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Permutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this…
We present an optimal O*(n^2) time algorithm for deciding if a metric space (X,d) on n points can be isometrically embedded into the plane endowed with the l_1-metric. It improves the O*(n^2 log^2 n) time algorithm of J. Edmonds (2008).…
In this paper, we obtain a number of new simple pseudo-polynomial time algorithms on the well-known knapsack problem, focusing on the running time dependency on the number of items $n$, the maximum item weight $w_\mathrm{max}$, and the…
In this paper, we study matrix scaling and balancing, which are fundamental problems in scientific computing, with a long line of work on them that dates back to the 1960s. We provide algorithms for both these problems that, ignoring…
We study the problem of estimating the covariance matrix of a high-dimensional distribution when a small constant fraction of the samples can be arbitrarily corrupted. Recent work gave the first polynomial time algorithms for this problem…
Let $P$ be an orthogonal polygon of $n$ vertices, without holes. The Orthogonal Polygon Covering with Squares (OPCS) problem takes as input such an orthogonal polygon $P$ with integral vertex coordinates, and asks to find the minimum number…
We show that the problem of finding the simplex of largest volume in the convex hull of $n$ points in $\mathbb{Q}^d$ can be approximated with a factor of $O(\log d)^{d/2}$ in polynomial time. This improves upon the previously best known…
Given a set $P$ of $n$ points in the plane, the unit-disk graph $G(P)$ is a graph with $P$ as its vertex set such that two points of $P$ have an edge if their Euclidean distance is at most $1$. We consider the problem of computing a maximum…
In the maximum independent set of convex polygons problem, we are given a set of $n$ convex polygons in the plane with the objective of selecting a maximum cardinality subset of non-overlapping polygons. Here we study a special case of the…
N-fold integer programming is a fundamental problem with a variety of natural applications in operations research and statistics. Moreover, it is universal and provides a new, variable-dimension, parametrization of all of integer…
Union volume estimation is a classical algorithmic problem. Given a family of objects $O_1,\ldots,O_n \subseteq \mathbb{R}^d$, we want to approximate the volume of their union. In the special case where all objects are boxes (also known as…
In the two-dimensional orthogonal colored range counting problem, we preprocess a set, $P$, of $n$ colored points on the plane, such that given an orthogonal query rectangle, the number of distinct colors of the points contained in this…
We study the Multiple Cluster Scheduling problem and the Multiple Strip Packing problem. For both problems, there is no algorithm with approximation ratio better than $2$ unless $P = NP$. In this paper, we present an algorithm with…
Packing is a classical problem where one is given a set of subsets of Euclidean space called objects, and the goal is to find a maximum size subset of objects that are pairwise non-intersecting. The problem is also known as the Independent…
A polygonal curve $P$ with $n$ vertices is $c$-packed, if the sum of the lengths of the parts of the edges of the curve that are inside any disk of radius $r$ is at most $cr$, for any $r>0$. Similarly, the concept of $c$-packedness can be…
We consider the problem of packing rectangles into bins that are unit squares, where the goal is to minimize the number of bins used. All rectangles have to be packed non-overlapping and orthogonal, i.e., axis-parallel. We present an…
In the Orthogonal Vectors problem (OV), we are given two families $A, B$ of subsets of $\{1,\ldots,d\}$, each of size $n$, and the task is to decide whether there exists a pair $a \in A$ and $b \in B$ such that $a \cap b = \emptyset$.…
Higher-dimensional orthogonal packing problems have a wide range of practical applications, including packing, cutting, and scheduling. Combining the use of our data structure for characterizing feasible packings with our new classes of…
$\renewcommand{\Re}{\mathbb{R}}$ We develop a general randomized technique for solving "implic it" linear programming problems, where the collection of constraints are defined implicitly by an underlying ground set of elements. In many…
Given in the plane a set of points and a set of halfplanes, we consider the problem of computing a smallest subset of halfplanes whose union covers all points. In this paper, we present an $O(n^{4/3}\log^{5/3}n\log^{O(1)}\log n)$-time…