Faster Algebraic Algorithms for Path and Packing Problems

We study the problem of deciding whether an n-variate polynomial, presented as an arithmetic circuit G, contains a degree ksquare-free term with an odd coefficient. We show that if Gcan be evaluated over the integers modulo 2k+ 1in time tand 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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