Fast algorithms for convex cost flow problems on circles, lines, and trees

Abstract

AbstractWe develop efficient algorithms to solve convex cost flow problems where the underlying graph is a circle, a line, or a tree. Each node i has an associated supply/demand b( i). The cost of sending flow on arc ( i, j) is a piecewise linear convex function fij defined over . Let n be the number of nodes and m = O(n) be the total number of pieces of all the convex functions. A flow x is feasible if the imbalances on all nodes are nonnegative. Excess stored on node i has an associated linear cost . We show that the problem on a circle can be transformed into an equivalent problem on a line in O( n) time. Thereafter, we develop an algorithm that solves the problem on a line in time, where sort( n) is the time to sort n real numbers and is the inverse Ackermann function. We also prove that when the nodes lie on a tree, the problem can be solved in time using the dynamic tree data structure. We describe applications in areas such as distributed computing, lot‐sizing, computational biology, computational music, and transportation. © 2013 Wiley Periodicals, Inc. NETWORKS, Vol. 62(4), 288–296 2013