Related papers: A better lower bound for Lower-Left Anchored Recta…
Kernel-based methods provide flexible and accurate algorithms for the reconstruction of functions from meshless samples. A major question in the use of such methods is the influence of the samples locations on the behavior of the…
We slightly improve the known lower bound on the asymptotic competitive ratio for online bin packing of rectangles. We present a complete proof for the new lower bound, whose value is above 1.91.
We study a variation of the classical Shortest Common Superstring (SCS) problem in which a shortest superstring of a finite set of strings $S$ is sought containing as a factor every string of $S$ or its reversal. We call this problem…
We study the problem of Covering Orthogonal Polygons with Rectangles. For polynomial-time algorithms, the best-known approximation factor is $O(\sqrt{\log n})$ when the input polygon may have holes [Kumar and Ramesh, STOC '99, SICOMP '03],…
We describe a greedy algorithm that approximates the Carleson constant of a collection of general sets. The approximation has a logarithmic loss in a general setting, but is optimal up to a constant with only mild geometric assumptions. The…
The set of 2-dimensional packing problems builds an important class of optimization problems and Strip Packing together with 2-dimensional Bin Packing and 2-dimensional Knapsack is one of the most famous of these problems. Given a set of…
We verify an explicit inequality conjectured recently by Gilmer, thus proving that for any nonempty union-closed family $F \subseteq 2^{[n]}$, some $i\in [n]$ is contained in at least a $\frac{3-\sqrt{5}}{2} \approx 0.38$ fraction of the…
This paper proves a corner occupying theorem for the two-dimensional integral rectangle packing problem, stating that if it is possible to orthogonally place n arbitrarily given integral rectangles into an integral rectangular container…
A subset $S$ of the unit sphere $\mathbb{S}^2$ is called orthogonal-pair-free if and only if there do not exist two distinct points $u, v \in S$ at distance $\frac{\pi}{2}$ from each other. Witsenhausen \cite{witsenhausen} asked the…
Given an edge-weighted metric complete graph with $n$ vertices, the maximum weight metric triangle packing problem is to find a set of $n/3$ vertex-disjoint triangles with the total weight of all triangles in the packing maximized. Several…
We study the properties of a simple greedy algorithm for the generation of data-adapted anisotropic triangulations. Given a function f, the algorithm produces nested triangulations and corresponding piecewise polynomial approximations of f.…
In the classic circle packing problem, one asks whether a given set of circles can be packed into a given container. Packing problems like this have been shown to be $\mathsf{NP}$-hard. In this paper, we present new sufficient conditions…
Set Cover is a classic NP-hard problem; as shown by Slav\'{i}k (1997) the greedy algorithm gives an approximation ratio of $\ln n - \ln \ln n + \Theta(1)$. A series of works by Lund \& Yannakakis (1994), Feige (1998), Moshkovitz (2015) have…
Boros and Furedi (for d=2) and Barany (for abritrary d) proved that there exists a positive real number c_d such that for every set P of n points in R^d in general position, there exists a point of R^d contained in at least c_d…
We study the problem of minimum enclosing rectangle with outliers, which asks to find, for a given set of $n$ planar points, a rectangle with minimum area that encloses at least $(n-t)$ points. The uncovered points are regarded as outliers.…
Almost $50$ years ago Erd\H{o}s and Purdy asked the following question: Given $n$ points in the plane, how many triangles can be approximate congruent to equilateral triangles? They pointed out that by dividing the points evenly into three…
In COCOA 2015, Korman et al. studied the following geometric covering problem: given a set $S$ of $n$ line segments in the plane, find a minimum number of line segments such that every cell in the arrangement of the line segments is…
We consider clustering problems with {\em non-uniform lower bounds and outliers}, and obtain the {\em first approximation guarantees} for these problems. We have a set $\F$ of facilities with lower bounds $\{L_i\}_{i\in\F}$ and a set $\D$…
In 1974, Witsenhausen asked for the maximum possible density $\alpha_n$ of a measurable subset $A$ of the unit sphere $\mathbb{S}^{n-1}\subset \mathbb{R}^n$ such that $A$ contains no pair of orthogonal vectors. For $n=3$, the best known…
We study the problem of guarding the boundary of a simple polygon with a minimum number of guards such that each guard covers a contiguous portion of the boundary. First, we present a simple greedy algorithm for this problem that returns a…