Related papers: Sparsifying Sums of Positive Semidefinite Matrices
The diameter of a graph is one if its most important parameters, being used in many real-word applications. In particular, the diameter dictates how fast information can spread throughout data and communication networks. Thus, it is a…
A polynomial matrix inequality is a formula asserting that a polynomial matrix is positive semidefinite. Polynomial matrix optimization concerns minimizing the smallest eigenvalue of a symmetric polynomial matrix subject to a tuple of…
Spectral Clustering is one of the most traditional methods to solve segmentation problems. Based on Normalized Cuts, it aims at partitioning an image using an objective function defined by a graph. Despite their mathematical attractiveness,…
Discrepancy theory provides powerful tools for producing higher-quality objects which "beat the union bound" in fundamental settings throughout combinatorics and computer science. However, this quality has often come at the price of more…
We prove that every graph has a spectral sparsifier with a number of edges linear in its number of vertices. As linear-sized spectral sparsifiers of complete graphs are expanders, our sparsifiers of arbitrary graphs can be viewed as…
Network embedding has numerous practical applications and has received extensive attention in graph learning, which aims at mapping vertices into a low-dimensional and continuous dense vector space by preserving the underlying structural…
We consider the problem of estimating the spectrum of a symmetric bounded entry (not necessarily PSD) matrix via entrywise sampling. This problem was introduced by [Bhattacharjee, Dexter, Drineas, Musco, Ray '22], where it was shown that…
We compute the spectral density for ensembles of of sparse symmetric random matrices using replica, managing to circumvent difficulties that have been encountered in earlier approaches along the lines first suggested in a seminal paper by…
We present a general framework of designing efficient dynamic approximate algorithms for optimization on undirected graphs. In particular, we develop a technique that, given any problem that admits a certain notion of vertex sparsifiers,…
Graph summarization via node grouping is a popular method to build concise graph representations by grouping nodes from the original graph into supernodes and encoding edges into superedges such that the loss of adjacency information is…
Network (or graph) sparsification compresses a graph by removing inessential edges. By reducing the data volume, it accelerates or even facilitates many downstream analyses. Still, the accuracy of many sparsification methods, with…
Network sparsification is the task of reducing the number of edges of a given graph while preserving some crucial graph property. In community-aware network sparsification, the preserved property concerns the subgraphs that are induced by…
The Sum-of-Squares (SoS) hierarchy of semidefinite programs is a powerful algorithmic paradigm which captures state-of-the-art algorithmic guarantees for a wide array of problems. In the average case setting, SoS lower bounds provide strong…
We study the problem of approximating the cone of positive semidefinite (PSD) matrices with a cone that can be described by smaller-sized PSD constraints. Specifically, we ask the question: "how closely can we approximate the set of…
In the pairwise weighted spanner problem, the input consists of an $n$-vertex-directed graph, where each edge is assigned a cost and a length. Given $k$ vertex pairs and a distance constraint for each pair, the goal is to find a…
Given a graph and an integer $k$, Densest $k$-Subgraph is the algorithmic task of finding the subgraph on $k$ vertices with the maximum number of edges. This is a fundamental problem that has been subject to intense study for decades, with…
Matroid intersection is a classical optimization problem where, given two matroids over the same ground set, the goal is to find the largest common independent set. In this paper, we show that there exists a certain "sparsifer": a subset of…
Designing dynamic graph algorithms against an adaptive adversary is a major goal in the field of dynamic graph algorithms. While a few such algorithms are known for spanning trees, matchings, and single-source shortest paths, very little…
We give an approximation algorithm for non-uniform sparsest cut with the following guarantee: For any $\epsilon,\delta \in (0,1)$, given cost and demand graphs with edge weights $C, D$ respectively, we can find a set $T\subseteq V$ with…
We prove decomposition theorems for sparse positive (semi)definite polynomial matrices that can be viewed as sparsity-exploiting versions of the Hilbert--Artin, Reznick, Putinar, and Putinar--Vasilescu Positivstellens\"atze. First, we…