Signal filtering to obtain number of Hamiltonian paths

TL;DR

This paper introduces a novel approach to count Hamiltonian paths by transforming the problem into a signal filtering task, employing a divide-and-conquer filtering strategy with polynomial extrapolation, linking computational complexity to P=NP conjecture.

Contribution

It presents a new reduction of the Hamiltonian path problem to a signal filtering framework and proposes a specific filtering strategy involving polynomial extrapolation.

Findings

Conditional P=NP implication based on filter design and sample complexity

Reduction of Hamiltonian path counting to signal amplitude analysis at zero frequency

Implementation details of a divide-and-conquer filtering approach

Abstract

This paper consists of two parts. First, the (undirected) Hamiltonian path problem is reduced to a signal filtering problem - number of Hamiltonian paths becomes amplitude at zero frequency for (a combination of) sinusoidal signal f(t) that encodes a graph. Then a 'divide and conquer' strategy to filtering out wide bandwidth components of a signal is suggested - one filters out angular frequency 1/2 to 1, then 1/4 to 1/2, then 1/8 to 1/4 and so on. An actual implementation of this strategy involves careful local polynomial extrapolation using numerical differentiation filters. When conjectures regarding required number of samples for specified filter designs and time complexity of obtaining filter coefficients hold, P=NP conditionally.