Mixing Time of Markov chain of the Knapsack Problem

TL;DR

This paper improves the upper bound on the mixing time of a Markov chain used for sampling solutions of the Knapsack Problem, making the sampling process more efficient.

Contribution

It introduces a canonical path method leveraging the solution set's lattice structure to significantly tighten the mixing time bound.

Findings

New mixing time bound: ^{3} \, \ln(16 \, \epsilon^{-1})

Improved analysis over previous ^{9/2+\, \epsilon} bound

Enhanced understanding of Markov chain convergence for combinatorial problems

Abstract

To find the number of assignments of zeros and ones satisfying a specific Knapsack Problem is $\#P$ hard, so only approximations are envisageable. A Markov chain allowing uniform sampling of all possible solutions is given by Luby, Randall and Sinclair. In 2005, Morris and Sinclair, by using a flow argument, have shown that the mixing time of this Markov chain is $\mathcal{O}(n^{9/2+\epsilon})$, for any $\epsilon > 0$. By using a canonical path argument on the distributive lattice structure of the set of solutions, we obtain an improved bound, the mixing time is given as $\tau_{_{x}}(\epsilon) \leq n^{3} \ln (16 \epsilon^{-1})$.