相关论文: Towards Resistance Sparsifiers

This work presents inGRASS, a novel algorithm designed for incremental spectral sparsification of large undirected graphs. The proposed inGRASS algorithm is highly scalable and parallel-friendly, having a nearly-linear time complexity for…

Motivated by the problem of routing reliably and scalably in a graph, we introduce the notion of a splicer, the union of spanning trees of a graph. We prove that for any bounded-degree n-vertex graph, the union of two random spanning trees…

We study the problem of compressing a weighted graph $G$ on $n$ vertices, building a "sketch" $H$ of $G$, so that given any vector $x \in \mathbb{R}^n$, the value $x^T L_G x$ can be approximated up to a multiplicative $1+\epsilon$ factor…

Effective resistances are ubiquitous in graph algorithms and network analysis. In this work, we study sublinear time algorithms to approximate the effective resistance of an adjacent pair $s$ and $t$. We consider the classical adjacency…

The interconnectedness and interdependence of modern graphs are growing ever more complex, causing enormous resources for processing, storage, communication, and decision-making of these graphs. In this work, we focus on the task graph…

We show that every locally sparse graph contains a linearly sized expanding subgraph. For constants $c_1>c_2>1$, $0<\alpha<1$, a graph $G$ on $n$ vertices is called a $(c_1,c_2,\alpha)$-graph if it has at least $c_1n$ edges, but every…

Spectral sparsification is a general technique developed by Spielman et al. to reduce the number of edges in a graph while retaining its structural properties. We investigate the use of spectral sparsification to produce good visual…

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…

In this paper, we revisit spectral sparsification for sums of arbitrary positive semidefinite (PSD) matrices. Concretely, for any collection of PSD matrices $\mathcal{A} = \{A_1, A_2, \ldots, A_r\} \subset \mathbb{R}^{n \times n}$, given…

Spectral graph sparsification aims to find ultra-sparse subgraphs which can preserve spectral properties of original graphs. In this paper, a new spectral criticality metric based on trace reduction is first introduced for identifying…

We present an algorithm that given any $n$-vertex, $m$-edge, rank $r$ hypergraph constructs a spectral sparsifier with $O(n \varepsilon^{-2} \log n \log r)$ hyperedges in nearly-linear $\widetilde{O}(mr)$ time. This improves in both size…

In graph sparsification, the goal has almost always been of {global} nature: compress a graph into a smaller subgraph ({sparsifier}) that maintains certain features of the original graph. Algorithms can then run on the sparsifier, which in…

Recently, Chalermsook et al. [SODA'21(arXiv:2007.07862)] introduces a notion of vertex sparsifiers for $c$-edge connectivity, which has found applications in parameterized algorithms for network design and also led to exciting dynamic…

Cuts in graphs are a fundamental object of study, and play a central role in the study of graph algorithms. The problem of sparsifying a graph while approximately preserving its cut structure has been extensively studied and has many…

We study vertex sparsification for preserving distances in planar graphs. Given an edge-weighted planar graph with $k$ terminals, the goal is to construct an emulator, which is a smaller edge-weighted planar graph that contains the…

We consider the problem of dynamically maintaining (approximate) all-pairs effective resistances in separable graphs, which are those that admit an $n^{c}$-separator theorem for some $c<1$. We give a fully dynamic algorithm that maintains…

A hypergraph spectral sparsifier of a hypergraph $G$ is a weighted subgraph $H$ that approximates the Laplacian of $G$ to a specified precision. Recent work has shown that similar to ordinary graphs, there exist $\widetilde{O}(n)$-size…

Graph sparsification is to approximate an arbitrary graph by a sparse graph and is useful in many applications, such as simplification of social networks, least squares problems, numerical solution of symmetric positive definite linear…

We introduce a new algorithmic framework for designing dynamic graph algorithms in minor-free graphs, by exploiting the structure of such graphs and a tool called vertex sparsification, which is a way to compress large graphs into small…

Preservers and additive spanners are sparse (hence cheap to store) subgraphs that preserve the distances between given pairs of nodes exactly or with some small additive error, respectively. Since real-world networks are prone to failures,…