Related papers: The Densest k-Subhypergraph Problem

We consider the $k$-Center problem and some generalizations. For $k$-Center a set of $k$ center vertices needs to be found in a graph $G$ with edge lengths, such that the distance from any vertex of $G$ to its nearest center is minimized.…

We consider the problem of finding small distance-preserving subgraphs of undirected, unweighted interval graphs with $k$ terminal vertices. To start with, we show that finding an optimal distance-preserving subgraph is $\mathsf{NP}$-hard…

The planted densest subgraph detection problem refers to the task of testing whether in a given (random) graph there is a subgraph that is unusually dense. Specifically, we observe an undirected and unweighted graph on $n$ vertices. Under…

It has long been known, since the classical work of (Arora, Karger, Karpinski, JCSS~99), that \MC\ admits a PTAS on dense graphs, and more generally, \kCSP\ admits a PTAS on "dense" instances with $\Omega(n^k)$ constraints. In this paper we…

We present several sparsification lower and upper bounds for classic problems in graph theory and logic. For the problems 4-Coloring, (Directed) Hamiltonian Cycle, and (Connected) Dominating Set, we prove that there is no polynomial-time…

We study the following two fixed-cardinality optimization problems (a maximization and a minimization variant). For a fixed $\alpha$ between zero and one we are given a graph and two numbers $k \in \mathbb{N}$ and $t \in \mathbb{Q}$. The…

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…

In the $k$-Edge Connected Spanning Subgraph ($k$-ECSS) problem we are given a (multi-)graph $G=(V,E)$ with edge costs and an integer $k$, and seek a min-cost $k$-edge-connected spanning subgraph of $G$. The problem admits a…

Network applications, such as multimedia streaming and video conferencing, impose growing requirements over Quality of Service (QoS), including bandwidth, delay, jitter, etc. Meanwhile, networks are expected to be load-balanced,…

Recently [Bhattacharya et al., STOC 2015] provide the first non-trivial algorithm for the densest subgraph problem in the streaming model with additions and deletions to its edges, i.e., for dynamic graph streams. They present a…

A subset $S$ of nodes in a graph $G$ is a $k$-connected $m$-dominating set ($(k,m)$-cds) if the subgraph $G[S]$ induced by $S$ is $k$-connected and every $v \in V \setminus S$ has at least $m$ neighbors in $S$. In the $k$-Connected…

Given a graph $G = (V, E)$ and an integer $k$, we study $k$-Vertex Seperator (resp. $k$-Edge Separator), where the goal is to remove the minimum number of vertices (resp. edges) such that each connected component in the resulting graph has…

We study the problem of approximating the cone of positive semidefinite (PSD) matrices with a cone that can be described by smaller-sized PSD constraints. Specifically, we ask the question: "how closely can we approximate the set of…

In a bipartite graph, a subgraph is an $s$-biplex if each vertex of the subgraph is adjacent to all but at most $s$ vertices on the opposite set. The enumeration of $s$-biplexes from a given graph is a fundamental problem in bipartite graph…

We consider the classical $k$-Center problem in undirected graphs. The problem is known to have a polynomial-time 2-approximation. There are even $(2+\varepsilon)$-approximations running in near-linear time. The conventional wisdom is that…

We consider the Hypergraph-$k$-cut problem. The input consists of a hypergraph $G=(V,E)$ with non-negative hyperedge-costs $c: E\rightarrow R_+$ and a positive integer $k$. The objective is to find a least-cost subset $F\subseteq E$ such…

We study the utility and limitations of using $k$-uniform hypergraphs $H = ([n], E)$ ($n \ge \mathrm{poly}(k)$) in the context of error reduction for randomized algorithms for decision problems with one- or two-sided error. Our error…

In the Min $k$-Cut problem, input is an edge weighted graph $G$ and an integer $k$, and the task is to partition the vertex set into $k$ non-empty sets, such that the total weight of the edges with endpoints in different parts is minimized.…

A $k$-dimensional box is the Cartesian product $R_1 \times R_2 \times ... \times R_k$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $\boxi(G)$, is the minimum integer $k$ such that $G$…

In distributed networks, it is often useful for the nodes to be aware of dense subgraphs, e.g., such a dense subgraph could reveal dense subtructures in otherwise sparse graphs (e.g. the World Wide Web or social networks); these might…