← 研踪 PaperTrailarXiv:2611.06657 · v1
Approximating #BIS in $\exp(\widetilde O(n^{1/3}))$ Time
首次提交 2026-11-04 23:33
版本更新 2026-11-04 23:33
雷达观测 2026-11-05 12:13
摘要
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.
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v1 · 2026-11-04 23:33 · 最新观测版本
Approximating #BIS in $\exp(\widetilde O(n^{1/3}))$ Time
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.
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