A polynomial time primal network simplex algorithm for minimum cost flows

In this work, we contribute a parallel implementation of the network simplex algorithm that is used for the solution of minimum cost flow problem. In the network simplex algorithm, finding an entering arc requires searching through many arcs to decide which one should be included in the spanning tree solution on the next iteration. We propose finding the entering arc in parallel as it often takes the majority of the execution time. A usual strategy is to pick the arc violating the optimality the most out of all possible candidates. Scanning all arcs can take quite some time, so it is common to consider only a fixed number of arcs which is referred as the block search pivoting rule. Arc scans can easily be done in parallel to find the best candidate as the calculations are independent of each other. We used shared memory parallelism using OpenMP along with vectorization using AVX instructions. We also tried adjusting block sizes to increase the parallel portion of the algorithm. Our dataset consists of various natural and synthetic graphs with sizes up to a billion arc. Our experiments show speedups up to four are possible, though they are typically lower.

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)).

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.

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.

Given a directed graph G = (N,A) with arc capacities u and a minimum cost flow problem defined on G, the capacity inverse minimum cost flow problem is to find a new capacity vector u' for the arc set A such that a given feasible flow x' is optimal with respect to the modified capacities. Among all capacity vectors u' satisfying this condition, we would like to find one with minimum ||u' - u|| value. We consider two distance measures for ||u' - u||, rectilinear and Chebyshev distances. By reduction from the feedback arc set problem we show that the capacity inverse minimum cost flow problem is NP-hard in the rectilinear case. On the other hand, it is polynomially solvable by a greedy algorithm for the Chebyshev norm. In the latter case we propose a heuristic for the bicriteria problem, where we minimize among all optimal solutions the number of affected arcs. We also present computational results for this heuristic.

A theoretical and experimental study of a new algorithm for minimum cost flow in dynamic graphs

A grid-based coverage approach for target tracking in hybrid sensor networks

The maximum flow and minimum cost-maximum flow problems are both concerned with determining flows through a network between a source and a destination. Maximum flow applies to any problem where the objective is move as many as possible goods/objects/people between two locations via intermediate locations. Maximum flow-minimum cost applies to flow problems where both capacities and costs are involved. When given information about a network (network flow diagram, capacities, costs), computing enables one to arrive at a solution to the problem. Once the solution becomes available, it has to be applied to a real world problem. The use of the following computer software in solving these problems will be discussed: R (several packages and functions), specially written Pascal programs and Excel SOLVER. The minimum cost-maximum flow solutions to the following problems will also be discussed: maximum flow, minimum cost-maximum flow, transportation problem, assignment problem, shortest path problem, caterer problem.

Chapter 1 Applications of network optimization

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.

A less-than-truckload carrier collaboration planning problem under dynamic capacities

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

An introduction to dynamic generative networks: Minimum cost flow

Decision support for truckload carriers in one-shot combinatorial auctions

Lexicography in evacuation planning refers to prioritizing and structuring the evacuation process in emergencies. It provides a systematic method for hierarchically addressing multiple objectives, ensuring that the most critical factors are resolved first. This work focuses on such an approach to the evacuation planning problem for the transit-based network. Lexicographically dynamic flow determines the optimal flow in a network like the maximum and minimum cost flow. The term lexicography is based on lexicographic ordering and refers to sorting or optimizing solutions in a way that respects a lexicographic priority. Such a priority can be considered based on the need of the network structures, objectives, demands, etc. Multi-objective optimization problems where secondary objectives are considered only after primary objectives are met also use such a technique.