TY - JOUR

T1 - Algebraic fingerprints for faster algorithms

AU - Koutis, Ioannis

AU - Williams, Ryan

N1 - Funding Information:
I. Koutis is supported by NSF CAREER award CCF-1149048. R. Williams is supported by a David Morgenthaler II Faculty Fellowship, NSF CCF-1212372, a Sloan Fellowship, and a Microsoft Research Fellowship. The final phase of this work occurred at the Simons Institute for the Theory of Computing.
Publisher Copyright:
© 2016 ACM.

PY - 2016/1

Y1 - 2016/1

N2 - 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. 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.

AB - 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. 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.

UR - http://www.scopus.com/inward/record.url?scp=84952360830&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84952360830&partnerID=8YFLogxK

U2 - 10.1145/2742544

DO - 10.1145/2742544

M3 - Review article

AN - SCOPUS:84952360830

SN - 0001-0782

VL - 59

SP - 98

EP - 105

JO - Communications of the ACM

JF - Communications of the ACM

IS - 1

ER -