Related papers: Improved Cheeger's Inequality: Analysis of Spectra…
The problem of multiway partitioning of an undirected graph is considered. A spectral method is used, where the k > 2 largest eigenvalues of the normalized adjacency matrix (equivalently, the k smallest eigenvalues of the normalized graph…
A basic fact in spectral graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph. In particular, the graph is disconnected if…
We prove two generalizations of the Cheeger's inequality. The first generalization relates the second eigenvalue to the edge expansion and the vertex expansion of the graph G, $\lambda_2 = \Omega(\phi^V(G) \phi(G))$, where $\phi^V(G)$…
Let G=(V,E) be an undirected graph, lambda_k be the k-th smallest eigenvalue of the normalized laplacian matrix of G. There is a basic fact in algebraic graph theory that lambda_k > 0 if and only if G has at most k-1 connected components.…
Spectral partitioning is a simple, nearly-linear time, algorithm to find sparse cuts, and the Cheeger inequalities provide a worst-case guarantee for the quality of the approximation found by the algorithm. Local graph partitioning…
Cheeger's fundamental inequality states that any edge-weighted graph has a vertex subset $S$ such that its expansion (a.k.a. conductance) is bounded as follows: \[ \phi(S) \defeq \frac{w(S,\bar{S})}{\min \set{w(S), w(\bar{S})}} \leq…
Given an undirected graph $G$, the classical Cheeger constant, $h_G$, measures the optimal partition of the vertices into 2 parts with relatively few edges between them based upon the sizes of the parts. The well-known Cheeger's inequality…
Partition problems in graphs are extremely important in applications, as shown in the Data science and Machine learning literature. One approach is spectral partitioning based on a Fiedler vector, i.e., an eigenvector corresponding to the…
We introduce the $st$-cut version the Sparsest-Cut problem, where the goal is to find a cut of minimum sparsity among those separating two distinguished vertices $s,t\in V$. Clearly, this problem is at least as hard as the usual (non-$st$)…
A basic fact in algebraic graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue 1 in the normalized adjacency matrix of the graph. In particular, the graph is…
We give new, improved bounds for approximating the sparsest cut value or in other words the conductance $\phi$ of a graph in the CONGEST model. As our main result, we present an algorithm running in $O(\log^2 n/\phi)$ rounds in which every…
Partitioning the vertices of a graph into two roughly equal parts while minimizing the number of edges crossing the cut is a fundamental problem (called Balanced Separator) that arises in many settings. For this problem, and variants such…
We consider the minimum-cut partitioning of a graph into more than two parts using spectral methods. While there exist well-established spectral algorithms for this problem that give good results, they have traditionally not been well…
We establish that a simple polynomial-time algorithm that we call reweighted spectral partitioning obtains small 2/3-balanced vertex-separators for a number of graph classes, including $O(\sqrt{n})$-sized separators for planar graphs,…
The celebrated Cheeger's Inequality establishes a bound on the edge expansion of a graph via its spectrum. This inequality is central to a rich spectral theory of graphs, based on studying the eigenvalues and eigenvectors of the adjacency…
Given a graph $G$, the sparsest-cut problem asks to find the set of vertices $S$ which has the least expansion defined as $$\phi_G(S) := \frac{w(E(S,\bar{S}))}{\min \set{w(S), w(\bar{S})}}, $$ where $w$ is the total edge weight of a subset.…
Cheeger's inequality states that a tightly connected subset can be extracted from a graph $G$ using an eigenvector of the normalized Laplacian associated with $G$. More specifically, we can compute a subset with conductance…
Motivated by applications of large-scale graph clustering, we study random-walk-based LOCAL algorithms whose running times depend only on the size of the output cluster, rather than the entire graph. All previously known such algorithms…
The Cheeger inequality for undirected graphs, which relates the conductance of an undirected graph and the second smallest eigenvalue of its normalized Laplacian, is a cornerstone of spectral graph theory. The Cheeger inequality has been…
Recent years have seen extensive research on directed graph sparsification. In this work, we initiate the study of fast fully dynamic spectral and cut sparsification algorithms for directed graphs. We introduce a new notion of spectral…