Related papers: Quantum Speedup for Graph Sparsification, Cut Appr…
For any undirected and weighted graph $G=(V,E,w)$ with $n$ vertices and $m$ edges, we call a sparse subgraph $H$ of $G$, with proper reweighting of the edges, a $(1+\varepsilon)$-spectral sparsifier if \[…
We study the problem of graph and hypergraph sparsification in insertion-only data streams. The input is a hypergraph $H=(V, E, w)$ with $n$ nodes, $m$ hyperedges, and rank $r$, and the goal is to compute a hypergraph $\widehat{H}$ that…
Quantum image processing is a growing field attracting attention from both the quantum computing and image processing communities. We propose a novel method in combining a graph-theoretic approach for optimal surface segmentation and hybrid…
The sparsest cut problem consists of identifying a small set of edges that breaks the graph into balanced sets of vertices. The normalized cut problem balances the total degree, instead of the size, of the resulting sets. Applications of…
We introduce a novel quantum algorithm for similar subgraph identification in form of an NP-hard cardinality-constrained binary quadratic optimization problem. Given a weighted reference graph with Laplacian $\boldsymbol{B}$, our algorithm…
Two kinds of approximation algorithms exist for the k-BALANCED PARTITIONING problem: those that are fast but compute unsatisfying approximation ratios, and those that guarantee high quality ratios but are slow. In this paper we prove that…
Spectral sparsification for directed Eulerian graphs is a key component in the design of fast algorithms for solving directed Laplacian linear systems. Directed Laplacian linear system solvers are crucial algorithmic primitives to fast…
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…
Reducing the running time of graph algorithms is vital for tackling real-world problems such as shortest paths and matching in large-scale graphs, where path information plays a crucial role. To address this critical challenge, this paper…
We show that the sparsified block elimination algorithm for solving undirected Laplacian linear systems from [Kyng-Lee-Peng-Sachdeva-Spielman STOC'16] directly works for directed Laplacians. Given access to a sparsification algorithm that,…
We initiate the study of dynamic algorithms for graph sparsification problems and obtain fully dynamic algorithms, allowing both edge insertions and edge deletions, that take polylogarithmic time after each update in the graph. Our three…
A seminal work of [Ahn-Guha-McGregor, PODS'12] showed that one can compute a cut sparsifier of an unweighted undirected graph by taking a near-linear number of linear measurements on the graph. Subsequent works also studied computing other…
The minimum cut problem in an undirected and weighted graph $G$ is to find the minimum total weight of a set of edges whose removal disconnects $G$. We completely characterize the quantum query and time complexity of the minimum cut problem…
Given an undirected graph $G$ and an error parameter $\epsilon > 0$, the {\em graph sparsification} problem requires sampling edges in $G$ and giving the sampled edges appropriate weights to obtain a sparse graph $G_{\epsilon}$ with the…
We give an algorithm for finding the arboricity of a weighted, undirected graph, defined as the minimum number of spanning forests that cover all edges of the graph, in $\sqrt{n} m^{1+o(1)}$ time. This improves on the previous best bound of…
This paper proposes a scalable algorithmic framework for spectral reduction of large undirected graphs. The proposed method allows computing much smaller graphs while preserving the key spectral (structural) properties of the original…
Uncertain graphs are prevalent in several applications including communications systems, biological databases and social networks. The ever increasing size of the underlying data renders both graph storage and query processing extremely…
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,…
A graph G'(V,E') is an \eps-sparsification of G for some \eps>0, if every (weighted) cut in G' is within (1\pm \eps) of the corresponding cut in G. A celebrated result of Benczur and Karger shows that for every undirected graph G, an…
For any undirected graph $G=(V,E)$ and a set $E_W$ of candidate edges with $E\cap E_W=\emptyset$, the $(k,\gamma)$-spectral augmentability problem is to find a set $F$ of $k$ edges from $E_W$ with appropriate weighting, such that the…