167 research outputs found
An Algorithm for Koml\'os Conjecture Matching Banaszczyk's bound
We consider the problem of finding a low discrepancy coloring for sparse set
systems where each element lies in at most t sets. We give an efficient
algorithm that finds a coloring with discrepancy O((t log n)^{1/2}), matching
the best known non-constructive bound for the problem due to Banaszczyk. The
previous algorithms only achieved an O(t^{1/2} log n) bound. The result also
extends to the more general Koml\'{o}s setting and gives an algorithmic
O(log^{1/2} n) bound
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Downlink and Uplink User Association in Dense Next-Generation Wireless Networks
5G, the next-generation cellular network, must serve an aggregate data rate of
1000 times that of current 4G networks while reducing data latency by a factor of ten. To meet these requirements, 5G networks will be far denser than existing networks, and small cells (femtocells and picocells) will augment network capacity. However, dense networks raise questions regarding interference, user association, and handoff between base stations. Where recent papers have demonstrated that interference from small cells will not be prohibitive under multi-slope path loss models, this thesis describes how the use of different path loss models affects the design of such dense, multi-tier networks. This thesis concludes that the gains realized by downlink biasing and uplink/downlink decoupling are strongly dependent on the path loss model assumed and the density differential between base station tiers. Furthermore, this thesis argues that the gains from uplink/downlink decoupling are reduced by a factor of 50% when optimal biasing for the downlink is used.Electrical and Computer Engineerin
A Matrix Expander Chernoff Bound
We prove a Chernoff-type bound for sums of matrix-valued random variables
sampled via a random walk on an expander, confirming a conjecture due to
Wigderson and Xiao. Our proof is based on a new multi-matrix extension of the
Golden-Thompson inequality which improves in some ways the inequality of
Sutter, Berta, and Tomamichel, and may be of independent interest, as well as
an adaptation of an argument for the scalar case due to Healy. Secondarily, we
also provide a generic reduction showing that any concentration inequality for
vector-valued martingales implies a concentration inequality for the
corresponding expander walk, with a weakening of parameters proportional to the
squared mixing time.Comment: Fixed a minor bug in the proof of Theorem 3.
Dual Half-Integrality for Uncrossable Cut Cover and Its Application to Maximum Half-Integral Flow
Given an edge weighted graph and a forest F, the 2-edge connectivity augmentation problem is to pick a minimum weighted set of edges, E\u27, such that every connected component of E\u27 ? F is 2-edge connected. Williamson et al. gave a 2-approximation algorithm (WGMV) for this problem using the primal-dual schema. We show that when edge weights are integral, the WGMV procedure can be modified to obtain a half-integral dual. The 2-edge connectivity augmentation problem has an interesting connection to routing flow in graphs where the union of supply and demand is planar. The half-integrality of the dual leads to a tight 2-approximate max-half-integral-flow min-multicut theorem
Equity in Resident Crowdsourcing: Measuring Under-reporting without Ground Truth Data
Modern city governance relies heavily on crowdsourcing (or "co-production")
to identify problems such as downed trees and power-lines. A major concern in
these systems is that residents do not report problems at the same rates,
leading to an inequitable allocation of government resources. However,
measuring such under-reporting is a difficult statistical task, as, almost by
definition, we do not observe incidents that are not reported. Thus,
distinguishing between low reporting rates and low ground-truth incident rates
is challenging. We develop a method to identify (heterogeneous) reporting
rates, without using external (proxy) ground truth data. Our insight is that
rates on reports about the same incident can be leveraged,
to turn the question into a standard Poisson rate estimation task -- even
though the full incident reporting interval is also unobserved. We apply our
method to over 100,000 resident reports made to the New York City Department of
Parks and Recreation, finding that there are substantial spatial and
socio-economic disparities in reporting rates, even after controlling for
incident characteristics
Improved Algorithmic Bounds for Discrepancy of Sparse Set Systems
We consider the problem of finding a low discrepancy coloring for sparse set
systems where each element lies in at most sets. We give an algorithm that
finds a coloring with discrepancy where is the
maximum cardinality of a set. This improves upon the previous constructive
bound of based on algorithmic variants of the partial
coloring method, and for small (e.g.) comes close to
the non-constructive bound due to Banaszczyk. Previously,
no algorithmic results better than were known even for . Our method is quite robust and we give several refinements and
extensions. For example, the coloring we obtain satisfies the stronger
size-sensitive property that each set in the set system incurs an discrepancy. Another variant can be used to
essentially match Banaszczyk's bound for a wide class of instances even where
is arbitrarily large. Finally, these results also extend directly to the
more general Koml\'{o}s setting
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