Approximately counting independent sets of a given size in boundeddegree graphs
Abstract
We determine the computational complexity of approximately counting and sampling independent sets of a given size in boundeddegree graphs. That is, we identify a critical density $\alpha_c(\Delta)$ and provide (i) for $\alpha < \alpha_c(\Delta)$ randomized polynomialtime algorithms for approximately sampling and counting independent sets of given size at most $\alpha n$ in $n$vertex graphs of maximum degree $\Delta$; and (ii) a proof that unless NP=RP, no such algorithms exist for $\alpha>\alpha_c(\Delta)$. The critical density is the occupancy fraction of the hard core model on the complete graph $K_{\Delta+1}$ at the uniqueness threshold on the infinite $\Delta$regular tree, giving $\alpha_c(\Delta)\sim\frac{e}{1+e}\frac{1}{\Delta}$ as $\Delta\to\infty$. Our methods apply more generally to antiferromagnetic 2spin systems and motivate new questions in extremal combinatorics.
 Publication:

arXiv eprints
 Pub Date:
 February 2021
 DOI:
 10.48550/arXiv.2102.04984
 arXiv:
 arXiv:2102.04984
 Bibcode:
 2021arXiv210204984D
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Computational Complexity;
 Mathematics  Combinatorics