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.
This work was partially supported by the National Science Foundation under grant number CCF-0635257.
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References
Alon, N., Yuster, R., Zwick, U.: Color coding. Journal of the ACM 42(4), 844–856 (1995)
Chen, J., Lu, S., Sze, S.-H., Zhang, F.: Improved algorithms for path, matching, and packing problems. In: SODA 2007: Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms, Philadelphia, PA, USA. Society for Industrial and Applied Mathematics, pp. 298–307 (2007)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)
Fellows, M.R., Knauer, C., Nishimura, N., Ragde, P., Rosamond, F.A., Stege, U., Thilikos, D.M., Whitesides, S.: Faster fixed-parameter tractable algorithms for matching and packing problems. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 311–322. Springer, Heidelberg (2004)
Jia, W., Zhang, C., Chen, J.: An efficient parameterized algorithm for m-set packing. J. Algorithms 50(1), 106–117 (2004)
Kneis, J., Mölle, D., Richter, S., Rossmanith, P.: Divide-and-color. In: WG: Graph-Theoretic Concepts in Computer Science, 32nd International Workshop, pp. 58–67 (2006)
Koutis, I.: A faster parameterized algorithm for set packing. Information Processing Letters 94(1), 4–7 (2005)
Liu, Y., Lu, S., Chen, J., Sze, S.-H.: Greedy localization and color-coding: Improved matching and packing algorithms. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 84–95. Springer, Heidelberg (2006)
Papadimitriou, C.H., Yannakakis, M.: On limited nondeterminism and the complexity of the V-C dimension. J. Comput. Syst. Sci. 53(2), 161–170 (1996)
Terras, A.: Fourier Analysis on Finite Groups and Applications. Cambridge University, Cambridge (1999)
Valiant, L.G.: Why is boolean complexity difficult? In: Boolean Function Complexity. Lond. Math. Soc. Lecure Note Ser, vol. 169
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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. https://doi.org/10.1007/978-3-540-70575-8_47
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DOI: https://doi.org/10.1007/978-3-540-70575-8_47
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