Signal filtering to obtain number of Hamiltonian paths

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.