Related papers: A Note on a Recent Algorithm for Minimum Cut
In this work we revisit the fundamental Single-Source Shortest Paths (SSSP) problem with possibly negative edge weights. A recent breakthrough result by Bernstein, Nanongkai and Wulff-Nilsen established a near-linear $O(m \log^8(n)…
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…
There exist many orthogonal graph drawing algorithms that minimize edge crossings or edge bends, however they produce unsatisfactory drawings in many practical cases. In this paper we present a grid-based algorithm for drawing orthogonal…
Let G = (V, E, L) be an edge-labeled graph such that V is the set of vertices, E is the set of edges, L is the set of labels (colors) and each edge e \in E has a label l(e) associated; The goal of the minimum labeling global cut problem…
We consider the problem of enumerating, for a given directed graph $G=(V,E)$ and a node $r\in V$, all directed spanning trees of $G$ rooted at $r$. For undirected graphs, the corresponding problem of enumerating all spanning trees has…
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…
The \emph{maximal $k$-edge-connected subgraphs} problem is a classical graph clustering problem studied since the 70's. Surprisingly, no non-trivial technique for this problem in weighted graphs is known: a very straightforward…
Let $H=(V,E)$ be an edge-weighted hypergraph of rank $r$. Kogan and Krauthgamer extended Bencz\'{u}r and Karger's random sampling scheme for cut sparsification from graphs to hypergraphs. The sampling requires an algorithm for computing the…
The minimum degree spanning tree (MDST) problem requires the construction of a spanning tree $T$ for graph $G=(V,E)$ with $n$ vertices, such that the maximum degree $d$ of $T$ is the smallest among all spanning trees of $G$. In this paper,…
In most of the shortest path problems like vehicle routing problems and network routing problems, we only need an efficient path between two points source and destination, and it is not necessary to calculate the shortest path from source…
In the minimum $k$-edge-connected spanning subgraph ($k$-ECSS) problem the goal is to find the minimum weight subgraph resistant to up to $k-1$ edge failures. This is a central problem in network design, and a natural generalization of the…
Given an edge-weighted directed graph $G=(V,E)$ on $n$ vertices and a set $T=\{t_1, t_2, \ldots, t_p\}$ of $p$ terminals, the objective of the \scss ($p$-SCSS) problem is to find an edge set $H\subseteq E$ of minimum weight such that $G[H]$…
We give an $\tilde{O}(n^{7/5} \log (nC))$-time algorithm to compute a minimum-cost maximum cardinality matching (optimal matching) in $K_h$-minor free graphs with $h=O(1)$ and integer edge weights having magnitude at most $C$. This improves…
We present improved approximation algorithms for some problems in the related areas of Capacitated Network Design and Flexible Graph Connectivity. In the Cap-$k$-ECSS problem, we are given a graph $G=(V,E)$ whose edges have non-negative…
In the decremental $(1+\epsilon)$-approximate Single-Source Shortest Path (SSSP) problem, we are given a graph $G=(V,E)$ with $n = |V|, m = |E|$, undergoing edge deletions, and a distinguished source $s \in V$, and we are asked to process…
This paper shows the weighted matching problem on general graphs can be solved in time $O(n(m + n\log n))$ for $n$ and $m$ the number of vertices and edges, respectively. This was previously known only for bipartite graphs. The crux is a…
We present an implementation of a recent algorithm to compute shortest-path trees in unit disk graphs in $O(n\log n)$ worst-case time, where $n$ is the number of disks. In the minimum-separation problem, we are given $n$ unit disks and two…
In this thesis, we present fast deterministic algorithm to find small cuts in distributed networks. Finding small min-cuts for a network is essential for ensuring the quality of service and reliability. Throughout this thesis, we use the…
We present an algorithm that on input of an $n$-vertex $m$-edge weighted graph $G$ and a value $k$, produces an {\em incremental sparsifier} $\hat{G}$ with $n-1 + m/k$ edges, such that the condition number of $G$ with $\hat{G}$ is bounded…
Consider the single-source shortest paths problem on a directed graph with real-valued edge weights. We solve this problem in $O(n^{2.5}\log^{4.5}n)$ time, improving on prior work of Fineman (STOC 2024) and Huang-Jin-Quanrud (SODA 2025,…