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We study the densest subgraph problem and its NP-hard densest at-most-$k$ subgraph variant through the lens of learning-augmented algorithms. We show that, given a reasonably accurate predictor that estimates whether a node belongs to the…

Data Structures and Algorithms · Computer Science 2026-04-16 Thai Bui , Luan Nguyen , Hoa T. Vu

We consider the densest $k$-subgraph problem, which seeks to identify the $k$-node subgraph of a given input graph with maximum number of edges. This problem is well-known to be NP-hard, by reduction to the maximum clique problem. We…

Optimization and Control · Mathematics 2019-04-09 Polina Bombina , Brendan Ames

Given a graph G = (V,E) and a parameter k, we consider the problem of finding a subset U in V of size k that maximizes the number of induced edges (DkS). We improve upon the previously best known approximation ratio for DkS, a ratio that…

Discrete Mathematics · Computer Science 2015-03-13 Doron Goldstein , Michael Langberg

The Densest $k$-Subgraph (D$k$S) problem, and its corresponding minimization problem Smallest $p$-Edge Subgraph (S$p$ES), have come to play a central role in approximation algorithms. This is due both to their practical importance, and…

Data Structures and Algorithms · Computer Science 2016-05-16 Eden Chlamtáč , Michael Dinitz , Christian Konrad , Guy Kortsarz , George Rabanca

Computing cohesive subgraphs is a central problem in graph theory. While many formulations of cohesive subgraphs lead to NP-hard problems, finding a densest subgraph can be done in polynomial time. As such, the densest subgraph model has…

Data Structures and Algorithms · Computer Science 2021-11-24 Riccardo Dondi , Danny Hermelin

Densest Subgraph Problem (DSP) is an important primitive problem with a wide range of applications, including fraud detection, community detection and DNA motif discovery. Edge-based density is one of the most common metrics in DSP.…

Databases · Computer Science 2023-10-31 Yugao Zhu , Shenghua Liu , Wenjie Feng , Xueqi Cheng

We consider a variant of the densest subgraph problem in networks with single or multiple edge attributes. For example, in a social network, the edge attributes may describe the type of relationship between users, such as friends, family,…

Social and Information Networks · Computer Science 2024-02-15 Lutz Oettershagen , Honglian Wang , Aristides Gionis

We study the Densest At-Least-$k$-Subgraph (DAL$k$S) problem, in which we are given an undirected graph $G$ and an integer $k$, and the goal is to find a subgraph of $G$ with at least $k$ vertices with maximum density. The best-known…

Data Structures and Algorithms · Computer Science 2026-05-26 Bundit Laekhanukit , Pasin Manurangsi , Ohad Trabelsi

Decomposing hypergraphs is a key task in hypergraph analysis with broad applications in community detection, pattern discovery, and task scheduling. Existing approaches such as $k$-core and neighbor-$k$-core rely on vertex degree…

Social and Information Networks · Computer Science 2026-04-10 Xiaoyu Leng , Hongchao Qin , Rong-Hua Li

This paper addresses the problem of finding the densest $k$-vertex subgraph in an arbitrary graph. This problem is NP-hard and has important applications in social network analysis, fraud detection, recommendation systems, and…

Quantum Physics · Physics 2026-05-01 Yu. A. Biriukov , R. D. Morozov , I. V. Dyakonov , S. S. Straupe

We consider the minimum vertex cover problem in hypergraphs in which every hyperedge has size k (also known as minimum hitting set problem, or minimum set cover with element frequency k). Simple algorithms exist that provide…

Data Structures and Algorithms · Computer Science 2010-12-14 Jean Cardinal , Marek Karpinski , Richard Schmied , Claus Viehmann

Many real-world networks can be modeled as graphs. Finding dense subgraphs is a key problem in graph mining with applications in diverse domains. In this paper, we consider two variants of the densest subgraph problem where multiple graph…

Data Structures and Algorithms · Computer Science 2025-02-04 Chamalee Wickrama Arachchi , Nikolaj Tatti

Given a connected graph $G$ on $n$ vertices and a positive integer $k\le n$, a subgraph of $G$ on $k$ vertices is called a $k$-subgraph in $G$. We design combinatorial approximation algorithms for finding a connected $k$-subgraph in $G$…

Discrete Mathematics · Computer Science 2015-01-30 Xujin Chen , Xiaodong Hu , Changjun Wang

Finding dense subgraphs is a core problem in graph mining with many applications in diverse domains. At the same time many real-world networks vary over time, that is, the dataset can be represented as a sequence of graph snapshots. Hence,…

Data Structures and Algorithms · Computer Science 2023-08-31 Chamalee Wickrama Arachchi , Nikolaj Tatti

Finding the densest subgraph (DS) from a graph is a fundamental problem in graph databases. The DS obtained, which reveals closely related entities, has been found to be useful in various application domains such as e-commerce, social…

Databases · Computer Science 2025-04-16 Yi Yang , Chenhao Ma , Reynold Cheng , Laks V. S. Lakshmanan , Xiaolin Han

The Densest Subgraph Problem requires to find, in a given graph, a subset of vertices whose induced subgraph maximizes a measure of density. The problem has received a great deal of attention in the algorithmic literature since the early…

Data Structures and Algorithms · Computer Science 2024-04-19 Tommaso Lanciano , Atsushi Miyauchi , Adriano Fazzone , Francesco Bonchi

Given a graph $G=(V,E)$, the dominating set problem asks for a minimum subset of vertices $D\subseteq V$ such that every vertex $u\in V\setminus D$ is adjacent to at least one vertex $v\in D$. That is, the set $D$ satisfies the condition…

Computational Geometry · Computer Science 2019-11-26 Sandip Banerjee , Sujoy Bhore

Given a simple graph $G = (V, E)$ and a constant integer $k \ge 2$, the $k$-path vertex cover problem ({\sc P$k$VC}) asks for a minimum subset $F \subseteq V$ of vertices such that the induced subgraph $G[V - F]$ does not contain any path…

Data Structures and Algorithms · Computer Science 2018-11-06 An Zhang , Yong Chen , Zhi-Zhong Chen , Guohui Lin

Given an undirected graph on a node set $V$ and positive integers $k$ and $m$, a $k$-connected $m$-dominating set ($(k,m)$-CDS) is defined as a subset $S$ of $V$ such that each node in $V \setminus S$ has at least $m$ neighbors in $S$, and…

Data Structures and Algorithms · Computer Science 2018-08-08 Takuro Fukunaga

{\sc Directed Feedback Vertex Set (DFVS)} is a fundamental computational problem that has received extensive attention in parameterized complexity. In this paper, we initiate the study of a wide generalization, the {\sc ${\cal H}$-free SCC…

Data Structures and Algorithms · Computer Science 2020-08-25 Rian Neogi , M. S. Ramanujan , Saket Saurabh , Roohani Sharma