Randomized Approximation Schemes for the Tutte Polynomial and Random Clustering in Subdense and Superdense Graphs

TL;DR

This paper develops randomized polynomial-time approximation schemes for the Tutte polynomial and related partition functions in subdense and superdense graphs, extending prior work and analyzing specific graph density regimes.

Contribution

It introduces new approximation schemes for the Tutte polynomial in subdense graphs and analyzes its asymptotic behavior in superdense graphs, expanding the understanding of these computations.

Findings

Polynomial-time approximation schemes for subdense graphs

Asymptotic equivalence of Tutte polynomial and Q in superdense graphs

Discussion on approximating Z in power law graphs

Abstract

Extending the work of Alon, Frieze abnd Welsh, we show that there are randomized polynomial time approximation schemes for computing the Tutte polynomial in subdense graphs with an minimal node degree of $\Omega\left ( \frac{n}{\sqrt{\log n}}\right )$ . The same holds for the partition function $Z$ in the random cluster model with uniform edge probabilities and for the associated distribution $\lambda (A),\: A \subseteq E$ whenever the underlying graph $G=(V,E)$ is $c\cdot\frac{n}{\sqrt{\log (n)}}$-subdense. In the superdense case with node degrees $n-o(n)$, we show that the Tutte polynomial $T_G(x,y)$ is asymptotically equal to $Q=(x-1)(y-1)$. Moreover, we briefly discuss the problem of approximating $Z$ in the case of $(\alpha, \beta )$-power law graphs.