIS, NCAA Basketball and VolleyTrack (newly collected) databases, and the experimental results show its advantages.

Facets of the Stochastic Network Flow Problem

We study a type of network flow problem that we call the minimum-cost F-graph flow problem. This problem generalizes the typical minimum-cost network flow problem by allowing the underlying network...

Convex Relaxations of the Network Flow Problem Under Cycle Constraints

In this paper, we consider the problem of optimizing the flows in a lossless flow network with additional nonconvex cycle constraints on nodal variables; such constraints appear in several applications, including electric power, water distribution, and natural gas networks. This problem is a nonconvex version of the minimum-cost network flow problem (NFP), and to solve it, we propose three different approaches. One approach is based on solving a convex approximation of the problem, obtained by augmenting the cost function with an entropy-like term to relax the nonconvex constraints. We show that the approximation error, for which we give an upper bound, can be made small enough for practical use. An alternative approach is to solve the classical NFP, that is, without the cycle constraints, and solve a separate optimization problem afterwards in order to recover the actual flows satisfying the cycle constraints; the solution of this separate problem maximizes the individual entropies of the cycles. The third approach is based on replacing the nonconvex constraint set with a convex inner approximation, which yields a suboptimal solution for the cyclic networks with each edge belonging to at most two cycles. We validate the practical usefulness of the theoretical results through numerical examples, in which we study the standard test systems for water and electric power distribution networks.

Facets of the Stochastic Network Flow Problem

We study a type of network flow problem that we call the minimum-cost F-graph flow problem. This problem generalizes the typical minimum-cost network flow problem by allowing the underlying network to be a directed hypergraph rather than a directed graph. This new problem is pertinent because it can be used to model network flow problems that occur in a dynamic, stochastic, environment. We formulate this problem as an integer program, and we study specifically the case where every node has at least one outgoing edge with no capacity constraint. We show that even with this restriction, the problem of finding an integral solution is NP-Hard. However, we can show that all of the inequality constraints of our formulation are either facet-defining or redundant.

The 2-allocation p-hub median problem and a modified Benders decomposition method for solving hub location problems

We study the uncapacitated 2-allocation p-hub median problem (U2ApHMP), which is a special case of the well-studied hub median problem. The hub median problem designs a hub network in which the location of p hubs needs to be decided (the hubs are fully interconnected). The other nodes (known as access nodes) in the hub median problem are then allocated to one or many hubs. In the U2ApHMP, each access node is allocated to exactly two hubs. We discuss how this problem provides an alternative network design option for well-known p-hub median problems. We show its relevance and usefulness in the context of survivable network design and show that it addresses network survivability, a feature that has often been largely overlooked in hub network design research to date. We show that U2ApHMP is NP-hard even for a fixed/known set of hubs. We propose a mathematical formulation and develop a modified Benders decomposition method for this problem. In this, we convert the corresponding subproblems to minimum cost network flow problems. This allows us to solve large instances efficiently. We believe that, while our resulting method solves the U2ApHMP efficiently, it is also generalisable and can potentially be employed for solving other classes and types of hub location problems too.

Optimizing under and out of warranty products’ decisions in the finite planning horizon

In this paper, we consider a manufacturer that produces products in a finite horizon time and sells products with non-renewing free replacement warranty policy. The manufacturer is responsible to provide spare parts for failed products, whether the products are under or out of warranty. Previous research on warranty optimization has focused on maximizing manufacturer profit without considering the spare part market for out-of-warranty products. This study proposes a novel nonlinear model that maximizes manufacturer profit by optimization of price, warranty length and spare part inventory for under- and out-of-warranty products in a manufacturing/remanufacturing system. Due to the model’s unique structure, we propose a new two-stage approach that combines metaheuristic and an exact method, in which the first stage is to determine product’s prices and warranty length with metaheuristic algorithm and in the second stage the remaining inventory related problem is transferred to a Minimum Cost Network Flow Problem which is solved for spare part inventory control. To illustrate the effectiveness of the suggested method, the model is solved for a case study of Iranian SANAM electronic company with two different metaheuristic algorithms and a sensitivity analysis is conducted to study the effect of various parameters on the optimal solution.

A Network Flow Model for Price-Responsive Control of Deferrable Load Profiles

This paper describes a minimum cost network flow model for the aggregated control of deferrable load profiles. The load aggregator responds to indicative energy price information and uses this model to formulate and submit a flexibility bid to a high-resolution real-time balancing market, as proposed by the SmartNet project. This bid represents the possibility of the cluster of deferrable loads to deviate from the scheduled consumption, in case the bid is accepted. When formulating this bid, the model is able to take into account the discretized power profiles of the individual loads. The solution of this type of aggregation problems is necessary for the participation of small loads in demand response programs, but scalability can be an issue. The minimum cost network flow problem belongs to a restricted class of discrete optimization problems for which efficient and scalable algorithms exist. Thanks to its scalability, this technique can be useful in the control of a large number of smart appliances in future real-time balancing markets. The technique is efficient enough to be employed by an aggregation module with limited computational resources. Alternatively, when indicative price information is not made available by the system operator, the technique can be combined with an external forecast in order to minimize possible imbalance costs.

Scheduling of Multiple Autonomous Guided Vehicles for an Assembly Line Using Minimum Cost Network Flow

This paper proposed a parallel automated assembly line system to produce multiple products having multiple autonomous guided vehicles (AGVs). Several assembly lines are configured to produce multiple products in which the technologies of machines are shared among the assembly lines when required. The transportation between the stations in an assembly line (intra assembly line) and among stations in different assembly lines (inter assembly line) are performed using AGVs. Scheduling of AGVs to service the assembly lines and the corresponding stations are purposed. In the proposed problem the assignment of multiple AGVs to different assembly lines and the stations are performed using minimum-cost network flow (MCF). It optimizes weighted completion time of tasks for each short-term window by formulating the task and resource assignment problem as MCF problem during each short-term scheduling window.

Quadratically Regularized Optimal Transport on Graphs

Optimal transportation provides a means of lifting distances between points on a geometric domain to distances between signals over the domain, expressed as probability distributions. On a graph, transportation problems can be used to express challenging tasks involving matching supply to demand with minimal shipment expense; in discrete language, these become minimum-cost network flow problems. Regularization typically is needed to ensure uniqueness for the linear ground distance case and to improve optimization convergence. In this paper, we characterize a quadratic regularizer for transport with linear ground distance over a graph. We theoretically analyze the behavior of quadratically regularized graph transport, characterizing how regularization affects the structure of flows in the regime of small but nonzero regularization. We further exploit elegant second-order structure in the dual of this problem to derive an easily implemented Newton-type optimization algorithm.