Related papers: Maximum Solow--Polasky Diversity Subset Selection …
A topological space has the fixed point property if every continuous self-map of that space has at least one fixed point. We demonstrate that there are serious restraints imposed by the requirement that there be a choice of fixed points…
We study the power of the bounded-width consistency algorithm in the context of the fixed-template Promise Constraint Satisfaction Problem (PCSP). Our main technical finding is that the template of every PCSP that is solvable in bounded…
In the subspace approximation problem, we seek a k-dimensional subspace F of R^d that minimizes the sum of p-th powers of Euclidean distances to a given set of n points a_1, ..., a_n in R^d, for p >= 1. More generally than minimizing sum_i…
Good approximations have been attained for the sparsest cut problem by rounding solutions to convex relaxations via low-distortion metric embeddings. Recently, Bryant and Tupper showed that this approach extends to the hypergraph setting by…
A standard type of uncertainty set in robust optimization is budgeted uncertainty, where an interval of possible values for each parameter is given and the total deviation from their lower bounds is bounded. In the two-stage setting,…
The Urysohn space is a separable complete metric space with two fundamental properties: (a) universality: every separable metric space can be isometrically embedded in it; (b) ultrahomogeneity: every finite isometry between two finite…
We propose the first general and scalable framework to design certifiable algorithms for robust geometric perception in the presence of outliers. Our first contribution is to show that estimation using common robust costs, such as truncated…
Thickenings of a metric space capture local geometric properties of the space. Here we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the…
In this paper we prove that the set of metrics conformal to the standard metric on $\mathbb{S}^{n}\backslash\{p_{1},\cdots,p_{l}\}$ is locally compact in $C^{m,\alpha}$ topology for any $m>0$, whenever the metrics have constant $\sigma_{k}$…
The concept of convex compactness, weaker than the classical notion of compactness, is introduced and discussed. It is shown that a large class of convex subsets of topological vector spaces shares this property and that is can be used in…
We introduce a model of the set of all Polish (=separable complete metric) spaces: the cone $\cal R$ of distance matrices, and consider geometric and probabilistic problems connected with this object. The notion of the universal distance…
A framework (a straight-line embedding of a graph into a normed space allowing edges to cross) is globally rigid if any other framework with the same edge lengths with respect to the chosen norm is an isometric copy. We investigate global…
Let $(M,g)$ be a closed Riemannian manifold of dimension $n$, and $k\geq 1$ an integer such that $n>2k$. We show that there exists $B_0>0$ such that for all $u \in H^{k}(M)$, \[\|u\|_{L^{2^\sharp}(M)}^2 \leq K_0^2 \int_M |\Delta_g^{k/2}…
The PL geometric category of a polyhedron $P$, denoted $\hbox{plgcat}(P)$, provides a natural upper bound for the Lusternik--Schnirelmann category and it is defined as the minimum number of PL collapsible subpolyhedra of $P$ that cover $P$.…
The basic input for many real objects is a finite cloud of unordered points. The strongest equivalence between objects in practice is rigid motion in a Euclidean space. A recent polynomial-time classification of point clouds required a…
A well-known result says that the Euclidean unit ball is the unique fixed point of the polarity operator. This result implies that if, in $\mathbb{R}^n$, the unit ball of some norm is equal to the unit ball of the dual norm, then the norm…
Maximum diversity problems arise in many practical settings from facility location to social networks, and constitute an important class of NP-hard problems in combinatorial optimization. There has been a growing interest in these problems…
We study the computational complexity of several problems connected with finding a maximal distance-$k$ matching of minimum cardinality or minimum weight in a given graph. We introduce the class of $k$-equimatchable graphs which is an edge…
A Euclidean noncrossing Steiner $(1+\epsilon)$-spanner for a point set $P\subset\mathbb{R}^2$ is a planar straight-line graph that, for any two points $a, b \in P$, contains a path whose length is at most $1+\epsilon$ times the Euclidean…
We prove that there is an absolute constant $ C$ such that for every $ n \geq 2 $ and $ N\geq 10^n, $ there exists a polytope $ P_{n,N} \subset \mathbb{R}^n $ with at most $ N $ facets that satisfies…