292 research outputs found
Reducing Revenue to Welfare Maximization: Approximation Algorithms and other Generalizations
It was recently shown in [http://arxiv.org/abs/1207.5518] that revenue
optimization can be computationally efficiently reduced to welfare optimization
in all multi-dimensional Bayesian auction problems with arbitrary (possibly
combinatorial) feasibility constraints and independent additive bidders with
arbitrary (possibly combinatorial) demand constraints. This reduction provides
a poly-time solution to the optimal mechanism design problem in all auction
settings where welfare optimization can be solved efficiently, but it is
fragile to approximation and cannot provide solutions to settings where welfare
maximization can only be tractably approximated. In this paper, we extend the
reduction to accommodate approximation algorithms, providing an approximation
preserving reduction from (truthful) revenue maximization to (not necessarily
truthful) welfare maximization. The mechanisms output by our reduction choose
allocations via black-box calls to welfare approximation on randomly selected
inputs, thereby generalizing also our earlier structural results on optimal
multi-dimensional mechanisms to approximately optimal mechanisms. Unlike
[http://arxiv.org/abs/1207.5518], our results here are obtained through novel
uses of the Ellipsoid algorithm and other optimization techniques over {\em
non-convex regions}
More Dynamic Data Structures for Geometric Set Cover with Sublinear Update Time
We study geometric set cover problems in dynamic settings, allowing insertions and deletions of points and objects. We present the first dynamic data structure that can maintain an O(1)-approximation in sublinear update time for set cover for axis-aligned squares in 2D . More precisely, we obtain randomized update time O(n^{2/3+?}) for an arbitrarily small constant ? > 0. Previously, a dynamic geometric set cover data structure with sublinear update time was known only for unit squares by Agarwal, Chang, Suri, Xiao, and Xue [SoCG 2020]. If only an approximate size of the solution is needed, then we can also obtain sublinear amortized update time for disks in 2D and halfspaces in 3D . As a byproduct, our techniques for dynamic set cover also yield an optimal randomized O(nlog n)-time algorithm for static set cover for 2D disks and 3D halfspaces, improving our earlier O(nlog n(log log n)^{O(1)}) result [SoCG 2020]
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