Related papers: A Nearly Tight Lower Bound for the $d$-Dimensional…
The optimal transport problem has many applications in machine learning, physics, biology, economics, etc. Although its goal is very clear and mathematically well-defined, finding its optimal solution can be challenging for large datasets…
A family of $k$ point sets in $d$ dimensions is well-separated if the convex hulls of any two disjoint subfamilies can be separated by a hyperplane. Well-separation is a strong assumption that allows us to conclude that certain kinds of…
In this paper we find an upper bound for the probability that a $3$ dimensional simple random walk covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_1$ ball of radius $N$. For $d\ge 4$, it has been shown in…
In the Traveling Salesperson Problem with Neighborhoods (TSPN), we are given a collection of geometric regions in some space. The goal is to output a tour of minimum length that visits at least one point in each region. Even in the…
Higher-dimensional orthogonal packing problems have a wide range of practical applications, including packing, cutting, and scheduling. In the context of a branch-and-bound framework for solving these packing problems to optimality, it is…
Optimal transport is a machine learning problem with applications including distribution comparison, feature selection, and generative adversarial networks. In this paper, we propose feature-robust optimal transport (FROT) for…
Given an $n \times d$ dimensional dataset $A$, a projection query specifies a subset $C \subseteq [d]$ of columns which yields a new $n \times |C|$ array. We study the space complexity of computing data analysis functions over such…
A landmark result of non-smooth convex optimization is that gradient descent is an optimal algorithm whenever the number of computed gradients is smaller than the dimension $d$. In this paper we study the extension of this result to the…
We study the d-dimensional hypercube knapsack problem where we are given a set of d-dimensional hypercubes with associated profits, and a knapsack which is a unit d-dimensional hypercube. The goal is to find an axis-aligned non-overlapping…
We study the online problem of assigning a moving point to a base-station region that contains it. For instance, the moving object could represent a cellular phone and the base station could represent the coverage zones of cell towers. Our…
In many robotic tasks, such as autonomous drone racing, the goal is to travel through a set of waypoints as fast as possible. A key challenge for this task is planning the time-optimal trajectory, which is typically solved by assuming…
We consider one of the simplest and best known buffer management architectures: the shared memory switch with multiple output queues and uniform packets. It was one of the first models studied by competitive analysis, with the Longest Queue…
Path Planning and target searching in a three-dimensional environment is a challenging task in the field of robotics. It is an optimization problem as the path from source to destination has to be optimal. This paper aims to generate a…
This paper presents the reachability analysis of curves in $\mathbb{R}^3$ with a prescribed curvature bound. Based on Pontryagin Maximum Principle, we leverage the existing knowledge on the structure of solutions to minimum-time problems,…
Consider the geometric range space $(X, \mathcal{H}_d)$ where $X \subset \mathbb{R}^d$ and $\mathcal{H}_d$ is the set of ranges defined by $d$-dimensional halfspaces. In this setting we consider that $X$ is the disjoint union of a red and…
Gap Hamming Distance is a well-studied problem in communication complexity, in which Alice and Bob have to decide whether the Hamming distance between their respective n-bit inputs is less than n/2-sqrt(n) or greater than n/2+sqrt(n). We…
We consider the optimal transport problem between a set of $n$ red points and a set of $n$ blue points subject to a concave cost function such as $c(x,y) = \|x-y\|^{p}$ for $0< p < 1$. Our focus is on a particularly simple matching…
The coordinate-wise median is a classic and most well-studied strategy-proof mechanism in social choice and facility location scenarios. Surprisingly, there is no systematic study of its approximation ratio in $d$-dimensional spaces. The…
We study the NP-complete Maximum Outerplanar Subgraph problem. The previous best known approximation ratio for this problem is 2/3. We propose a new approximation algorithm which improves the ratio to 7/10.
We present a partial-differential-equation-based optimal path-planning framework for curvature constrained motion, with application to vehicles in 2- and 3-spatial-dimensions. This formulation relies on optimal control theory, dynamic…