Algebraic Fingerprints For Faster Algorithms

It was a major surprise when, in 2010, Andreas Björklund discovered what many previously thought impossible: a significantly improved algorithm for the famous Hamiltonian path problem, also known as Hamiltonicity.⁴ Hamiltonicity asks if a given graph contains a path that goes through each vertex exactly once, as illustrated in Figure 1.

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Hamiltonicity was one of the first problems shown to be NP-complete by Karp.²⁰ The only known algorithms for NP-complete problems require time scaling exponentially with the size of the input. It is believed they cannot be solved much faster in general, although the possibility has not been ruled out.¹⁷ Given an undirected graph on n vertices, Björklund's algorithm can find a Hamiltonian path or report that no such path exists in O*(1.657ⁿ) time.^a The algorithm still runs in exponential time but it is much faster than the O*(2ⁿ) running time of the previously fastest algorithm, known since the 1960s.^3,19