We study a multi-player one-round game termed Stackelberg
Network Pricing Game, in which a leader can set prices for a subset of m
pricable edges in a graph. The other edges have a fixed cost. Based on
the leader's decision one or more followers optimize a polynomial-time
solvable combinatorial minimization problem and choose a minimum cost
solution satisfying their requirements based on the fixed costs and the
leader's prices. The leader receives as revenue the total amount of
prices paid by the followers for pricable edges in their solutions. Our
model extends several known pricing problems, including single-minded
and unit-demand pricing, as well as Stackelberg pricing for certain
follower problems like shortest path or minimum spanning tree. Our first
main result is a tight analysis of a single-price algorithm for the
single follower game, which provides a (1+eps)log m approximation for
any eps > 0. This can be extended to provide a (1+eps)(log k + log
m)-approximation for the general problem and k followers. The latter
result is essentially best possible, as the problem is shown to be hard
to approximate within O(log^eps k + log^eps m) for some eps > 0. If
followers have demands, the single-price algorithm provides a
(1+eps)m^2-approximation, and the problem is hard to approximate within
O(m^eps) for some eps > 0. Our second main result is a polynomial time
algorithm for revenue maximization in the special case of Stackelberg
bipartite vertex cover, which is based on non-trivial max-flow and
LP-duality techniques. Our results can be extended to provide
constant-factor approximations for any constant number of followers.

We study a multi-player one-round game termed Stackelberg Network Pricing Game, in which a leader can set prices for a subset of m pricable edges in a graph. The other edges have a fixed cost. Based on the leader's decision one or more followers optimize a polynomial-time solvable combinatorial minimization problem and choose a minimum cost solution satisfying their requirements based on the fixed costs and the
leader's prices. The leader receives as revenue the total amount of prices paid by the followers for pricable edges in their solutions. Our model extends several known pricing problems, including single-minded and unit-demand pricing, as well as Stackelberg pricing for certain follower problems like shortest path or minimum spanning tree. Our first main result is a tight analysis of a single-price algorithm for the single follower game, which provides a (1+\eps)log m approximation for any \eps > 0. This can be extended to provide a (1+\eps)(log k + log m)-approximation for the general problem and k followers. The latter result is essentially best possible, as the problem is shown to be hard to approximate within O(log^\eps k + log^\eps m) for some \eps > 0. If followers have demands, the single-price algorithm provides a (1+\eps)m^2-approximation, and the problem is hard to approximate within O(m^\eps) for some \eps > 0.

Our second main result is a polynomial time algorithm for revenue maximization in the special case of Stackelberg bipartite vertex cover, which is based on non-trivial max-flow and LP-duality techniques. Our results can be extended to provide constant-factor approximations for any constant number of followers.