Approximate Pricing in Networks: How to Boost the Betweenness and Revenue of a Node

We introduce and study two new pricing problems in networks: Suppose we are given a directed graph G = (V, E) with non-negative edge costs (c_e)_{e in E}, k commodities (s_i, t_i, w_i)_{i in [k]} and a designated node u in V. Each commodity i in [k] is represented by a source-target pair (s_i, t_i) in V x V and a demand w_i>0, specifying that w_i units of flow are sent from s_i to t_i along shortest s_i, t_i-paths (with respect to (c_e)_{e in E}). The demand of each commodity is split evenly over all shortest paths. Assume we can change the edge costs of some of the outgoing edges of u, while the costs of all other edges remain fixed; we also say that we price (or tax) the edges of u. We study the problem of pricing the edges of u with respect to the following two natural objectives: (i) max-flow: maximize the total flow passing through u, and (ii) max-revenue: maximize the total revenue (flow times tax) through u. Both variants have various applications in practice. For example, the max flow objective is equivalent to maximizing the betweenness centrality of u, which is one of the most popular measures for the influence of a node in a (social) network. We prove that (except for some special cases) both problems are NP-hard and inapproximable in general and therefore resort to approximation algorithms. We derive approximation algorithms for both variants and show that the derived approximation guarantees are best possible.