Improved Hardness of Approximation for Stackelberg Shortest-Path Pricing

We consider the Stackelberg shortest-path pricing problem, which is defined as follows. Given a graph G with fixed-cost and pricable edges and two distinct vertices s and t, we may assign prices to the pricable edges. Based on the predefined fixed costs and our prices, a customer purchases a cheapest s-t-path in G and we receive payment equal to the sum of prices of pricable edges belonging to the path. Our goal is to find prices maximizing the payment received from the customer. While Stackelberg shortest-path pricing was known to be APX-hard before, we provide the first explicit approximation threshold and prove hardness of approximation within 2−o(1). We also argue that the nicely structured type of instance resulting from our reduction captures most of the challenges we face in dealing with the problem in general and, in particular, we show that the gap between the revenue of an optimal pricing and the only known general upper bound can still be logarithmically large

Specializations and Generalizations of the Stackelberg Minimum Spanning Tree Game

Let be given a graph $G=(V,E)$ whose edge set is partitioned into a set $R$ of \emph{red} edges and a set $B$ of \emph{blue} edges, and assume that red edges are weighted and form a spanning tree of $G$. Then, the \emph{Stackelberg Minimum Spanning Tree} (\stack) problem is that of pricing (i.e., weighting) the blue edges in such a way that the total weight of the blue edges selected in a minimum spanning tree of the resulting graph is maximized. \stack \ is known to be \apx-hard already when the number of distinct red weights is 2. In this paper we analyze some meaningful specializations and generalizations of \stack, which shed some more light on the computational complexity of the problem. More precisely, we first show that if $G$ is restricted to be \emph{complete}, then the following holds: (i) if there are only 2 distinct red weights, then the problem can be solved optimally (this contrasts with the corresponding \apx-hardness of the general problem); (ii) otherwise, the problem can be approximated within $7/4 + \epsilon$, for any $\epsilon > 0$. Afterwards, we define a natural extension of \stack, namely that in which blue edges have a non-negative \emph{activation cost} associated, and it is given a global \emph{activation budget} that must not be exceeded when pricing blue edges. Here, after showing that the very same approximation ratio as that of the original problem can be achieved, we prove that if the spanning tree of red edges can be rooted so as that any root-leaf path contains at most $h$ edges, then the problem admits a $(2h+\epsilon)$-approximation algorithm, for any $\epsilon > 0$.Comment: 22 pages, 7 figure

Revenue Maximization in Stackelberg Pricing Games: Beyond the Combinatorial Setting

In a Stackelberg Pricing Game a distinguished player, the leader, chooses prices for a set of items, and the other players, the followers, each seeks to buy a minimum cost feasible subset of the items. The goal of the leader is to maximize her revenue, which is determined by the sold items and their prices. Most previously studied cases of such games can be captured by a combinatorial model where we have a base set of items, some with fixed prices, some priceable, and constraints on the subsets that are feasible for each follower. In this combinatorial setting, Briest et al. and Balcan et al. independently showed that the maximum revenue can be approximated to a factor of H_k ~ log(k), where k is the number of priceable items. Our results are twofold. First, we strongly generalize the model by letting the follower minimize any continuous function plus a linear term over any compact subset of R_(n>=0); the coefficients (or prices) in the linear term are chosen by the leader and determine her revenue. In particular, this includes the fundamental case of linear programs. We give a tight lower bound on the revenue of the leader, generalizing the results of Briest et al. and Balcan et al. Besides, we prove that it is strongly NP-hard to decide whether the optimum revenue exceeds the lower bound by an arbitrarily small factor. Second, we study the parameterized complexity of computing the optimal revenue with respect to the number k of priceable items. In the combinatorial setting, given an efficient algorithm for optimal follower solutions, the maximum revenue can be found by enumerating the 2^k subsets of priceable items and computing optimal prices via a result of Briest et al., giving time O(2^k|I|^c ) where |I| is the input size. Our main result here is a W[1]-hardness proof for the case where the followers minimize a linear program, ruling out running time f(k)|I|^c unless FPT = W[1] and ruling out time |I|^o(k) under the Exponential-Time Hypothesis

An Approximation Algorithm for Stackelberg Network Pricing

We consider the problem of maximizing the revenue raised from tolls set on the arcs of a transportation network, under the constraint that users are assigned to toll-compatible shortest paths. We first prove that this problem is strongly NP-hard. We then provide a polynomial time algorithm with a worst-case precision guarantee of ${1/2}\log_2 m_T+1$, where $m_T$ denotes the number of toll arcs. Finally we show that the approximation is tight with respect to a natural relaxation by constructing a family of instances for which the relaxation gap is reached.Comment: 38 page

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

Stackelberg Max Closure with Multiple Followers

In a Stackelberg max closure game, we are given a digraph whose vertices correspond to projects from which firms can choose and whose arcs represent precedence constraints. Some projects are under the control of a leader who sets prices in the first stage of the game, while in the second stage, the firms choose a feasible subset of projects of maximum value. For a single follower, the leader’s problem of finding revenue-maximizing prices can be solved in strongly polynomial time. In this paper, we focus on the setting with multiple followers and distinguish two situations. In the case in which only one copy of each project is available (limited supply), we show that the two-follower problem is solvable in strongly polynomial time, whereas the problem with three or more followers is NP-hard. In the case of unlimited supply, that is, when sufficient copies of each project are available, we show that the two-follower problem is already APX-hard. As a side result, we prove that Stackelberg min vertex cover on bipartite graphs with a single follower is APX-hard

Pricing Network Edges to Cross a River.

We consider a Stackelberg pricing problem in directed networks:Tariffs (prices) have to be defined by an operator, the leader, for a subset of the arcs. Clients, the followers, choose paths to route their demand through the network selfishly and independently of each other, on the basis of minimal total cost. The problem is to find tariffs such as to maximize the operator''s revenue. We consider the case where each client takes at most one tariff arc to route the demand.The problem, which we refer to as the river tarification problem, is a special case of problems studied previously in the literature.We prove that the problem is strongly NP-hard.Moreover, we show that the polynomially solvable case of uniform tarification yields an m--approximation algorithm, and this is tight. We suggest a new type of analysis that allows to improve the result to \bigO{\log m}, whenever the input data is polynomially bounded. We furthermore derive an \bigO{m^{1-\varepsilon}}--inapproximability result for problems where the operator must serve all clients, and we discuss some polynomial special cases. Finally, a computational study with instances from France Telecom suggests that uniform pricing performs better in practice than theory would suggest.operations research and management science;