Keywords: approximation scheme, matrix permanent, practice, rejection sampling
TL;DR: We present a technique for boosting the acceptance–rejection method with superior empirical performance and analytic bounds.
Abstract: We present a randomized approximation scheme for the permanent of a matrix with nonnegative entries. Our scheme extends a recursive rejection sampling method of Huber and Law (SODA 2008) by replacing the permanent upper bound with a linear combination of the subproblem bounds at a moderately large depth of the recursion tree. This method, we call deep rejection sampling, is empirically shown to outperform the basic, depth-zero variant, as well as a related method by Kuck et al. (NeurIPS 2019). We analyze the expected running time of the scheme on random $(0, 1)$-matrices where each entry is independently $1$ with probability $p$. Our bound is superior to a previous one for $p$ less than $1/5$, matching another bound that was only known to hold when every row and column has density exactly $p$.
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