Abstract
We study the problem of computing an ensemble of multiple sums where the summands in each sum are indexed by subsets of size p of an n-element ground set. More precisely, the task is to compute, for each subset of size q of the ground set, the sum over the values of all subsets of size p that are disjoint from the subset of size q. We present an arithmetic circuit that, without subtraction, solves the problem using O((n p + n q)logn) arithmetic gates, all monotone; for constant p, q this is within the factor logn of the optimal. The circuit design is based on viewing the summation as a “set nucleation” task and using a tree-projection approach to implement the nucleation. Applications include improved algorithms for counting heaviest k-paths in a weighted graph, computing permanents of rectangular matrices, and dynamic feature selection in machine learning.
This research was supported in part by the Academy of Finland, Grants 252083 (P.K.), 256287 (P.K.), and 125637 (M.K.), and by the Helsinki Doctoral Programme in Computer Science - Advanced Computing and Intelligent Systems (J.K.).
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References
Alon, N., Yuster, R., Zwick, U.: Finding and counting given length cycles. Algorithmica 17(3), 209–223 (1997)
Bellman, R.: Combinatorial processes and dynamic programming. In: Combinatorial Analysis. Proceedings of Symposia in Applied Mathematics, vol. 10, pp. 217–249. ACM (1960)
Bellman, R.: Dynamic programming treatment of the travelling salesman problem. J. ACM 9(1), 61–63 (1962)
Björklund, A.: Determinant sums for undirected Hamiltonicity. In: 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010), pp. 173–182. IEEE Computer Society, Los Alamitos (2010)
Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Fourier meets möbius: fast subset convolution (manuscript submitted for publication)
Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Computing the Tutte polynomial in vertex-exponential time. In: 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2008), pp. 677–686. IEEE Computer Society, Los Alamitos (2008)
Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Counting Paths and Packings in Halves. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 578–586. Springer, Heidelberg (2009)
Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Evaluation of permanents in rings and semirings. Information Processing Letters 110(20), 867–870 (2010)
Björklund, A., Husfeldt, T., Koivisto, M.: Set partitioning via inclusion-exclusion. SIAM Journal on Computing 39(2), 546–563 (2009)
de Campos, C.P., Ji, Q.: Efficient structure learning of bayesian networks using constraints. Journal of Machine Learning Research 12, 663–689 (2011)
Cohn, H., Kleinberg, R., Szegedy, B., Umans, C.: Group-theoretic algorithms for matrix multiplication. In: 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), pp. 379–388. IEEE Computer Society, Los Alamitos (2005)
Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Mathematics of Computation 19, 297–301 (1965)
Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation 9(3), 251–280 (1990)
Cygan, M., Nederlof, J., Pilipczuk, M., Pilipczuk, M., van Rooij, J.M.M., Wojtaszczyk, J.O.: Solving connectivity problems parameterized by treewidth in single exponential time (2011) (manuscript)
Garey, M.R., Johnson, D.S.: Computers and Intractability, A Guide to the Theory of NP-Completeness. W.H. Freeman and Company (1979)
Held, M., Karp, R.M.: A dynamic programming approach to sequencing problems. Journal of the Society for Industrial and Applied Mathematics 10(1), 196–210 (1962)
Itai, A., Rodeh, M.: Finding a minimum circuit in a graph. SIAM Journal on Computing 7(4), 413–423 (1978)
Kerr, L.R.: The effect of algebraic structure on the computational complexity of matrix multiplications. Ph.D. thesis, Cornell University (1970)
Knuth, D.E.: The Art of Computer Programming, 3rd edn. Seminumerical Algorithms, vol. 2. Addison–Wesley, Upper Saddle River (1998)
Koutis, I., Williams, R.: Limits and Applications of Group Algebras for Parameterized Problems. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 653–664. Springer, Heidelberg (2009)
Lokshtanov, D., Nederlof, J.: Saving space by algebraization. In: 2010 ACM International Symposium on Theory of Computing (STOC 2010), pp. 321–330. ACM, New York (2010)
Valiant, L.G.: Negation is powerless for boolean slice functions. SIAM Journal on Computing 15, 531–535 (1986)
Vassilevska, V., Williams, R.: Finding, minimizing, and counting weighted subgraphs. In: 2009 ACM International Symposium on Theory of Computing (STOC 2009), pp. 455–464. ACM, New York (2009)
Vassilevska, V., Williams, R., Yuster, R.: Finding heaviest H-subgraphs in real weighted graphs, with applications. ACM Transactions on Algorithms 6(3), Art. 44, 23 (2010)
Williams, R.: Finding paths of length k in O *(2k) time. Information Processing Letters 109(6), 315–318 (2009)
Yates, F.: The Design and Analysis of Factorial Experiments. Imperial Bureau of Soil Science, Harpenden, England (1937)
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Kaski, P., Koivisto, M., Korhonen, J.H. (2012). Fast Monotone Summation over Disjoint Sets. In: Thilikos, D.M., Woeginger, G.J. (eds) Parameterized and Exact Computation. IPEC 2012. Lecture Notes in Computer Science, vol 7535. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33293-7_16
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DOI: https://doi.org/10.1007/978-3-642-33293-7_16
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