583 research outputs found
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]
Packing Cars into Narrow Roads: PTASs for Limited Supply Highway
In the Highway problem, we are given a path with n edges (the highway), and a set of m drivers, each one characterized by a subpath and a budget. For a given assignment of edge prices (the tolls), the highway owner collects from each driver the total price of the associated path when it does not exceed drivers\u27s budget, and zero otherwise. The goal is to choose the prices to maximize the total profit. A PTAS is known for this (strongly NP-hard) problem [Grandoni,Rothvoss-SODA\u2711, SICOMP\u2716].
In this paper we study the limited supply generalization of Highway, that incorporates capacity constraints. Here the input also includes a capacity u_e >= 0 for each edge e; we need to select, among drivers that can afford the required price, a subset such that the number of drivers that use each edge e is at most u_e (and we get profit only from selected drivers). To the best of our knowledge, the only approximation algorithm known for this problem is a folklore O(log m) approximation based on a reduction to the related Unsplittable Flow on a Path problem (UFP). The main result of this paper is a PTAS for limited supply highway.
As a second contribution, we study a natural generalization of the problem where each driver i demands a different amount d_i of capacity. Using known techniques, it is not hard to derive a QPTAS for this problem. Here we present a PTAS for the case that drivers have uniform budgets. Finding a PTAS for non-uniform-demand limited supply highway is left as a challenging open problem
3-D Statistical Channel Model for Millimeter-Wave Outdoor Mobile Broadband Communications
This paper presents an omnidirectional spatial and temporal 3-dimensional
statistical channel model for 28 GHz dense urban non-line of sight
environments. The channel model is developed from 28 GHz ultrawideband
propagation measurements obtained with a 400 megachips per second broadband
sliding correlator channel sounder and highly directional, steerable horn
antennas in New York City. A 3GPP-like statistical channel model that is easy
to implement in software or hardware is developed from measured power delay
profiles and a synthesized method for providing absolute propagation delays
recovered from 3-D ray-tracing, as well as measured angle of departure and
angle of arrival power spectra. The extracted statistics are used to implement
a MATLAB-based statistical simulator that generates 3-D millimeter-wave
temporal and spatial channel coefficients that reproduce realistic impulse
responses of measured urban channels. The methods and model presented here can
be used for millimeter-wave system-wide simulations, and air interface design
and capacity analyses.Comment: 7 pages, 6 figures, ICC 2015 (London, UK, to appear
Connectivity Oracles for Graphs Subject to Vertex Failures
We introduce new data structures for answering connectivity queries in graphs
subject to batched vertex failures. A deterministic structure processes a batch
of failed vertices in time and thereafter
answers connectivity queries in time. It occupies space . We develop a randomized Monte Carlo version of our data structure
with update time , query time , and space
for any failure bound . This is the first connectivity oracle for
general graphs that can efficiently deal with an unbounded number of vertex
failures.
We also develop a more efficient Monte Carlo edge-failure connectivity
oracle. Using space , edge failures are processed in time and thereafter, connectivity queries are answered in
time, which are correct w.h.p.
Our data structures are based on a new decomposition theorem for an
undirected graph , which is of independent interest. It states that
for any terminal set we can remove a set of
vertices such that the remaining graph contains a Steiner forest for with
maximum degree
On Measuring the Value of a Nonmarket Good Using Market Data
Our purpose is to present in detail numerical methods of measuring the value of nonmarket goods using market data, under either weak neutrality, weak complementarity, or any other preference restriction meeting the requirements discussed in this paper. It has been claimed in a number of places in the literature that numerical methods cannot be used to measure the value of nonmarket goods unless the very restrictive Willig conditions are satisfied. We show that this claim is mistaken, and that numerical methods can be used whether or not the Willig conditions are satisfied. Our numerical methods are more flexible than the existing analytical method because ours can be used with any Marshallian demand system.Resource /Energy Economics and Policy,
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