Counting cliques and clique covers in random graphs

TL;DR

This paper develops efficient randomized approximation schemes for counting specific subgraphs like cliques and independent sets in random graphs, especially for small sizes relative to the number of vertices, using novel variance analysis techniques.

Contribution

It introduces new approximation algorithms with polynomial runtime for counting small cliques, independent sets, and clique covers in random graphs, with a novel variance bounding approach.

Findings

Provides FPRAS for counting k-cliques and k-independent sets when k ≤ (1+o(1)) log n.

Achieves polynomial bounds on the variance of estimators.

Offers an alternative derivation of the binomial moments using variance recurrence techniques.

Abstract

We study the problem of counting the number of {\em isomorphic} copies of a given {\em template} graph, say $H$, in the input {\em base} graph, say $G$. In general, it is believed that polynomial time algorithms that solve this problem exactly are unlikely to exist. So, a lot of work has gone into designing efficient {\em approximation schemes}, especially, when $H$ is a perfect matching. In this work, we present efficient approximation schemes to count $k$-Cliques, $k$-Independent sets and $k$-Clique covers in random graphs. We present {\em fully polynomial time randomized approximation schemes} (fpras) to count $k$-Cliques and $k$-Independent sets in a random graph on $n$ vertices when $k$ is at most $(1+o(1))\log n$, and $k$-Clique covers when $k$ is a constant. [Grimmett and McDiarmid, 1975] present a simple greedy algorithm that {\em detects} a clique (independent set) of size $(1+o(1))\log_2 n$ in $G\in \mathcal{G}(n,\frac{1}{2})$ with high probability. No algorithm is known to detect a clique or an independent set of larger size with non-vanishing probability. Furthermore, [Coja-Oghlan and Efthymiou, 2011] present some evidence that one cannot hope to easily improve a similar, almost 40 years old bound for sparse random graphs. Therefore, our results are unlikely to be easily improved. We use a novel approach to obtain a recurrence corresponding to the variance of each estimator. Then we upper bound the variance using the corresponding recurrence. This leads us to obtain a polynomial upper bound on the critical ratio. As an aside, we also obtain an alternate derivation of the closed form expression for the $k$-th moment of a binomial random variable using our techniques. The previous derivation [Knoblauch (2008)] was based on the moment generating function of a binomial random variable.