Approximately counting independent sets in bipartite graphs via graph containers

By implementing algorithmic versions of Sapozhenko's graph container methods, we give new algorithms for approximating the number independent sets in bipartite graphs. Our first algorithm applies to d $$ -regular, graphs satisfying a weak expansion condition: when is constant, and Ω ( log 2 / ) \Omega \left({\log}^2d/d\right) -expander, obtain an FPTAS sets. Previously such result > 5 d>5 was known only much stronger conditions random The also weighted sets: α \alpha with 0 >0 fixed, hard-core model partition function at fugacity λ = 1 4 \lambda =\Omega \left(\log d/{d}^{1/4}\right) . Finally present that all graphs, runs time exp O n · 3 \exp \left(O\left(n\cdotp \frac{\log^3d}{d}\right)\right) , outputs + o \left(1+o(1)\right) -approximation