Covering the complete graph by partitions

AbstractA (D, c)-coloring of the complete graph Kn is a coloring of the edges with c colors such that all monochromatic connected subgraphs have at most D vertices. Resolvable block designs with c parallel classes and with block size D are natural examples of (D, c)-colorings. However, (D, c)-colorings are more relaxed structures. We investigate the largest n such that Kn has a (D, c)-coloring. Our main tool is the fractional matching theory of hypergraphs

Combinatorics

Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combiantorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session

Discrete Geometry and Convexity in Honour of Imre Bárány

This special volume is contributed by the speakers of the Discrete Geometry and Convexity conference, held in Budapest, June 19–23, 2017. The aim of the conference is to celebrate the 70th birthday and the scientific achievements of professor Imre Bárány, a pioneering researcher of discrete and convex geometry, topological methods, and combinatorics. The extended abstracts presented here are written by prominent mathematicians whose work has special connections to that of professor Bárány. Topics that are covered include: discrete and combinatorial geometry, convex geometry and general convexity, topological and combinatorial methods. The research papers are presented here in two sections. After this preface and a short overview of Imre Bárány’s works, the main part consists of 20 short but very high level surveys and/or original results (at least an extended abstract of them) by the invited speakers. Then in the second part there are 13 short summaries of further contributed talks. We would like to dedicate this volume to Imre, our great teacher, inspiring colleague, and warm-hearted friend

Real Analysis, Harmonic Analysis, and Applications

The workshop focused on important developments within the last few years in real and harmonic analysis, including polynomial partitioning and decoupling as well as significant concurrent progress in the application of these for example to number theory and partial differential equations

Geometric partitioning algorithms for fair division of geographic resources

University of Minnesota Ph.D. dissertation. July 2014. Major: Industrial and Systems Engineering. Advisor: John Gunnar Carlsson. 1 computer file (PDF): vi, 140 pages, appendices p. 129-140.This dissertation focuses on a fundamental but under-researched problem: how does one divide a piece of territory into smaller pieces in an efficient way? In particular, we are interested in \emph{map segmentation problem} of partitioning a geographic region into smaller subregions for allocating resources or distributing a workload among multiple agents. This work would result in useful solutions for a variety of fundamental problems, ranging from congressional districting, facility location, and supply chain management to air traffic control and vehicle routing. In a typical map segmentation problem, we are given a geographic region $R$, a probability density function defined on $R$ (representing, say population density, distribution of a natural resource, or locations of clients) and a set of points in $R$ (representing, say service facilities or vehicle depots). We seek a \emph{partition} of $R$ that is a collection of disjoint sub-regions $\{R_1, . . . , R_n\}$ such that $\bigcup_i R_i = R$, that optimizes some objective function while satisfying a shape condition. As examples of shape conditions, we may require that all sub-regions be compact, convex, star convex, simply connected (not having holes), connected, or merely measurable.Such problems are difficult because the search space is infinite-dimensional (since we are designing boundaries between sub-regions) and because the shape conditions are generally difficult to enforce using standard optimization methods. There are also many interesting variants and extensions to this problem. It is often the case that the optimal partition for a problem changes over time as new information about the region is collected. In that case, we have an \emph{online} problem and we must re-draw the sub-region boundaries as time progresses. In addition, we often prefer to construct these sub-regions in a \emph{decentralized} fashion: that is, the sub-region assigned to agent $i$ should be computable using only local information to agent $i$ (such as nearby neighbors or information about its surroundings), and the optimal boundary between two sub-regions should be computable using only knowledge available to those two agents.This dissertation is an attempt to design geometric algorithms aiming to solve the above mentioned problems keeping in view the various design constraints. We describe the drawbacks of the current approach to solving map segmentation problems, its ineffectiveness to impose geometric shape conditions and its limited utility in solving the online version of the problem. Using an intrinsically interdisciplinary approach, combining elements from variational calculus, computational geometry, geometric probability theory, and vector space optimization, we present an approach where we formulate the problems geometrically and then use a fast geometric algorithm to solve them. We demonstrate our success by solving problems having a particular choice of objective function and enforcing certain shape conditions. In fact, it turns out that such methods actually give useful insights and algorithms into classical location problems such as the continuous $k$-medians problem, where the aim is to find optimal locations for facilities. We use a map segmentation technique to present a constant factor approximation algorithm to solve the continuous $k$-medians problem in a convex polygon. We conclude this thesis by describing how we intend to build on this success and develop algorithms to solve larger classes of these problems

Active cellular mechanics and its consequences for animal development

A central goal of developmental biology is to understand how an organism shapes itself, a process referred to as morphogenesis. While the molecular components critical to determining the initial body plan have been well characterized, the control of the subsequent dynamics of cellular rearrangements which ultimately shape the organism are far less understood. A major roadblock to a more complete picture of morphogenesis is the inability to measure tissue-scale mechanics throughout development and thus answer fundamental questions: How is the mechanical state of the cell regulated by local protein expression and global pattering? In what way does stress feedback onto the larger developmental program?In this dissertation, we begin to approach these questions through the introduction and analysis of a multi-scale model of epithelial mechanics which explicitly connects cytoskeletal protein activity to tissue-level stress. In Chapter 2, we introduce the discrete Active Tension Network (ATN) model of cellular mechanics. ATNs are tissues that satisfy two primary assumptions: that the mechanical balance of cells is dominated by cortical tension and that myosin actively remodels the actin cytoskeleton in a stress-dependent manner. Remarkably, the interplay of these features allows for angle-preserving, i.e. `isogonal', dilations or contractions of local cell geometry that do not generate stress. Asymptotically this model is stabilized provided there is mechanical feedback on expression of myosin within the cell; we take this to be a strong prediction to be tested. The ATN model exposes a fundamental connection between equilibrium cell geometry and its underlying force network. In Chapter 3, we relax the tension-net approximation and demonstrate that at equilibrium, epithelial tissues with non-uniform pressure have non-trivial geometric constraints that imply the network is described by a weighted `dual' triangulation. We show that the dual triangulation encodes all information about the mechanical state of an epithelial tissue. Utilizing the stress-geometry ‘duality’, we formulate a local "Mechanical Inference" of cellular-level stress using solely cell geometry that dramatically improves over past image-based inference techniques.In Chapter 4, we generalize the ATN model to explore the controlled re-arrangement of cells within epithelial tissues. This requires us to explicitly consider the effects of cadherin mediated adhesion, and its regulation, on tissue morphogenesis. We find that positive feedback between myosin and cortical tension, along with traction-dependent depletion of cytoskeletal cadherin is sufficient to recapitulate the morphogenetic movement of cells observed during convergent extension of the lateral ectoderm during Drosophila embryogenesis. Statistical analyses of live-imaging data supports the fundamentals of the model.Chapter 5 focuses on morphogenesis at a mesoscopic scale by coarse-graining the cellular ATN model. Under this limit, we expect an epithelial tissue should behave as an effective viscous, compressible fluid driven by myosin gradients on intermediate time-scales. Theoretical predictions are empirically tested against in-toto microscopy data obtained during early Drosophila embryogenesis

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum