Estimating clique size via discarding subgraphs

Discrete Optimization, 2008

We consider the problem of determining the size of a maximum clique in a graph, also known as the clique number. Given any method that computes an upper bound on the clique number of a graph, we present a sequential elimination algorithm which is guaranteed to improve upon that upper bound. Computational experiments on DIMACS instances show that, on average, this algorithm can reduce the gap between the upper bound and the clique number by about 60%. We also show how to use this sequential elimination algorithm to improve the computation of lower bounds on the clique number of a graph.

In this text we provide an algorithm giving an upper bound on the size of the maximum clique in a graph. The algorithm uses induced subgraphs and rules designed to decrease a set of properties as much as possible. Time and memory requirements are O(n^5) and O(n^3) respectively. Evaluation on random graphs suggests that the upper bounds depend linearly on the graph orders, and the slope depends on the probability that certain edges exist in a random graph. This paper also provides an introduction to variants of the clique problem and discusses a memory ecient data structure representing undirected graphs. 1. Introduction Many different algorithms can be constructed to solve one specic problem. Some might be more ecient than others. An algorithm's eciency is determined by its time and memory consumption. Often these measures are analyzed by theoretical means, but when an algorithm's behavior is too complex to understand, simulations and statistics might come in handy. Computat...

International Journal of Advanced Computer Science and Applications, 2019

A large dataset network is considered for computation of maximal clique size (MC). Additionally, its link with popular centrality metrics to decrease uncertainty and complexity and for finding influential points of any network has also been investigated. Previous studies focus on centrality metrics like degree centrality (DC), closeness centrality (CC), betweenness centrality (BC) and Eigenvector centrality (EVC) and compare them with maximal clique size however, in this study Katz centrality measure is also considered and shows a pretty robust relation with maximal clique size (MC). Secondly, maximal clique size (MC) algorithm is also revised for network analysis to avoid complexity in computation. Association between MC and five centrality metrics has been evaluated through recognized methods that are Pearson's correlation coefficient (PCC), Spearman's correlation coefficient (SCC) and Kendall's correlation coefficient (KCC). The strong strength of association between them is seen through all three correlation coefficients measure.

Acta Universitatis Sapientiae, Informatica

It is a common practice to find upper bound for clique number via legal coloring of the nodes of the graph. We will point out that with a little extra work we may lower this bound. Applying this procedure to a suitably constructed auxiliary graph one may further improve the clique size estimate of the original graph.

Proceedings of the 26th International Conference on World Wide Web, 2017

Clique counts reveal important properties about the structure of massive graphs, especially social networks. The simple setting of just 3-cliques (triangles) has received much attention from the research community. For larger cliques (even, say 6-cliques) the problem quickly becomes intractable because of combinatorial explosion. Most methods used for triangle counting do not scale for large cliques, and existing algorithms require massive parallelism to be feasible. We present a new randomized algorithm that provably approximates the number of k-cliques, for any constant k. The key insight is the use of (strengthenings of) the classic Turán's theorem: this claims that if the edge density of a graph is sufficiently high, the k-clique density must be non-trivial. We define a combinatorial structure called a Turán shadow, the construction of which leads to fast algorithms for clique counting. We design a practical heuristic, called Turán-shadow, based on this theoretical algorithm, and test it on a large class of test graphs. In all cases, Turán-shadow has less than 2% error, in a fraction of the time used by well-tuned exact algorithms. We do detailed comparisons with a range of other sampling algorithms, and find that Turán-shadow is generally much faster and more accurate. For example, Turán-shadow estimates all cliques numbers up to size 10 in social network with over a hundred million edges. This is done in less than three hours on a single commodity machine.

Mathematica Pannonica, 2021

In many clique search algorithms well coloring of the nodes is employed to find an upper bound of the clique number of the given graph. In an earlier work a non-traditional edge coloring scheme was proposed to get upper bounds that are typically better than the one provided by the well coloring of the nodes. In this paper we will show that the same scheme for well coloring of the edges can be used to find lower bounds for the clique number of the given graph. In order to assess the performance of the procedure we carried out numerical experiments.

Int. Conf. on Modeling, Simulation & Visualization Methods, 2004

We tackle the problem of counting the number q k of k-cliques in large-scale graphs, for any constant k ≥ 3. Clique counting is essential in a variety of applications, among which social network analysis. Our algorithms make it possible to compute q k for several real-world graphs and shed light on its growth rate as a function of k. Even for small values of k, the number q k of k-cliques can be in the order of tens or hundreds of trillions. As k increases, different graph instances show different behaviors: while on some graphs q k+1 < q k , on other benchmarks q k+1 q k , up to two orders of magnitude in our observations. Graphs with steep clique growth rates represent particularly tough instances in practice.

Discrete Mathematics, 1986

For each natural number n, denote by G(n) the set of all numbers c such that there exists a graph with exactly c cliques (i.e., complete subgraphs) and n vertices. We prove the asymptotic estimate

Progress in Artificial Intelligence, 2009

In social network analysis, a k-clique is a relaxed clique, i.e., a kclique is a quasi-complete sub-graph. A k-clique in a graph is a sub-graph where the distance between any two vertices is no greater than k. The visualization of a small number of vertices can be easily performed in a graph. However, when the number of vertices and edges increases the visualization becomes incomprehensible. In this paper, we propose a new graph mining approach based on k-cliques. The concept of relaxed clique is extended to the whole graph, to achieve a general view, by covering the network with k-cliques. The sequence of k-clique covers is presented, combining small world concepts with community structure components. Computational results and examples are presented.