A Simple Algorithm for Hamiltonicity

Abstract

We develop a new algebraic technique that solves the following problem: Given a black box that contains an arithmetic circuit $f$ over a field of characteristic $2$ of degree~$d$. Decide whether $f$, expressed as an equivalent multivariate polynomial, contains a multilinear monomial of degree $d$. This problem was solved by Williams \cite{W} and Bj\"orklund et. al. \cite{BHKK} for a white box (the circuit is given as an input) that contains arithmetic circuit. We show a simple black box algorithm that solves the problem with the same time complexity. This gives a simple randomized algorithm for the simple $k$-path problem for directed graphs of the same time complexity\footnote{$O^*(f(k))$ is $O(poly(n)\cdot f(k))$} $O^*(2^k)$ as in \cite{W} and with reusing the same ideas from \cite{BHKK} with the above gives another algorithm (probably not simpler) for undirected graphs of the same time complexity $O^*(1.657^k)$ as in \cite{B10,BHKK}.