Below All Subsets for Some Permutational Counting Problems

TL;DR

This paper introduces the first deterministic algorithms for computing the permanent and counting Hamiltonian cycles that run faster than exponential time, specifically in subexponential $2^{n- ext{Omega}( ext{sqrt}(n/ ext{log} n))}$ time, improving over classic methods.

Contribution

It presents novel reductions that enable subexponential algorithms for permanent and Hamiltonian cycle counting, surpassing classical exponential algorithms.

Findings

First deterministic subexponential algorithms for the problems.

Algorithms run in $2^{n- ext{Omega}( ext{sqrt}(n/ ext{log} n))}$ time.

Achieved reductions to smaller instances of the problems.

Abstract

We show that the two problems of computing the permanent of an $n\times n$ matrix of $\operatorname{poly}(n)$-bit integers and counting the number of Hamiltonian cycles in a directed $n$-vertex multigraph with $\operatorname{exp}(\operatorname{poly}(n))$ edges can be reduced to relatively few smaller instances of themselves. In effect we derive the first deterministic algorithms for these two problems that run in $o(2^n)$ time in the worst case. Classic $\operatorname{poly}(n)2^n$ time algorithms for the two problems have been known since the early 1960's. Our algorithms run in $2^{n-\Omega(\sqrt{n/\log n})}$ time.