2 research outputs found
Specializations and Generalizations of the Stackelberg Minimum Spanning Tree Game
Let be given a graph whose edge set is partitioned into a set
of \emph{red} edges and a set of \emph{blue} edges, and assume that red
edges are weighted and form a spanning tree of . 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 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 , for any .
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 edges,
then the problem admits a -approximation algorithm, for any
.Comment: 22 pages, 7 figure
Hardness of an asymmetric 2-player Stackelberg Network Pricing Game
Consider a communication network represented by a directed graph G=(V,E) of n nodes and m edges. Assume that edges in E are partitioned into two sets: a set C of edges with a fixed non-negative real cost, and a set P of edges whose costs are instead priced by a leader. This is done with the final intent of maximizing a revenue that will be returned for their use by a follower, whose goal in turn is to select for his communication purposes a subnetwork of Gminimizing a given objective function of the edge costs. In this paper, we study the natural setting in which the follower computes a single-source shortest paths tree of G, and then returns to the leader a payment equal to the sum of the selected priceable edges. Thus, the problem can be modeled as a one-round two-player Stackelberg Network Pricing Game, but with the novelty that the objective functions of the two players are asymmetric, in that the revenue returned to the leader for any of her selected edges is not equal to the cost of such an edge in the follower’s solution. As is shown, for any ϵ>0 and unless P=NP, the leader’s problem of finding an optimal pricing is not approximable within n1/2−ϵ, while, if G is unweighted and the leader can only decide which of her edges enter in the solution, then the problem is not approximable within n1/3−ϵ. On the positive side, we devise a strongly polynomial-time O(n)-approximation algorithm, which favorably compares against the classic approach based on a single-price algorithm. Finally, motivated by practical applications, we consider the special cases in which edges in C are unweighted and happen to form two popular network topologies, namely stars and chains, and we provide a comprehensive characterization of their computational tractability