7 research outputs found
On Profit-Maximizing Pricing for the Highway and Tollbooth Problems
In the \emph{tollbooth problem}, we are given a tree \bT=(V,E) with
edges, and a set of customers, each of whom is interested in purchasing a
path on the tree. Each customer has a fixed budget, and the objective is to
price the edges of \bT such that the total revenue made by selling the paths
to the customers that can afford them is maximized. An important special case
of this problem, known as the \emph{highway problem}, is when \bT is
restricted to be a line.
For the tollbooth problem, we present a randomized -approximation,
improving on the current best -approximation. We also study a
special case of the tollbooth problem, when all the paths that customers are
interested in purchasing go towards a fixed root of \bT. In this case, we
present an algorithm that returns a -approximation, for any
, and runs in quasi-polynomial time. On the other hand, we rule
out the existence of an FPTAS by showing that even for the line case, the
problem is strongly NP-hard. Finally, we show that in the \emph{coupon model},
when we allow some items to be priced below zero to improve the overall profit,
the problem becomes even APX-hard
Prizing on Paths: A PTAS for the Highway Problem
In the highway problem, we are given an n-edge line graph (the highway), and
a set of paths (the drivers), each one with its own budget. For a given
assignment of edge weights (the tolls), the highway owner collects from each
driver the weight of the associated path, when it does not exceed the budget of
the driver, and zero otherwise. The goal is choosing weights so as to maximize
the profit.
A lot of research has been devoted to this apparently simple problem. The
highway problem was shown to be strongly NP-hard only recently
[Elbassioni,Raman,Ray-'09]. The best-known approximation is O(\log n/\log\log
n) [Gamzu,Segev-'10], which improves on the previous-best O(\log n)
approximation [Balcan,Blum-'06].
In this paper we present a PTAS for the highway problem, hence closing the
complexity status of the problem. Our result is based on a novel randomized
dissection approach, which has some points in common with Arora's quadtree
dissection for Euclidean network design [Arora-'98]. The basic idea is
enclosing the highway in a bounding path, such that both the size of the
bounding path and the position of the highway in it are random variables. Then
we consider a recursive O(1)-ary dissection of the bounding path, in subpaths
of uniform optimal weight. Since the optimal weights are unknown, we construct
the dissection in a bottom-up fashion via dynamic programming, while computing
the approximate solution at the same time. Our algorithm can be easily
derandomized. We demonstrate the versatility of our technique by presenting
PTASs for two variants of the highway problem: the tollbooth problem with a
constant number of leaves and the maximum-feasibility subsystem problem on
interval matrices. In both cases the previous best approximation factors are
polylogarithmic [Gamzu,Segev-'10,Elbassioni,Raman,Ray,Sitters-'09]
On Revenue Maximization with Sharp Multi-Unit Demands
We consider markets consisting of a set of indivisible items, and buyers that
have {\em sharp} multi-unit demand. This means that each buyer wants a
specific number of items; a bundle of size less than has no value,
while a bundle of size greater than is worth no more than the most valued
items (valuations being additive). We consider the objective of setting
prices and allocations in order to maximize the total revenue of the market
maker. The pricing problem with sharp multi-unit demand buyers has a number of
properties that the unit-demand model does not possess, and is an important
question in algorithmic pricing. We consider the problem of computing a revenue
maximizing solution for two solution concepts: competitive equilibrium and
envy-free pricing.
For unrestricted valuations, these problems are NP-complete; we focus on a
realistic special case of "correlated values" where each buyer has a
valuation v_i\qual_j for item , where and \qual_j are positive
quantities associated with buyer and item respectively. We present a
polynomial time algorithm to solve the revenue-maximizing competitive
equilibrium problem. For envy-free pricing, if the demand of each buyer is
bounded by a constant, a revenue maximizing solution can be found efficiently;
the general demand case is shown to be NP-hard.Comment: page2
A PTAS for the Highway Problem
In the highway problem, we are given an n-edge line graph (the highway), and a set of paths (the drivers), each one with its own budget. For a given assignment of edge weights (the tolls), the highway owner collects from each driver the weight of the associated path, when it does not exceed the budget of the driver, and zero otherwise. The goal is choosing weights so as to maximize the profit. A lot of research has been devoted to this apparently simple problem. The highway problem was shown to be strongly NP-hard only recently [Elbassioni,Raman,Ray,Sitters-'09]. The best-known approximation is O(log n / log log n) [Gamzu,Segev-'10], which improves on the previous-best O(log n) approximation [Balcan,Blum-'06]. Better approximations are known for a number of special cases. Finding a constant (or better!) approximation algorithm for the general case is a challenging open problem. In this paper we present a PTAS for the highway problem, hence closing the complexity status of the problem. Our result is based on a novel randomized dissection approach, which has some points in common with Arora's quadtree dissection for Euclidean network design [Arora-'98]. The basic idea is enclosing the highway in a bounding path, such that both the size of the bounding path and the position of the highway in it are random variables. Then we consider a recursive O(1)-ary dissection of the bounding path, in subpaths of uniform optimal weight. Since the optimal weights are unknown, we construct the dissection in a bottom-up fashion via dynamic programming, while computing the approximate solution at the same time. Our algorithm can be easily derandomized. The same basic approach provides PTASs also for two generalizations of the problem: the tollbooth problem with a constant number of leaves and the \emph{maximum-feasibility subsystem} problem on interval matrices. In both cases the previous best approximation factors are polylogarithmic [Gamzu,Segev-'10,Elbassioni,Raman,Ray,Sitters-'09]