Optimal Random Matchings, Tours, and Spanning Trees in Hierarchically Separated Trees

We derive tight bounds on the expected weights of several combinatorial optimization problems for random point sets of size $n$ distributed among the leaves of a balanced hierarchically separated tree. We consider {\it monochromatic} and {\it bichromatic} versions of the minimum matching, minimum spanning tree, and traveling salesman problems. We also present tight concentration results for the monochromatic problems.Comment: 24 pages, to appear in TC

Online Duet between Metric Embeddings and Minimum-Weight Perfect Matchings

Low-distortional metric embeddings are a crucial component in the modern algorithmic toolkit. In an online metric embedding, points arrive sequentially and the goal is to embed them into a simple space irrevocably, while minimizing the distortion. Our first result is a deterministic online embedding of a general metric into Euclidean space with distortion $O(\log n)\cdot\min\{\sqrt{\log\Phi},\sqrt{n}\}$ (or, $O(d)\cdot\min\{\sqrt{\log\Phi},\sqrt{n}\}$ if the metric has doubling dimension $d$), solving a conjecture by Newman and Rabinovich (2020), and quadratically improving the dependence on the aspect ratio $\Phi$ from Indyk et al.\ (2010). Our second result is a stochastic embedding of a metric space into trees with expected distortion $O(d\cdot \log\Phi)$, generalizing previous results (Indyk et al.\ (2010), Bartal et al.\ (2020)). Next, we study the \emph{online minimum-weight perfect matching} problem, where a sequence of $2n$ metric points arrive in pairs, and one has to maintain a perfect matching at all times. We allow recourse (as otherwise the order of arrival determines the matching). The goal is to return a perfect matching that approximates the \emph{minimum-weight} perfect matching at all times, while minimizing the recourse. Our third result is a randomized algorithm with competitive ratio $O(d\cdot \log \Phi)$ and recourse $O(\log \Phi)$ against an oblivious adversary, this result is obtained via our new stochastic online embedding. Our fourth result is a deterministic algorithm against an adaptive adversary, using $O(\log^2 n)$ recourse, that maintains a matching of weight at most $O(\log n)$ times the weight of the MST, i.e., a matching of lightness $O(\log n)$. We complement our upper bounds with a strategy for an oblivious adversary that, with recourse $r$, establishes a lower bound of $\Omega(\frac{\log n}{r \log r})$ for both competitive ratio and lightness.Comment: 53 pages, 8 figures, to be presented at the ACM-SIAM Symposium on Discrete Algorithms (SODA24

Problems in extremal and combinatorial geometry

This thesis deals with three families of optimization problems: (1) Euclidean optimization problems on random point sets; (2) independent sets in hypergraphs; and (3) packings in point lattices. First, we consider bounds on several monochromatic and bichromatic optimization problems including minimum matching, minimum spanning trees, and the travelling salesman problem. Many of these problems lend themselves to representations in terms of hierarchically separated trees | trees with uniform branching factor and depth, and having edge weights exponential in the depth of the edge in the tree. In the second part, we consider the independent set problem on uniform hypergraphs, in anticipation of applying it to the third part, packing problems on point lattices. In these problems we wish to select a subset of points from an n n ::: n grid avoiding particular patterns. We also study several generalizations of these problems that have not been handled previously.M.S., Computer Science -- Drexel University, 201

Inferring Geodesic Cerebrovascular Graphs: Image Processing, Topological Alignment and Biomarkers Extraction

A vectorial representation of the vascular network that embodies quantitative features - location, direction, scale, and bifurcations - has many potential neuro-vascular applications. Patient-specific models support computer-assisted surgical procedures in neurovascular interventions, while analyses on multiple subjects are essential for group-level studies on which clinical prediction and therapeutic inference ultimately depend. This first motivated the development of a variety of methods to segment the cerebrovascular system. Nonetheless, a number of limitations, ranging from data-driven inhomogeneities, the anatomical intra- and inter-subject variability, the lack of exhaustive ground-truth, the need for operator-dependent processing pipelines, and the highly non-linear vascular domain, still make the automatic inference of the cerebrovascular topology an open problem. In this thesis, brain vessels’ topology is inferred by focusing on their connectedness. With a novel framework, the brain vasculature is recovered from 3D angiographies by solving a connectivity-optimised anisotropic level-set over a voxel-wise tensor field representing the orientation of the underlying vasculature. Assuming vessels joining by minimal paths, a connectivity paradigm is formulated to automatically determine the vascular topology as an over-connected geodesic graph. Ultimately, deep-brain vascular structures are extracted with geodesic minimum spanning trees. The inferred topologies are then aligned with similar ones for labelling and propagating information over a non-linear vectorial domain, where the branching pattern of a set of vessels transcends a subject-specific quantized grid. Using a multi-source embedding of a vascular graph, the pairwise registration of topologies is performed with the state-of-the-art graph matching techniques employed in computer vision. Functional biomarkers are determined over the neurovascular graphs with two complementary approaches. Efficient approximations of blood flow and pressure drop account for autoregulation and compensation mechanisms in the whole network in presence of perturbations, using lumped-parameters analog-equivalents from clinical angiographies. Also, a localised NURBS-based parametrisation of bifurcations is introduced to model fluid-solid interactions by means of hemodynamic simulations using an isogeometric analysis framework, where both geometry and solution profile at the interface share the same homogeneous domain. Experimental results on synthetic and clinical angiographies validated the proposed formulations. Perspectives and future works are discussed for the group-wise alignment of cerebrovascular topologies over a population, towards defining cerebrovascular atlases, and for further topological optimisation strategies and risk prediction models for therapeutic inference. Most of the algorithms presented in this work are available as part of the open-source package VTrails

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

Adaptive Approximation Algorithms for Ranking, Routing and Classification

This dissertation aims to consider different problems in the area of stochastic optimization, where we are provided with more information about the instantiation of the stochastic parameters over time. With uncertainty being an inseparable part of every industry, several applications can be modeled as discussed. In this dissertation we focus on three main areas of applications: 1) ranking problems, which can be helpful for modeling product ranking, designing recommender systems, etc., 2) routing problems which can cover applications in delivery, transportation and networking, and 3) classification problems with possible applications in medical diagnosis and chemical identification. We consider three types of solutions for these problems based on how we want to deal with the observed information: static, adaptive and a priori solutions. In Chapter II, we study two general stochastic submodular optimization problems that we call Adaptive Submodular Ranking and Adaptive Submodular Routing. In the ranking version, we want to provide an adaptive sequence of weighted elements to cover a random submodular function with minimum expected cost. In the routing version, we want to provide an adaptive path of vertices to cover a random scenario with minimum expected length. We provide (poly)logarithmic approximation algorithms for these problems that (nearly) match or improve the best-known results for various special cases. We also implemented different variations of the ranking algorithm and observed that it outperforms other practical algorithms on real-world and synthetic data sets. In Chapter III, we consider the Optimal Decision Tree problem: an identification task that is widely used in active learning. We study this problem in presence of noise, where we want to perform a sequence of tests with possible noisy outcomes to identify a random hypothesis. We give different static (non-adaptive) and adaptive algorithms for this task with almost logarithmic approximation ratios. We also implemented our algorithms on real-world and synthetic data sets and compared our results with an information theoretic lower bound to show that in practice, our algorithms' value is very close to this lower bound. In Chapter IV, we focus on a stochastic vehicle routing problem called a priori traveling repairman, where we are given a metric and probabilities of each vertices being active. We want to provide an a priori master tour originating from the root that is shortcut later over the observed active vertices. Our objective is to minimize the expected total wait time of active vertices, where the wait time of a vertex is defined as the length of the path from the root to this vertex. We consider two benchmarks to evaluate the performance of an algorithm for this problem: optimal a priori solution and the re-optimization solution. We provide two algorithms to compare with each of these benchmarks that have constant and logarithmic approximation ratios respectively.PHDIndustrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155058/1/navidi_1.pd

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum