A Fast Parallel Algorithm for Minimum-Cost Small Integral Flows

We present an algorithm that computes exact maximum flows and minimum-cost flows on directed graphs with m edges and polynomially bounded integral demands, costs, and capacities in m 1+ o (1) time. Our algorithm builds the flow through a sequence of m 1+ o (1) approximate undirected minimum-ratio cycles, each of which is computed and processed in amortized m o (1) time using a new dynamic graph data structure. Our framework extends to algorithms running in m 1+ o (1) time for computing flows that minimize general edge-separable convex functions to high accuracy. This gives almost-linear time algorithms for several problems including entropy-regularized optimal transport, matrix scaling, p -norm flows, and p -norm isotonic regression on arbitrary directed acyclic graphs.

Decomposition based approaches are recalled from primal and dual point of view. The possibility of building partially disaggregated reduced master problems is investigated. This extends the idea of aggregated-versus-disaggregated formulation to a gradual choice of alternative level of aggregation. Partial aggregation is applied to the linear multicommodity minimum cost flow problem. The possibility of having only partially aggregated bundles opens a wide range of alternatives with different trade-offs between the number of iterations and the required computation for solving it. This trade-off is explored for several sets of instances and the results are compared with the ones obtained by directly solving the natural node-arc formulation. An iterative solution process to the route assignment problem is proposed, based on the well-known Frank Wolfe algorithm. In order to provide a first feasible solution to the Frank Wolfe algorithm, a linear multicommodity min-cost flow problem is solved to optimality by using the decomposition techniques mentioned above. Solutions of this problem are useful for network orientation and design, especially in relation with public transportation systems as the Personal Rapid Transit. A single-commodity robust network design problem is addressed. In this, an undirected graph with edge costs is given together with a discrete set of balance matrices, representing different supply/demand scenarios. The goal is to determine the minimum cost installation of capacities on the edges such that the flow exchange is feasible for every scenario. A set of new instances that are computationally hard for the natural flow formulation are solved by means of a new heuristic algorithm. Finally, an efficient decomposition-based heuristic approach for a large scale stochastic unit commitment problem is presented. The addressed real-world stochastic problem employs at its core a deterministic unit commitment planning model developed by the California Independent System Operator (ISO).

In this paper, we present a new strongly polynomial time algorithm for the minimum cost flow problem, based on a refinement of the Edmonds-Karp scaling technique. Our algorithm solves the uncapacitated minimum cost flow problem as a sequence of O(n log n) shortest path problems on networks with n nodes and m arcs and runs in O(n log n(m + n log n)) time. Using a standard transformation, this approach yields an O(m log n(m + n log n)) algorithm for the capacitated minimum cost flow problem. This algorithm improves the best previous strongly polynomial time algorithm, due to Z. Galil and E. Tardos, by a factor of n2/m. Our algorithm for the capacitated minimum cost flow problem is even more efficient if the number of arcs with finite upper bounds, say m′, is much less than m. In this case, the running time of the algorithm is O((m′ + n) log n(m + n log n)).

Preemptive priority-based algorithms for fuzzy minimal cost flow problem: An application in hazardous materials transportation

We consider the quickest flow problem in dynamic networks with a single source s and a single sink t: given an amount of flow F, find the minimum time needed to send it from s to t, and the corresponding optimal flow over time.We introduce new mathematical formulations and derive optimality conditions for the quickest flow problem. Based on the optimality conditions, we develop a new cost-scaling algorithm that leverages the parametric nature of the problem. The algorithm solves the quickest flow problem with integer arc costs in O(nm log(n2/m) log(nC)) time, where n, m, and C are the number of nodes, arcs, and the maximum arc cost, respectively.Our algorithm runs in the same time bound as the cost-scaling algorithm by Goldberg and Tarjan [10, 11] for solving the min-cost flow problem. This result shows for the first time that the quickest flow problem can be solved within the same time bound as one of the fastest algorithms for the min-cost flow problem. As a consequence, our algorithm will remain one of the fastest unless the quickest flow problem can be shown to be simpler than the min-cost flow problem.

Decision support for truckload carriers in one-shot combinatorial auctions

It is known that minimum cost flow problems can be solved by successive augmentations along shortest paths. In this paper the issues of implementing shortest path algorithms in this context are examined. Of particular interest is the dynamic topology that the flow networks exhibit. We develop a network generator capable of emulating such topology. Strategies for exploiting the special structures in such networks are discussed. A set of 9000 test problems is offered, from which a particular strategy/algorithm combination is shown to consistently produce superior results when compared to the other combinations.

In rider demand responsive systems riders submit requests to demand transit services and the incoming requests and vehicles are matched by the system. This demand-responsive transport problem is viewed by existing research as a kind of spatial matching problem between vehicles and riders. However, existing schemes mainly focus on maximizing the number of matching pairs. It neglects other factors like the length of pickup trajectories and riders’ waiting time. And the matching is not revocable, which loses the chance for further optimizations. Moreover, there is still not much work on handling and matching the appointment-based requests that play a key role in the demand-responsive transport market. In this article, we propose an algorithm called BMCF ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Bipartite Minimal-Cost Flow</i> ) to solve the taxi-rider matching problem with appointment-based rider requests on a time-dependent road network. Unlike existing solutions that map the problem into the max flow problem, we transform the optimal matching to a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">minimal cost flow problem</i> which aims to maximize a utility value and could be solved efficiently. Riders and vehicles are modeled as vertices in a bipartite graph, and factors like the length of pickup trajectories, riders’ waiting time, etc. are abstracted as utilities and denoted as weights of the edges which are taken into account during the matching process. To the best of our knowledge, the proposed scheme is the first to efficiently integrate and process both the real-time and appointment-based requests within the same framework, and the assignments between vehicles and requests could be dynamically adjusted and revoked to maximise the overall utility. Experimental results show that the proposed algorithm can effectively increase the appointment-based matching ratio and decrease the riders’ waiting time (> 10.4%) and vacant vehicles’ picking up time (> 60.9%) compared with existing schemes, while at the cost of acceptable increase on the running time.

Dynamic network models are widely employed for system analysis and performance evaluation of computer and communication systems. In dynamic networks each arc is weighted with a capacity and a transit time. The former is the maximum amount of flow that can travel through the arc per unit time, the later is the time period needed for flow to travel through the arc. In this article, an additional weight operation probability, is assigned to each arc to note the capability of the arc's successful transition. The performance of dynamic networks is defined by the probability that the maximum dynamic flow from source to sink within required time horizon is no less than specified threshold. To the best of our knowledge, no existing article had considered dynamic flow and network reliability simultaneously. Based on facts that there is always a maximum dynamic flow among the temporally repeated flows and the maximum temporally repeated flow can be obtained via an equivalent minimum cost maximum flow problem, we present a pioneering algorithm to compute performance of dynamic networks by solving minimum cost maximum flow and two-terminal capacity reliability at the same time.

As users drive through the vehicular networks with high speed, they frequently associate with different access points in succession to maintain connectivity. Without careful designed association control policy, it may lead to large number of handoffs and inefficient connections, which can potentially incur unacceptable delays and poor throughput. In this paper, we propose a new association control scheme with the goal of reducing the frequency of handoffs and improving throughput for all vehicular users. The defined problem HMTE (Handoff Minimization and Throughput Efficiency) is mathematically described as a min-max programming. Since the HMTE problem is NP-hard, we propose an alpha-approximation(alpha >= 2) algorithm CHMTE to resolve it. CHMTE employs a novel method to simplify the original problem to the Minimum cost Maximum flow problem, which is solved in polynomial time. Extensive evaluations show that the proposed algorithm outperforms the existing schemes in terms of the handoff frequency and network throughput.

System optimal dynamic traffic assignment: Properties and solution procedures in the case of a many-to-one network

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...

In this paper, we establish conditions for the existence of an equilibrium in the equilibrium flow problem studied by Galichon et al. (2024). The problem nests several classical economic models such as bipartite matching models, hedonic pricing models, shortest-path and minimum-cost flow problems, and time-dependent routing problems.

Chapter 1 Applications of network optimization

An introduction to dynamic generative networks: Minimum cost flow