31 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
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
Stackelberg Network Pricing is Hard to Approximate
In the Stackelberg Network Pricing problem, one has to assign tariffs to a
certain subset of the arcs of a given transportation network. The aim is to
maximize the amount paid by the user of the network, knowing that the user will
take a shortest st-path once the tariffs are fixed. Roch, Savard, and Marcotte
(Networks, Vol. 46(1), 57-67, 2005) proved that this problem is NP-hard, and
gave an O(log m)-approximation algorithm, where m denote the number of arcs to
be priced. In this note, we show that the problem is also APX-hard
Stackelberg Network Pricing Games
We study a multi-player one-round game termed Stackelberg Network Pricing
Game, in which a leader can set prices for a subset of priceable 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 priceable edges in their
solutions, and the problem is to find revenue maximizing prices. 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 -approximation for any . This can
be extended to provide a -approximation for the
general problem and followers. The latter result is essentially best
possible, as the problem is shown to be hard to approximate within
\mathcal{O(\log^\epsilon k + \log^\epsilon m). If followers have demands, the
single-price algorithm provides a -approximation, and the
problem is hard to approximate within \mathcal{O(m^\epsilon) for some
. 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
Computational comparisons of different formulations for the Bilevel Minimum Spanning Tree Problem
International audienceLet be given a graph G = (V, E) whose edge set is partitioned into a set R of red edges and a set B of blue edges, and assume that red edges are weighted and contain a spanning tree of G. Then, the Bilevel Minimum Spanning Tree Problem (BMSTP) 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. In this paper we present different mathematical formulations for the BMSTP based on the properties of the Minimum Spanning Tree Problem and the bilevel optimization. We establish a theoretical and empirical comparison between these new formulations and we also provide reinforcements that together with a proper formulation are able to solve medium to big size instances random instances. We also test our models in instances already existing in the literature
Graph Pricing Problem on Bounded Treewidth, Bounded Genus and k-partite graphs
Consider the following problem. A seller has infinite copies of products
represented by nodes in a graph. There are consumers, each has a budget and
wants to buy two products. Consumers are represented by weighted edges. Given
the prices of products, each consumer will buy both products she wants, at the
given price, if she can afford to. Our objective is to help the seller price
the products to maximize her profit.
This problem is called {\em graph vertex pricing} ({\sf GVP}) problem and has
resisted several recent attempts despite its current simple solution. This
motivates the study of this problem on special classes of graphs. In this
paper, we study this problem on a large class of graphs such as graphs with
bounded treewidth, bounded genus and -partite graphs.
We show that there exists an {\sf FPTAS} for {\sf GVP} on graphs with bounded
treewidth. This result is also extended to an {\sf FPTAS} for the more general
{\em single-minded pricing} problem. On bounded genus graphs we present a {\sf
PTAS} and show that {\sf GVP} is {\sf NP}-hard even on planar graphs.
We study the Sherali-Adams hierarchy applied to a natural Integer Program
formulation that -approximates the optimal solution of {\sf GVP}.
Sherali-Adams hierarchy has gained much interest recently as a possible
approach to develop new approximation algorithms. We show that, when the input
graph has bounded treewidth or bounded genus, applying a constant number of
rounds of Sherali-Adams hierarchy makes the integrality gap of this natural
{\sf LP} arbitrarily small, thus giving a -approximate solution
to the original {\sf GVP} instance.
On -partite graphs, we present a constant-factor approximation algorithm.
We further improve the approximation factors for paths, cycles and graphs with
degree at most three.Comment: Preprint of the paper to appear in Chicago Journal of Theoretical
Computer Scienc
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