On the complexity of partial derivatives

The method of partial derivatives is one of the most successful lower bound methods for arithmetic circuits. It uses as a complexity measure the dimension of the span of the partial derivatives of a polynomial. In this paper, we consider this complexity measure as a computational problem: for an input polynomial given as the sum of its nonzero monomials, what is the complexity of computing the dimension of its space of partial derivatives? We show that this problem is #P-hard and we ask whether it belongs to #P. We analyze the "trace method", recently used in combinatorics and in algebraic complexity to lower bound the rank of certain matrices. We show that this method provides a polynomial-time computable lower bound on the dimension of the span of partial derivatives, and from this method we derive closed-form lower bounds. We leave as an open problem the existence of an approximation algorithm with reasonable performance guarantees.A slightly shorter version of this paper was presented at STACS'17. In this new version we have corrected a typo in Section 4.1, and added a reference to Shitov's work on tensor rank.

[1]  D. Gottlieb.  A certain class of incidence matrices , 1966 .

[2]  Leslie G. Valiant,et al.  The Complexity of Computing the Permanent , 1979, Theor. Comput. Sci..

[3]  J. Scott Provan,et al.  The Complexity of Counting Cuts and of Computing the Probability that a Graph is Connected , 1983, SIAM J. Comput..

[4]  Catherine S. Greenhill.  The complexity of counting colourings and independent sets in sparse graphs and hypergraphs , 2000, computational complexity.

[5]  Martin E. Dyer,et al.  On Markov Chains for Independent Sets , 2000, J. Algorithms.

[6]  Salil P. Vadhan,et al.  The Complexity of Counting in Sparse, Regular, and Planar Graphs , 2002, SIAM J. Comput..

[7]  Martin E. Dyer,et al.  On Counting Independent Sets in Sparse Graphs , 2002, SIAM J. Comput..

[8]  Noam Nisan,et al.  Lower bounds on arithmetic circuits via partial derivatives , 2005, computational complexity.

[9]  Noga Alon,et al.  Perturbed Identity Matrices Have High Rank: Proof and Applications , 2009, Combinatorics, Probability and Computing.

[10]  Amir Yehudayoff,et al.  Arithmetic Circuits: A survey of recent results and open questions , 2010, Found. Trends Theor. Comput. Sci..

[11]  A. Razborov.  Communication Complexity , 2011 .

[12]  Neeraj Kayal,et al.  Affine projections of polynomials , 2011, Electron. Colloquium Comput. Complex..

[13]  A. Wigderson,et al.  Partial Derivatives in Arithmetic Complexity and Beyond (Foundations and Trends(R) in Theoretical Computer Science) , 2011 .

[14]  Avi Wigderson,et al.  Partial Derivatives in Arithmetic Complexity and Beyond , 2011, Found. Trends Theor. Comput. Sci..

[15]  Eduardo Sáenz-de-Cabezón,et al.  Complexity and algorithms for Euler characteristic of simplicial complexes , 2013, J. Symb. Comput..

[16]  Christopher J. Hillar,et al.  Most Tensor Problems Are NP-Hard , 2009, JACM.

[17]  Nutan Limaye,et al.  An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[18]  Meena Mahajan,et al.  The Shifted Partial Derivative Complexity of Elementary Symmetric Polynomials , 2017, Theory Comput..

[19]  Y. Shitov.  How hard is the tensor rank , 2016, 1611.01559.

[20]  Neeraj Kayal,et al.  Lower Bounds for Depth-Three Arithmetic Circuits with small bottom fanin , 2016, computational complexity.