Faster Algebraic Algorithms for Path and Packing Problems

  • Ioannis Koutis
Conference paper

DOI: 10.1007/978-3-540-70575-8_47

Part of the Lecture Notes in Computer Science book series (LNCS, volume 5125)
Cite this paper as:
Koutis I. (2008) Faster Algebraic Algorithms for Path and Packing Problems. In: Aceto L., Damgård I., Goldberg L.A., Halldórsson M.M., Ingólfsdóttir A., Walukiewicz I. (eds) Automata, Languages and Programming. ICALP 2008. Lecture Notes in Computer Science, vol 5125. Springer, Berlin, Heidelberg

Abstract

We study the problem of deciding whether an n-variate polynomial, presented as an arithmetic circuit G, contains a degree k square-free term with an odd coefficient. We show that if G can be evaluated over the integers modulo 2k + 1 in time t and space s, the problem can be decided with constant probability in O((kn + t)2k) time and O(kn + s) space. Based on this, we present new and faster algorithms for two well studied problems: (i) an O*(2mk) algorithm for the m-set k-packing problem and (ii) an O*(23k/2) algorithm for the simple k-path problem, or an O*(2k) algorithm if the graph has an induced k-subgraph with an odd number of Hamiltonian paths. Our algorithms use poly(n) random bits, comparing to the 2O(k) random bits required in prior algorithms, while having similar low space requirements.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Ioannis Koutis
    • 1
  1. 1.Computer Science DepartmentCarnegie Mellon UniversityPittsburgh 

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