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

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 $\textit{duplicate}$ 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 $t$ sets. We give an algorithm that finds a coloring with discrepancy $O((t \log n \log s)^{1/2})$ where $s$ is the maximum cardinality of a set. This improves upon the previous constructive bound of $O(t^{1/2} \log n)$ based on algorithmic variants of the partial coloring method, and for small $s$ (e.g.$s=\textrm{poly}(t)$) comes close to the non-constructive $O((t \log n)^{1/2})$ bound due to Banaszczyk. Previously, no algorithmic results better than $O(t^{1/2}\log n)$ were known even for $s = O(t^2)$. 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 $S$ in the set system incurs an $O((t \log n \log |S|)^{1/2})$ discrepancy. Another variant can be used to essentially match Banaszczyk's bound for a wide class of instances even where $s$ is arbitrarily large. Finally, these results also extend directly to the more general Koml\'{o}s setting