1. The average cost of a directed cycle is AvgCost(C) = ω(C) /t = e∈C ω(e) /t. (v) denote the minimum length of a walk with exactly k edges, ending at v. So, for each v, we have d 0 (v) = 0 and d k+1 (v) = min e=(u→v)∈E d k (u) + ω(e) .

In this article, we devise two dual based methods for obtaining very good solution to a single stage un-capacitated minimum cost flow problem. These methods are an improvement to the methods already developed by Sharma and Saxena [1]. We further develop a method to extract a very good primal solution from a given dual solution. We later demonstrate the efficacies and the significance of these m...

We consider the capacitated minimum cost flow problem on directed hypergraphs. We define spanning hypertrees so generalizing the spanning tree of a standard graph, and show that, like in the standard and in the generalized minimum cost flow problems, a correspondence exists between bases and spanning hypertrees. Then, we show that, like for the network simplex algorithms for the standard and fo...

The cost scaling push-relabel method has been shown to be efficient for solving minimum-cost flow problems. In this paper we apply the method to the assignment problem and investigate implementations of the method that take advantage of assignment's special structure. The results show that the method is very promising for practical use.

It is well-known how to use maximum flow to decide when a flow problem with demands, lower bounds, and upper bounds is infeasible. Less well-known is how to compute a flow that is least infeasible. This paper considers many possible ways to define “least infeasible” and shows how to compute optimal flows for each definition. For each definition it also gives a dual characterization in terms of ...

Given an instance of the minimum cost flow problem, a version of the corresponding inverse problem, called the capacity inverse problem, is to modify the upper and lower bounds on arc flows as little as possible so that a given feasible flow becomes optimal to the modified minimum cost flow problem. The modifications can be measured by different distances. In this article, we consider the capac...

Given a directed graph G = (N,A) with arc capacities uij and a minimum cost flow problem defined on G, the capacity inverse minimum cost flow problem is to find a new capacity vector û for the arc set A such that a given feasible flow x̂ is optimal with respect to the modified capacities. Among all capacity vectors û satisfying this condition, we would like to find one with minimum ‖û− u‖ value....

We give an upper bound for the number of different basic feasible solutions generated by Dantzig’s simplex method (the simplex method with the most negative pivoting rule) for LP with bounded variables by extending the result of Kitahara and Mizuno (2010). We refine the analysis by them and improve an upper bound for a standard form of LP. Then we utilize the improved bound for an LP with bound...

Periodic global updates of dual variables have been shown to yield a substantial speed advantage in implementations of push-relabel algorithms for the maximum flow and minimum cost flow problems. In this paper, we show that in the context of the bipartite matching and assignment problems, global updates yield a theoretical improvement as well. For bipartite matching, a pushrelabel algorithm tha...