Abstract
In this paper, we study the computational complexity of the commutative determinant polynomial computed by a class of set-multilinear circuits which we call regular set-multilinear circuits. Regular set-multilinear circuits are commutative circuits with a restriction on the order in which they can compute polynomials. A regular circuit can be seen as the commutative analogue of the ordered circuit defined by Hrubes, Wigderson and Yehudayoff [5]. We show that if the commutative determinant polynomial has small representation in the sum of constantly many regular set-multilinear circuits, then the commutative permanent polynomial also has a small arithmetic circuit.
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References
Arvind, V., Raja, S.: Some lower bound results for set-multilinear arithmetic computations. Chicago J. Theor. Comput. Sci. 2016(6)
Arvind, V., Srinivasan, S.: On the hardness of the noncommutative determinant. In: Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5–8 June 2010, pp. 677–686 (2010)
Berkowitz, S.J.: On computing the determinant in small parallel time using a small number of processors. Inf. Process. Lett. 18(3), 147–150 (1984)
Erdös, P., Szekeres, G.: A combinatorial problem in geometry. Compos. Math. 2, 463–470 (1935)
Hrubes, P., Wigderson, A., Yehudayoff, A.: Non-commutative circuits and the sum-of-squares problem. In: Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5–8 June 2010, pp. 667–676 (2010)
Nisan, N.: Lower bounds for non-commutative computation (extended abstract). In: STOC, pp. 410–418 (1991)
Nisan, N., Wigderson, A.: Lower bounds for arithmetic circuits via partial derivatives (preliminary version). In: 36th Annual Symposium on Foundations of Computer Science, Milwaukee, Wisconsin, 23–25 October 1995, pp. 16–25 (1995)
Raz, R.: Multi-linear formulas for permanent and determinant are of super-polynomial size. J. ACM 56(2), 1–17 (2009)
Valiant, L.G.: Completeness classes in algebra. In: Proceedings of the 11h Annual ACM Symposium on Theory of Computing, 30 April–2 May 1979, Atlanta, Georgia, USA, pp. 249–261 (1979)
Valiant, L.G.: The complexity of computing the permanent. Theor. Comput. Sci. 8, 189–201 (1979)
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Raja, S., Bharadwaj, G.V.S. (2021). On the Hardness of the Determinant: Sum of Regular Set-Multilinear Circuits. In: Bampis, E., Pagourtzis, A. (eds) Fundamentals of Computation Theory. FCT 2021. Lecture Notes in Computer Science(), vol 12867. Springer, Cham. https://doi.org/10.1007/978-3-030-86593-1_30
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DOI: https://doi.org/10.1007/978-3-030-86593-1_30
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