Related papers: Short Cycles via Low-Diameter Decompositions
This paper presents new deterministic and distributed low-diameter decomposition algorithms for weighted graphs. In particular, we show that if one can efficiently compute approximate distances in a parallel or a distributed setting, one…
We give the first truly subquadratic time algorithm, with $O^*(n^{2-1/18})$ running time, for computing the diameter of an $n$-vertex unit-disk graph, resolving a central open problem in the literature. Our result is obtained as an instance…
Graph sparsification serves as a foundation for many algorithms, such as approximation algorithms for graph cuts and Laplacian system solvers. As its natural generalization, hypergraph sparsification has recently gained increasing…
Graph-based nearest neighbor search methods have seen a surge of popularity in recent years, offering state-of-the-art performance across a wide variety of applications. Central to these methods is the task of constructing a sparse…
Graph coloring is one of the central problems in distributed graph algorithms. Much of the research on this topic has focused on coloring with $\Delta+1$ colors, where $\Delta$ denotes the maximum degree. Using $\Delta+1$ colors may be…
Dynamically maintaining the minimum cut in a graph $G$ under edge insertions and deletions is a fundamental problem in dynamic graph algorithms for which no conditional lower bound on the time per operation exists. In an $n$-node graph the…
We prove that for any integer $k\geq 2$ and $\varepsilon>0$, there is an integer $\ell_0\geq 1$ such that any $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $(1/2+\varepsilon)n$ has a fractional decomposition into…
We present a dynamic algorithm for maintaining the connected and 2-edge-connected components in an undirected graph subject to edge deletions. The algorithm is Monte-Carlo randomized and processes any sequence of edge deletions in $O(m + n…
We present an $\tilde O(m+n^{1.5})$-time randomized algorithm for maximum cardinality bipartite matching and related problems (e.g. transshipment, negative-weight shortest paths, and optimal transport) on $m$-edge, $n$-node graphs. For…
We make progress on three long standing conjectures from the 1960s about path and cycle decompositions of graphs. Gallai conjectured that any connected graph on $n$ vertices can be decomposed into at most $\left\lceil…
Graph sparsification is an area of interest in computer science and applied mathematics. Sparsification of a graph, in general, aims to reduce the number of edges in the network while preserving specific properties of the graph, like cuts…
We consider the problem of computing the diameter of a unicycle graph (i.e., a graph with a unique cycle). We present an O(n) time algorithm for the problem, where n is the number of vertices of the graph. This improves the previous best…
We present the first polynomial-time algorithm for computing a near-optimal \emph{flow}-expander decomposition. Given a graph $G$ and a parameter $\phi$, our algorithm removes at most a $\phi\log^{1+o(1)}n$ fraction of edges so that every…
We prove that a complete bipartite graph can be decomposed into cycles of arbitrary specified lengths provided that the obvious necessary conditions are satisfied, the length of each cycle is at most the size of the smallest part, and the…
We consider the problem of sampling and approximately counting an arbitrary given motif $H$ in a graph $G$, where access to $G$ is given via queries: degree, neighbor, and pair, as well as uniform edge sample queries. Previous algorithms…
Expander decompositions form the basis of one of the most flexible paradigms for close-to-linear-time graph algorithms. Length-constrained expander decompositions generalize this paradigm to better work for problems with lengths, distances…
We study deterministic algorithms for computing graph cuts, with focus on two fundamental problems: balanced sparse cut and $k$-vertex connectivity for small $k$ ($k=O(\polylog n)$). Both problems can be solved in near-linear time with…
We develop a sparse graph regularity method that applies to graphs with few 4-cycles, including new counting and removal lemmas for 5-cycles in such graphs. Some applications include: * Every $n$-vertex graph with no 5-cycle can be made…
We use exponential start time clustering to design faster and more work-efficient parallel graph algorithms involving distances. Previous algorithms usually rely on graph decomposition routines with strict restrictions on the diameters of…
In recent years, hypergraph generalizations of many graph cut problems have been introduced and analyzed as a way to better explore and understand complex systems and datasets characterized by multiway relationships. Recent work has made…