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Automata, Languages and Programming

Volume 5125 of the series Lecture Notes in Computer Science pp 575-586

Faster Algebraic Algorithms for Path and Packing Problems

  • Ioannis KoutisAffiliated withComputer Science Department, Carnegie Mellon University

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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)2 k ) time and O(kn + s) space. Based on this, we present new and faster algorithms for two well studied problems: (i) an O *(2 mk ) algorithm for the m-set k-packing problem and (ii) an O *(23k/2) algorithm for the simple k-path problem, or an O *(2 k ) 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.