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Proximity graph-based methods have emerged as a leading paradigm for approximate nearest neighbor (ANN) search in the system community. This paper presents fresh insights into the theoretical foundation of these methods. We describe an…

Data Structures and Algorithms · Computer Science 2025-09-10 Shangqi Lu , Yufei Tao

Let $H$ be a fixed undirected graph on $k$ vertices. The $H$-hitting set problem asks for deleting a minimum number of vertices from a given graph $G$ in such a way that the resulting graph has no copies of $H$ as a subgraph. This problem…

Data Structures and Algorithms · Computer Science 2020-12-01 Noah Brüstle , Tal Elbaz , Hamed Hatami , Onur Kocer , Bingchan Ma

One of the simplest problems on directed graphs is that of identifying the set of vertices reachable from a designated source vertex. This problem can be solved easily sequentially by performing a graph search, but efficient parallel…

Data Structures and Algorithms · Computer Science 2017-11-07 Jeremy T. Fineman

In the pairwise weighted spanner problem, the input consists of an $n$-vertex-directed graph, where each edge is assigned a cost and a length. Given $k$ vertex pairs and a distance constraint for each pair, the goal is to find a…

Data Structures and Algorithms · Computer Science 2023-07-10 Elena Grigorescu , Nithish Kumar , Young-San Lin

The directed graph reachability problem takes as input an $n$-vertex directed graph $G=(V,E)$, and two distinguished vertices $s$ and $t$. The problem is to determine whether there exists a path from $s$ to $t$ in $G$. This is a canonical…

Data Structures and Algorithms · Computer Science 2019-09-23 Ryo Ashida , Kotaro Nakagawa

Max-k-Cut and correlation clustering are fundamental graph partitioning problems. For a graph with G=(V,E) with n vertices, the methods with the best approximation guarantees for Max-k-Cut and the Max-Agree variant of correlation clustering…

Optimization and Control · Mathematics 2021-10-28 Nimita Shinde , Vishnu Narayanan , James Saunderson

We give an algorithm to find a minimum cut in an edge-weighted directed graph with $n$ vertices and $m$ edges in $\tilde O(n\cdot \max(m^{2/3}, n))$ time. This improves on the 30 year old bound of $\tilde O(nm)$ obtained by Hao and Orlin…

Data Structures and Algorithms · Computer Science 2021-11-18 Ruoxu Cen , Jason Li , Danupon Nanongkai , Debmalya Panigrahi , Kent Quanrud , Thatchaphol Saranurak

Consider a generalization of the classical binary search problem in linearly sorted data to the graph-theoretic setting. The goal is to design an adaptive query algorithm, called a strategy, that identifies an initially unknown target…

Data Structures and Algorithms · Computer Science 2020-05-04 Dariusz Dereniowski , Aleksander Łukasiewicz , Przemysław Uznański

We improve on random sampling techniques for approximately solving problems that involve cuts and flows in graphs. We give a near-linear-time construction that transforms any graph on n vertices into an O(n\log n)-edge graph on the same…

Data Structures and Algorithms · Computer Science 2007-05-23 Andras Benczur , David R. Karger

We provide a deterministic algorithm that finds, in $\epsilon^{-O(1)} n^2$ time, an $\epsilon$-regular Frieze-Kannan partition of a graph on $n$ vertices. The algorithm outputs an approximation of a given graph as a weighted sum of…

Combinatorics · Mathematics 2019-08-21 Jacob Fox , László Miklós Lovász , Yufei Zhao

For a family of graphs $\cal F$, the canonical Weighted $\cal F$ Vertex Deletion problem is defined as follows: given an $n$-vertex undirected graph $G$ and a weight function $w: V(G)\rightarrow\mathbb{R}$, find a minimum weight subset…

Data Structures and Algorithms · Computer Science 2017-07-18 Akanksha Agrawal , Daniel Lokshtanov , Pranabendu Misra , Saket Saurabh , Meirav Zehavi

We show that there exists an adjacency labelling scheme for planar graphs where each vertex of an $n$-vertex planar graph $G$ is assigned a $(1+o(1))\log_2 n$-bit label and the labels of two vertices $u$ and $v$ are sufficient to determine…

Data Structures and Algorithms · Computer Science 2021-12-03 Vida Dujmović , Louis Esperet , Gwenaël Joret , Cyril Gavoille , Piotr Micek , Pat Morin

An instance of the Connected Maximum Cut problem consists of an undirected graph G = (V, E) and the goal is to find a subset of vertices S $\subseteq$ V that maximizes the number of edges in the cut \delta(S) such that the induced graph…

Data Structures and Algorithms · Computer Science 2015-07-03 MohammadTaghi Hajiaghayi , Guy Kortsarz , Robert MacDavid , Manish Purohit , Kanthi Sarpatwar

We study partial and budgeted versions of the well studied connected dominating set problem. In the partial connected dominating set problem, we are given an undirected graph G = (V,E) and an integer n', and the goal is to find a minimum…

Data Structures and Algorithms · Computer Science 2013-11-12 Samir Khuller , Manish Purohit , Kanthi Sarpatwar

Graph labeling problems have been widely studied in the last decades and have a vast area of application. In this work, we study the recently introduced S-labeling problem, in which the nodes get labeled using labels from 1 to |V | and for…

Optimization and Control · Mathematics 2019-09-12 Markus Sinnl

The Euler genus of a graph is a fundamental and well-studied parameter in graph theory and topology. Computing it has been shown to be NP-hard by [Thomassen '89 & '93], and it is known to be fixed-parameter tractable. However, the…

Data Structures and Algorithms · Computer Science 2013-11-04 Chandra Chekuri , Anastasios Sidiropoulos

A distributed network is modeled by a graph having $n$ nodes (processors) and diameter $D$. We study the time complexity of approximating {\em weighted} (undirected) shortest paths on distributed networks with a $O(\log n)$ {\em bandwidth…

Data Structures and Algorithms · Computer Science 2014-05-23 Danupon Nanongkai

In this paper we deal with the Bin Packing Problem with Conflicts on interval graphs: given an interval graph, a nonnegative integer weight for each vertex, and a nonnegative integer B, find a partition of the vertex set of the graph into k…

Combinatorics · Mathematics 2017-09-15 Tiziano Bacci , Sara Nicoloso

We propose an efficient $\epsilon$-differentially private algorithm, that given a simple {\em weighted} $n$-vertex, $m$-edge graph $G$ with a \emph{maximum unweighted} degree $\Delta(G) \leq n-1$, outputs a synthetic graph which…

Data Structures and Algorithms · Computer Science 2023-10-02 Jingcheng Liu , Jalaj Upadhyay , Zongrui Zou

We develop a new framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common when working with real-world…