Related papers: Unique Set Cover on Unit Disks and Unit Squares

We propose faster algorithms for the following three optimization problems on $n$ collinear points, i.e., points in dimension one. The first two problems are known to be NP-hard in higher dimensions. 1- Maximizing total area of disjoint…

Given a polygon $P$, for two points $s$ and $t$ contained in the polygon, their \emph{geodesic distance} is the length of the shortest $st$-path within $P$. A \emph{geodesic disk} of radius $r$ centered at a point $v \in P$ is the set of…

We present algorithms for the online minimum hitting set problem in geometric range spaces: given a set $P$ of $n$ points in the plane and a sequence of geometric objects that arrive one-by-one, we need to maintain a hitting set at all…

Retraction note: After posting the manuscript on arXiv, we were informed by Erik Jan van Leeuwen that both results were known and they appeared in his thesis[vL09]. A PTAS for MDS is at Theorem 6.3.21 on page 79 and A PTAS for MCDS is at…

We study the complexity of Maximum Clique in intersection graphs of convex objects in the plane. On the algorithmic side, we extend the polynomial-time algorithm for unit disks [Clark '90, Raghavan and Spinrad '03] to translates of any…

The optimization version of the Unique Label Cover problem is at the heart of the Unique Games Conjecture which has played an important role in the proof of several tight inapproximability results. In recent years, this problem has been…

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for \textsc{Maximum Clique} on unit disk graphs [Clark, Colbourn, Johnson; Discrete…

Let $G=(V, E)$ be a simple undirected graph. A closed neighbourhood of an edge $e=uv$ between two vertices $u$ and $v$ of $G$, denoted by $N_G[e]$, is the set of vertices in the neighbourhood of $u$ and $v$ including $\{u,v\}$. A subset $L$…

A system of sets forms an {\em $m$-fold covering} of a set $X$ if every point of $X$ belongs to at least $m$ of its members. A $1$-fold covering is called a {\em covering}. The problem of splitting multiple coverings into several coverings…

We consider variants of the following multi-covering problem with disks. We are given two point sets $Y$ (servers) and $X$ (clients) in the plane, a coverage function $\kappa :X \rightarrow \mathcal{N}$, and a constant $\alpha \geq 1$.…

In the classical partial vertex cover problem, we are given a graph $G$ and two positive integers $R$ and $L$. The goal is to check whether there is a subset $V'$ of $V$ of size at most $R$, such that $V'$ covers at least $L$ edges of $G$.…

In the stochastic set cover problem (Grandoni et al., FOCS '08), we are given a collection $\mathcal{S}$ of $m$ sets over a universe $\mathcal{U}$ of size $N$, and a distribution $D$ over elements of $\mathcal{U}$. The algorithm draws $n$…

We introduce a new technique for designing fixed-parameter algorithms for cut problems, namely randomized contractions. We apply our framework to obtain the first FPT algorithm for the Unique Label Cover problem and new FPT algorithms with…

We give a review of results on the minimum convex cover and maximum hidden set problems. In addition, we give some new results. First we show that it is NP-hard to determine whether a polygon has the same convex cover number as its hidden…

Consider the following variant of the set cover problem. We are given a universe $U=\{1,...,n\}$ and a collection of subsets $\mathcal{C} = \{S_1,...,S_m\}$ where $S_i \subseteq U$. For every element $u \in U$ we need to find a set $\phi(u)…

For a set of n points in the plane, we consider the axis--aligned (p,k)-Box Covering problem: Find p axis-aligned, pairwise-disjoint boxes that together contain n-k points. In this paper, we consider the boxes to be either squares or…

We describe a new numerical procedure for generating dense packings of disks and spheres inside various geometric shapes. We believe that in some of the smaller cases, these packings are in fact optimal. When applied to the previously…

We consider the problem of cutting a set of edges on a polyhedral manifold surface, possibly with boundary, to obtain a single topological disk, minimizing either the total number of cut edges or their total length. We show that this…

In this paper we discuss a result similar to the polynomial version of the Alon-F\"uredi theorem. We prove that if you want to cover the vertices of the $n$-dimensional unit cube, except those of weight at most $r$ then you need an…

Unique-sink orientations (USOs) are an abstract class of orientations of the n-cube graph. We consider some classes of USOs that are of interest in connection with the linear complementarity problem. We summarise old and show new lower and…