Related papers: Efficient Splitting of Measures and Necklaces
The dispersion problem has been widely studied in computational geometry and facility location, and is closely related to the packing problem. The goal is to locate n points (e.g., facilities or persons) in a k-dimensional polytope, so that…
Clustering is an important technique for identifying structural information in large-scale data analysis, where the underlying dataset may be too large to store. In many applications, recent data can provide more accurate information and…
We resolve a long-standing open question, about the existence of a constant-factor approximation algorithm for the average-case \textsc{Decision Tree} problem with uniform probability distribution over the hypotheses. We answer the question…
$k$-center is one of the most popular clustering models. While it admits a simple 2-approximation in polynomial time in general metrics, the Euclidean version is NP-hard to approximate within a factor of 1.93, even in the plane, if one…
Sampling a diverse set of high-quality solutions for hard optimization problems is of great practical relevance in many scientific disciplines and applications, such as artificial intelligence and operations research. One of the main open…
We propose a simple and efficient clustering method for high-dimensional data with a large number of clusters. Our algorithm achieves high-performance by evaluating distances of datapoints with a subset of the cluster centres. Our…
We address the solution of time-varying optimization problems characterized by the sum of a time-varying strongly convex function and a time-invariant nonsmooth convex function. We design an online algorithmic framework based on…
Many online platforms, ranging from online retail stores to social media platforms, employ algorithms to optimize their offered assortment of items (e.g., products and contents). These algorithms often focus exclusively on achieving the…
Given a finite metric space $(X\cup Y, \mathbf{d})$ the $k$-median problem is to find a set of $k$ centers $C\subseteq Y$ that minimizes $\sum_{p\in X} \min_{c\in C} \mathbf{d}(p,c)$. In general metrics, the best polynomial time algorithm…
The well-known "necklace splitting theorem" of Alon asserts that every $k$-colored necklace can be fairly split into $q$ parts using at most $t$ cuts, provided $k(q-1)\leq t$. In a joint paper with Alon et al. we studied a kind of opposite…
This paper studies the sample complexity (aka number of comparisons) bounds for the active best-$k$ items selection from pairwise comparisons. From a given set of items, the learner can make pairwise comparisons on every pair of items, and…
In the $\varepsilon$-Consensus-Halving problem, we are given $n$ probability measures $v_1, \dots, v_n$ on the interval $R = [0,1]$, and the goal is to partition $R$ into two parts $R^+$ and $R^-$ using at most $n$ cuts, so that $|v_i(R^+)…
In this work, we study diversity-aware clustering problems where the data points are associated with multiple attributes resulting in intersecting groups. A clustering solution needs to ensure that the number of chosen cluster centers from…
The Euclidean $k$-means problem is a classical problem that has been extensively studied in the theoretical computer science, machine learning and the computational geometry communities. In this problem, we are given a set of $n$ points in…
Confidence intervals are a standard technique for analyzing data. When applied to time series, confidence intervals are computed for each time point separately. Alternatively, we can compute confidence bands, where we are required to find…
The $k$-median and $k$-means clustering objectives are classic objectives for modeling clustering in a metric space. Given a set of points in a metric space, the goal of the $k$-median (resp. $k$-means) problem is to find $k$ representative…
Cake cutting is a classic model for studying fair division of a heterogeneous, divisible resource among agents with individual preferences. Addressing cake division under a typical requirement that each agent must receive a connected piece…
We explore Cluster Editing and its generalization Correlation Clustering with a new operation called permissive vertex splitting which addresses finding overlapping clusters in the face of uncertain information. We determine that both…
Convex nonsmooth optimization problems, whose solutions live in very high dimensional spaces, have become ubiquitous. To solve them, the class of first-order algorithms known as proximal splitting algorithms is particularly adequate: they…
Max-k-Cut and correlation clustering are fundamental graph partitioning problems. For a graph with G=(V,E) with n vertices, the methods with the best approximation guarantees for Max-k-Cut and the Max-Agree variant of correlation clustering…