A Faster Strongly Polynomial Minimum Cost Flow Algorithm

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

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.

Developing a polynomial time primal network simplex algorithm for the minimum cost flow problem has been a long standing open problem. In this paper, we develop one such algorithm that runs in O(min(n 2m lognC, n 2m2 logn)) time, wheren is the number of nodes in the network,m is the number of arcs, andC denotes the maximum absolute arc costs if arc costs are integer and ∞ otherwise. We first introduce a pseudopolynomial variant of the network simplex algorithm called the “premultiplier algorithm”. We then develop a cost-scaling version of the premultiplier algorithm that solves the minimum cost flow problem in O(min(nm lognC, nm 2 logn)) pivots. With certain simple data structures, the average time per pivot can be shown to be O(n). We also show that the diameter of the network polytope is O(nm logn).

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.

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.

An introduction to dynamic generative networks: Minimum cost flow

Wasserstein distributionally robust shortest path problem

Decision support for truckload carriers in one-shot combinatorial auctions

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

Chapter 1 Applications of network optimization

The author describes a formulation of National Airspace System (NAS) traffic management problem as a multicommodity minimum cost flow problem over a network in space-time. It is a formulation of the multidestination delay minimization problem of managing traffic in the NAS. The model's potential for improving over present practice is described in terms of six assumptions of present practice. The space-time network model can address uncertainty in capacity values, restrictive airways capacities, and hubbing. The minimum cost flow formulation further assumes that attempted entries into the system are deterministic, not Poisson, and that actual entries are controllable. The formulation specifically allows dynamic capacities and possesses good potential for short execution time. >

There are various situations in which real-world problems can be modeled and solved as minimum flow problems. Sometimes, in these situations, minor data changes may occur, leading to corresponding changes of the networks in which the practical problems are modeled as flow problems, such as slight variations in capacity or lower bound. For instance, the capacity or the lower bound of an arc may increase or decrease in time, leaving one with no other choice than finding the new minimum network flow. Given both the various ways in which the networks can be changed and the high frequency of these changes, it is desirable to find as fast a computation method for minimum flow as possible. This paper is focused on the cases that concern increasing and decreasing the capacity or the lower bound of an arc. For these cases, both the minimum flow algorithms and the dynamic minimum flow algorithms that are already known are inefficient. Our incremental algorithms for determining minimum flow in the modified network are more efficient than both the above-mentioned types of algorithms. The proposed method starts from the initial network minimum flow and solves the minimum flow problem in a significantly faster way than recalculating the new network minimum flow starting from scratch.

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.

Physarum polycephalum (true slime mold) has recently emerged as a fascinating example of biological computation through morphogenesis. Despite being a single cell organism, experiments have observed that through its growth process, the Physarum is able to solve various minimum cost flow problems. This paper analyzes a mathematical model of the Physarum growth dynamics. We show how to encode general linear programming (LP) problems as instances of the Physarum. We prove that under the growth dynamics, the Physarum is guaranteed to converge to the optimal solution of the LP. We further derive an efficient discrete algorithm based on the Physarum model, and experimentally verify its performance on assignment problems.KeywordsAssignment ProblemLinear Program ProblemShort Path ProblemSlime MoldPhysarum PolycephalumThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We address the energy-efficient data redistribution problem in data intensive sensor networks (DISNs). The key question in sensor networks with large volumes of sensory data is how to redistribute the data efficiently under limited storage and energy constraints at the sensor nodes. The goal of the redistribution scheme is to minimize the energy consumption during the process, while guaranteeing full utilization of the distributed storage capacity in the DISNs. We formulate this problem as a minimum cost flow problem, which can be solved optimally. However, the optimal solution's centralized nature makes it unsuitable for large-scale distributed sensor networks. We thus design a distributed algorithm for the data redistribution problem which performs very close to the optimal, and compare its performance with various intuitive heuristics. Our proposed algorithm relies on potential function based computations, incurs limited message and computational overhead at both the sensor nodes and data generator nodes, and is easily implementable in a distributed manner. We analytically show the convergence of our algorithm, and demonstrate its near-optimal performance and scalability under various network scenarios considered. Finally, we implement our distributed algorithm in TinyOS and evaluate it using TOSSIM simulator, and show that it outperforms EnviroStore, the only existing scheme for data redistribution in sensor networks, in both solution quality and overhead messages.