On the Total Perimeter of Homothetic Convex Bodies in a Convex Container

For two planar convex bodies, C and D, consider a packing S of n positive homothets of C contained in D. We estimate the total perimeter of the bodies in S, denoted per(S), in terms of per(D) and n. When all homothets of C touch the boundary of the container D, we show that either per(S) = O(log n) or per(S) = O(1), depending on how C and D “fit together”. Apart from the constant factors, these bounds are the best possible. Specifically, we prove that per(S) = O(1) if D is a convex polygon and every side of D is parallel to a corresponding segment on the boundary of C (for short, D is parallel to C) and per(S) = O(log n) otherwise. When D is parallel to C but the homothets of C may lie anywhere in D, we show that per(S) = O((1 + esc(S)) log n / log log n), where esc(S) denotes the total distance of the bodies in S from the boundary of D. Apart from the constant factor, this bound is also the best possible

Constant-Factor Approximation for TSP with Disks

We revisit the traveling salesman problem with neighborhoods (TSPN) and present the first constant-ratio approximation for disks in the plane: Given a set of $n$ disks in the plane, a TSP tour whose length is at most $O(1)$ times the optimal can be computed in time that is polynomial in $n$. Our result is the first constant-ratio approximation for a class of planar convex bodies of arbitrary size and arbitrary intersections. In order to achieve a $O(1)$-approximation, we reduce the traveling salesman problem with disks, up to constant factors, to a minimum weight hitting set problem in a geometric hypergraph. The connection between TSPN and hitting sets in geometric hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure

The traveling salesman problem for lines, balls and planes

We revisit the traveling salesman problem with neighborhoods (TSPN) and propose several new approximation algorithms. These constitute either first approximations (for hyperplanes, lines, and balls in $\mathbb{R}^d$, for $d\geq 3$) or improvements over previous approximations achievable in comparable times (for unit disks in the plane). \smallskip (I) Given a set of $n$ hyperplanes in $\mathbb{R}^d$, a TSP tour whose length is at most $O(1)$ times the optimal can be computed in $O(n)$ time, when $d$ is constant. \smallskip (II) Given a set of $n$ lines in $\mathbb{R}^d$, a TSP tour whose length is at most $O(\log^3 n)$ times the optimal can be computed in polynomial time for all $d$. \smallskip (III) Given a set of $n$ unit balls in $\mathbb{R}^d$, a TSP tour whose length is at most $O(1)$ times the optimal can be computed in polynomial time, when $d$ is constant.Comment: 30 pages, 9 figures; final version to appear in ACM Transactions on Algorithm

Finding heavy hitters from lossy or noisy data

Abstract. Motivated by Dvir et al. and Wigderson and Yehudayoff [3

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Shape optimization problems of higher codimension

The field of shape optimization problems has received a lot of attention in recent years, particularly in relation to a number of applications in physics and engineering that require a focus on shapes instead of parameters or functions. In general for ap- plications the aim is to deform and modify the admissible shapes in order to optimize a given cost function. The fascinating feature is that the variables are shapes, i.e., domains of R^{d}, instead of functions. This choice often produces additional dicul- ties for the existence of a classical solution (that is an optimizing domain) and the introduction of suitable relaxed formulation of the problem is needed in order to get a solution which is in this case a measure. However, we may obtain a classical solution by imposing some geometrical constraint on the class of competing domains or requiring the cost functional verifies some particular conditions. The shape optimization problem is in general an optimization problem of the form min\{F(\Omega): \Omega \in {\cal O} \}; where F is a given cost functional and {\cal O} a class of domains in R^{d}. They are many books written on shape optimization problems. The thesis is organized as follows: the first chapter is dedicated to a brief introduction and presentation of some examples. In Academic examples, we present the isoperimetric problems, minimal and capillary surface problems and the spectral optimization problems while in applied examples the Newton's problem of optimal aerodinamical profile and optimal mixture of two con- ductors are considered. The second chapter is concerned with some basics elements of geometric measure theory that will be used in the sequel. After recalling some notions of abstract measure theory, we deal with the Hausdorff measures which are important for defining the notion of approximate tangent space. Finally we introduce the notion of approximate tangent space to a measure and to a set and also some differential op- erators like tangential differential, tangential gradient and tangential divergence. The third chapter is devoted to the topologies on the set of domains in R^{d}. Three topologies induced by convergence of domains are presented namely the convergence of charac- teristic functions, the convergence in the sense of Hausdorff and the convergence in the sense of compacts as well as the relationship between those different topologies. In the fourth chapter we present a shape optimization problem governed by linear state equations. After dealing with the continuity of the solution of the Laplacian problem with respect to the domain variation (including counter-examples to the continuity and the introduction to a new topology: the gamma-convergence), we analyse the existence of optimal shapes and the necessary condition of optimality in the case where an optimal shape exists. The shape optimization problems governed by nonlinear state equations are treated in chapter ve. The plan of study is the same as in chapter four that is continuity with respect to the domain variation of the solution of the p-Laplacian problem (and more general operator in divergence form), the existence of optimal shapes and the necessary condition of optimality in the case where an optimal shape exists. The last chapter deals with asymptotical shapes. After recalling the notion of Gamma-convergence, we study the asymptotic of the compliance functional in different situations. First we study the asymptotic of an optimal p-compliance-networks which is the compliance associated to p-Laplacian problem with control variables running in the class of one dimensional closed connected sets with assigned length. We provide also the connection with other asymptotic problems like the average distance problem. The asymptotic of the p-compliance-location which deal with the compliance associ- atied to the p-Laplacian problem with control variables running in the class of sets of finite numbers of points, is deduced from the study of the asymptotic of p-compliance- networks. Secondly we study the asymptotic of an optimal compliance-location. In this case we deal with the compliance associated to the classical Laplacian problem and the class of control variables is the class of identics n balls with radius depending on n and with fixed capacity

Hadron models and related New Energy issues

The present book covers a wide-range of issues from alternative hadron models to their likely implications in New Energy research, including alternative interpretation of lowenergy reaction (coldfusion) phenomena. The authors explored some new approaches to describe novel phenomena in particle physics. M Pitkanen introduces his nuclear string hypothesis derived from his Topological Geometrodynamics theory, while E. Goldfain discusses a number of nonlinear dynamics methods, including bifurcation, pattern formation (complex GinzburgLandau equation) to describe elementary particle masses. Fu Yuhua discusses a plausible method for prediction of phenomena related to New Energy development. F. Smarandache discusses his unmatter hypothesis, and A. Yefremov et al. discuss Yang-Mills field from Quaternion Space Geometry. Diego Rapoport discusses theoretical link between Torsion fields and Hadronic Mechanic. A.H. Phillips discusses semiconductor nanodevices, while V. and A. Boju discuss Digital Discrete and Combinatorial methods and their likely implications in New Energy research. Pavel Pintr et al. describe planetary orbit distance from modified Schrödinger equation, and M. Pereira discusses his new Hypergeometrical description of Standard Model of elementary particles. The present volume will be suitable for researchers interested in New Energy issues, in particular their link with alternative hadron models and interpretation

Advances in Discrete Differential Geometry

Differential Geometr

Advances in Discrete Differential Geometry

Differential Geometr