Related papers: Maximum Weight Independent Sets and Matchings in S…
The maximum independent set problem is one of the most important problems in graph algorithms and has been extensively studied in the line of research on the worst-case analysis of exact algorithms for NP-hard problems. In the weighted…
Computing maximum weight independent sets in graphs is an important NP-hard optimization problem. The problem is particularly difficult to solve in large graphs for which data reduction techniques do not work well. To be more precise,…
Computing the maximum size of an independent set in a graph is a famously hard combinatorial problem that has been well-studied for various classes of graphs. When it comes to random graphs, only the classical Erd\H{o}s-R\'enyi-Gilbert…
In this paper, we develop efficient exact and approximate algorithms for computing a maximum independent set in random graphs. In a random graph $G$, each pair of vertices are joined by an edge with a probability $p$, where $p$ is a…
The maximum independent set problem is a classic optimization problem that has also been studied quite intensively in the distributed setting. While the problem is hard to approximate in general, there are good approximation algorithms…
Let $V$ be a set of $n$ vertices, ${\cal M}$ a set of $m$ labels, and let $\mathbf{R}$ be an $m \times n$ matrix of independent Bernoulli random variables with success probability $p$. A random instance $G(V,E,\mathbf{R}^T\mathbf{R})$ of…
The size of a largest independent set of vertices in a given graph $G$ is denoted by $\alpha(G)$ and is called its independence number (or stability number). Given a graph $G$ and an integer $K,$ it is NP-complete to decide whether…
We consider the infinite directed graph with vertices the set of integers ...,-2,-1,0,1,2,... . Let v be a random variable taking either finite values or value "minus infinity". Consider random weights v(j,k), indexed by pairs (j,k) of…
The random greedy algorithm for finding a maximal independent set in a graph constructs a maximal independent set by inspecting the graph's vertices in a random order, adding the current vertex to the independent set if it is not adjacent…
In this paper, we consider the task of computing an independent set of maximum weight in a given $d$-claw free graph $G=(V,E)$ equipped with a positive weight function $w:V\rightarrow\mathbb{R}^+$. In doing so, $d\geq 2$ is considered a…
We prove an asymptotically tight lower bound on the average size of independent sets in a triangle-free graph on $n$ vertices with maximum degree $d$, showing that an independent set drawn uniformly at random from such a graph has expected…
Finding independent sets of maximum size in fixed graphs is well known to be an NP-hard task. Using scaling limits, we characterise the asymptotics of sequential degree-greedy explorations and provide sufficient conditions for this…
Let m(G) denote the number of maximal independent sets of vertices in a graph G and let c(n,r) be the maximum value of m(G) over all connected graphs with n vertices and at most r cycles. A theorem of Griggs, Grinstead, and Guichard gives a…
Determining the size of a maximum independent set of a graph $G$, denoted by $\alpha(G)$, is an NP-hard problem. Therefore, many attempts are made to find upper and lower bounds, or exact values of $\alpha (G)$ for special classes of…
Given a set of vertices $V$ with $|V| = n$, a weight vector $w \in (\mathbb{R}^+ \cup \{ 0 \})^{\binom{V}{2}}$, and a probability vector $x \in [0, 1]^{\binom{V}{2}}$ in the matching polytope, we study the quantity $\frac{E_{G}[…
Consider a set of $n$ vertices, where each vertex has a location in $\mathbb{R}^d$ that is sampled uniformly from the unit cube in $\mathbb{R}^d$, and a weight associated to it. Construct a random graph by placing edges independently for…
We present improved results for approximating maximum-weight independent set ($\MaxIS$) in the CONGEST and LOCAL models of distributed computing. Given an input graph, let $n$ and $\Delta$ be the number of nodes and maximum degree,…
A 2-packing set for an undirected, weighted graph G=(V,E,w) is a subset S of the vertices V such that any two vertices are not adjacent and have no common neighbors. The Maximum Weight 2-Packing Set problem that asks for a 2-packing set of…
Finding maximum-weight independent sets in graphs is an important NP-hard optimization problem. Given a vertex-weighted graph $G$, the task is to find a subset of pairwise non-adjacent vertices of $G$ with maximum weight. Most recently…
We study the algorithmic problem of finding a large independent set in the Erd{\"o}s-R\'{e}nyi random graph $G(n,p)$. For constant $p$ and $b=1/(1-p)$, the largest independent set has size $2\log_b n$, while a simple greedy algorithm -…