A network flow approach to cost allocation for rooted trees

Abstract

In the game theory approach to cost allocation, the main computational issue is an algorithm for finding solutions such as the Shapley value and the nucleolus. In this article, we consider the problem of allocating the maintenance cost of a tree network that connects the supply source at the root to the users at the leaves. We show that the core of the game can be expressed in terms of network flows. Based on this observation, we present O(n log n) algorithms for computing the nucleolus and the egalitarian allocation.

Structured Optimal Transport

“…Simple means that the minimum norm point problem can be solved fast. For the functions defined in (11), and more generally, for certain hierarchical functions [Hochbaum andHong, 1995, Iwata andZuiki, 2004], coverage functions [Stobbe and Krause, 2010] and graph cuts on lines (equivalent to Total Variation), this can be solved in O(m log m) time, where m is the support size of the respective F i . We provide an O(m log m) algorithm for our cluster functions in the Appendix.…”

Reflection methods for user-friendly submodular optimization

“…Here, the ground set corresponds to the leaves of a rooted, undirected tree. Each node has a weight, and the cost of a set of nodes S ⊆ V is the sum of the weights of all nodes in the smallest subtree (including the root) that spans S. This class of functions too admits to solve the proximal problem in O(n log n) time [24,25]. Related tree functions have been considered in [27], where the elements v of the ground set are arranged in a tree of height d and each have a weight w(v).…”

“…We discussed the more general problem (25) because it contains the smoothed primal as a special case, namely with y = 0 in (25), f = f 1 , and h = f 2 , we obtain min f 1 (x) + f 2 (x) + 1 2 x 2 2 , for which BCD yields the proximal-Dykstra method that was previously used in [3] for twodimensional TV optimization.…”