We give randomized algorithms for approximately counting and nearly uniformly sampling independent sets in arbitrary bipartite graphs (#BIS). At fixed accuracy, both algorithms run in $\exp(O(n^{1/3}\log^{4/3}n))$ time on $n$-vertex graphs. More generally, the same running-time dependence on the number of variables holds for approximate counting and sampling with efficiently evaluable positive log-supermodular weights on Boolean variables. A weighted Sinclair-Jerrum walk on the self-reduction tree reduces these tasks to coarse lower estimates of residual log-partition functions. We obtain these estimates by optimizing a concave relaxation of an exact variational principle on the tree; threshold smoothing certifies that the low-degree search suffices.
We resolve longstanding questions about connectivity, reconstruction,and looseness in random $k$-SAT, for sufficiently large fixed $k$ as the number $n$ of variables tends to infinity. $\textbf{Single-variable connectivity up to reconstruction.}$ At clause densities up to \[\frac{2^k}{k}\bigl(\ln k+\ln\ln k+γ_*-o_k(1)\bigr),\] a polynomial-time algorithm joins a typical pair of satisfying assignments with high probability, using $O_k(n)$ single-variable flips that preserve satisfiability. Here $γ_*\approx0.871$ arises from a refined tree-reconstruction analysis. $\textbf{Settling the reconstruction threshold.}$ Building on the tools developed for connectivity, we determine the clustering threshold defined by the reconstruction criterion to be \[\frac{2^k}{k}\bigl(\ln k+\ln\ln k+γ_*+o_k(1)\bigr),\] below the classical cavity prediction with constant term $1+O(\ln\ln k/\ln k)$. $\textbf{Shattering before rigidity.}$ We locate the looseness--rigidity transition at a higher density, matching the classical cavity prediction for clustering: \[\frac{2^k}{k} \left(\ln k+\ln\ln k+1+Θ\left(\frac{\ln\ln k}{\ln k}\right)\right).\] Below the transition, we establish a stronger form of looseness: any variable of a typical solution can be reversed by logarithmically many single-variable flips that preserve satisfiability. Above it, reversing most variables of a typical solution requires changing linearly many variables (rigidity). Yet throughout an interval between reconstruction and rigidity, the solution space already splits into groups of exponentially small relative size separated by linear Hamming distance and linear energy barriers (shattering), while logarithmic looseness persists.
How does a computational problem transition from easy to hard? In the hard-core model, approximate counting and sampling are polynomial-time tractable up to tree uniqueness, but worst-case intractable at any constant relative distance above it unless $\mathsf{NP}=\mathsf{RP}$. We quantify the transition from polynomial to exponential mixing for classical Glauber dynamics. On $n$-vertex graphs of maximum degree $d\ge3$, at fugacity $λ=(1+δ)λ_{\mathrm c}(d)$ with sufficiently small $δ\ge0$ that may vanish with $n$, its worst-case mixing time has the form $\exp\{Θ(nδ^2/d)\}n^{1+Θ(1)}$. The upper bound is degree-uniform, and the matching lower bound holds for every fixed $d$. Annealing gives approximate counting with the same running-time upper bound at fixed accuracy. For general sampling and counting algorithms, we establish a robust fine-grained reduction from SAT without assuming a lower bound on SAT's running time. For any fixed $d\ge3$ and $0<γ<1$, a sampling or approximate-counting algorithm running in $\exp\{O((nδ^2/d)^{1-γ})\}\operatorname{poly}(n)$ time along any one suitable inverse-polynomial distance sequence $δ=δ(n)$, yields a randomized SAT algorithm running in $\exp\{\widetilde O(s^{1-γ})\}$ time on inputs of binary length $s$. This power-saving transfer identifies $nδ^2/d$ as the analogue of SAT input length for hard-core sampling and counting at relative distance $δ$ above criticality: beating the Glauber exponent by a fixed power through any sampling or counting method would yield a corresponding power improvement for SAT. At criticality, we establish $O(\sqrt{n/d})$ spectral independence, optimal in $n$ at fixed degree, and degree-uniform Glauber mixing in $\widetilde O(n(n/d)^{1+1/(d-2)})$ time, improving the previous $\widetilde O(n^{4+4/(d-2)})$ bound.