Related papers: Few Cuts Meet Many Point Sets
The Ham-Sandwich theorem is a well-known result in geometry. It states that any $d$ mass distributions in $\mathbb{R}^d$ can be simultaneously bisected by a hyperplane. The result is tight, that is, there are examples of $d+1$ mass…
We investigate the partitioning of partial orders into a minimal number of heapable subsets. We prove a characterization result reminiscent of the proof of Dilworth's theorem, which yields as a byproduct a flow-based algorithm for computing…
We prove a common generalization to several mass partition results using hyperplane arrangements to split $\mathbb{R}^d$ into two sets. Our main result implies the ham-sandwich theorem, the necklace splitting theorem for two thieves, a…
The classic Ham-Sandwich theorem states that for any $d$ measurable sets in $\mathbb{R}^d$, there is a hyperplane that bisects them simultaneously. An extension by B\'ar\'any, Hubard, and Jer\'onimo [DCG 2008] states that if the sets are…
The linear-time ham-sandwich cut algorithm of Lo, Matou\v{s}ek, and Steiger for bi-chromatic finite point sets in the plane works by appropriately selecting crossings of the lines in the dual line arrangement with a set of well-chosen…
Multicuts enable to conveniently represent discrete graphical models for unsupervised and supervised image segmentation, in the case of local energy functions that exhibit symmetries. The basic Potts model and natural extensions thereof to…
Consider the Hitting Set problem where, for a given universe $\mathcal{X} = \left\{ 1, ... , n \right\}$ and a collection of subsets $\mathcal{S}_1, ... , \mathcal{S}_m$, one seeks to identify the smallest subset of $\mathcal{X}$ which has…
We study some measure partition problems: Cut the same positive fraction of $d+1$ measures in $\mathbb R^d$ with a hyperplane or find a convex subset of $\mathbb R^d$ on which $d+1$ given measures have the same prescribed value. For both…
Let $\D$ be a set of $n$ pairwise disjoint unit balls in $\R^d$ and $P$ the set of their center points. A hyperplane $\Hy$ is an \emph{$m$-separator} for $\D$ if each closed halfspace bounded by $\Hy$ contains at least $m$ points from $P$.…
Graphs are a natural representation of data from various contexts, such as social connections, the web, road networks, and many more. In the last decades, many of these networks have become enormous, requiring efficient algorithms to cut…
As two fundamental problems, graph cuts and graph matching have been investigated over decades, resulting in vast literature in these two topics respectively. However the way of jointly applying and solving graph cuts and matching receives…
Identifying breakpoints in piecewise regression is critical in enhancing the reliability and interpretability of data fitting. In this paper, we propose novel algorithms based on the greedy algorithm to accurately and efficiently identify…
In this paper we study the problem of locating a given number of hyperplanes minimizing an objective function of the closest distances from a set of points. We propose a general framework for the problem in which norm-based distances…
Balanced partitioning is often a crucial first step in solving large-scale graph optimization problems, e.g., in some cases, a big graph can be chopped into pieces that fit on one machine to be processed independently before stitching the…
Most state-of-the-art motion segmentation algorithms draw their potential from modeling motion differences of local entities such as point trajectories in terms of pairwise potentials in graphical models. Inference in instances of minimum…
The minimum cut problem for an undirected edge-weighted graph asks us to divide its set of nodes into two blocks while minimizing the weight sum of the cut edges. Here, we introduce a linear-time algorithm to compute near-minimum cuts. Our…
The minimum $s$-$t$ cut problem in graphs is one of the most fundamental problems in combinatorial optimization, and graph cuts underlie algorithms throughout discrete mathematics, theoretical computer science, operations research, and data…
In connection with the needs of solving optimization problems, the development of conditional minimization methods with convenient numerical implementation continues to attract the attention of mathematicians. In this monograph we propose…
Motivated by federated learning, we consider the hub-and-spoke model of distributed optimization in which a central authority coordinates the computation of a solution among many agents while limiting communication. We first study some past…
To cope with the high level of ambiguity faced in domains such as Computer Vision or Natural Language processing, robust prediction methods often search for a diverse set of high-quality candidate solutions or proposals. In structured…