12,194 research outputs found
On the complexity of skew arithmetic
13 pagesIn this paper, we study the complexity of several basic operations on linear differential operators with polynomial coefficients. As in the case of ordinary polynomials, we show that these complexities can be expressed in terms of the cost of multiplication
Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits
We deploy algebraic complexity theoretic techniques for constructing
symmetric determinantal representations of for00504925mulas and weakly skew
circuits. Our representations produce matrices of much smaller dimensions than
those given in the convex geometry literature when applied to polynomials
having a concise representation (as a sum of monomials, or more generally as an
arithmetic formula or a weakly skew circuit). These representations are valid
in any field of characteristic different from 2. In characteristic 2 we are led
to an almost complete solution to a question of B\"urgisser on the
VNP-completeness of the partial permanent. In particular, we show that the
partial permanent cannot be VNP-complete in a finite field of characteristic 2
unless the polynomial hierarchy collapses.Comment: To appear in the AMS Contemporary Mathematics volume on
Randomization, Relaxation, and Complexity in Polynomial Equation Solving,
edited by Gurvits, Pebay, Rojas and Thompso
On the expressive power of planar perfect matching and permanents of bounded treewidth matrices
Valiant introduced some 25 years ago an algebraic model of computation along
with the complexity classes VP and VNP, which can be viewed as analogues of the
classical classes P and NP. They are defined using non-uniform sequences of
arithmetic circuits and provides a framework to study the complexity for
sequences of polynomials. Prominent examples of difficult (that is,
VNP-complete) problems in this model includes the permanent and hamiltonian
polynomials. While the permanent and hamiltonian polynomials in general are
difficult to evaluate, there have been research on which special cases of these
polynomials admits efficient evaluation. For instance, Barvinok has shown that
if the underlying matrix has bounded rank, both the permanent and the
hamiltonian polynomials can be evaluated in polynomial time, and thus are in
VP. Courcelle, Makowsky and Rotics have shown that for matrices of bounded
treewidth several difficult problems (including evaluating the permanent and
hamiltonian polynomials) can be solved efficiently. An earlier result of this
flavour is Kasteleyn's theorem which states that the sum of weights of perfect
matchings of a planar graph can be computed in polynomial time, and thus is in
VP also. For general graphs this problem is VNP-complete. In this paper we
investigate the expressive power of the above results. We show that the
permanent and hamiltonian polynomials for matrices of bounded treewidth both
are equivalent to arithmetic formulas. Also, arithmetic weakly skew circuits
are shown to be equivalent to the sum of weights of perfect matchings of planar
graphs.Comment: 14 page
Arithmetic circuits: the chasm at depth four gets wider
In their paper on the "chasm at depth four", Agrawal and Vinay have shown
that polynomials in m variables of degree O(m) which admit arithmetic circuits
of size 2^o(m) also admit arithmetic circuits of depth four and size 2^o(m).
This theorem shows that for problems such as arithmetic circuit lower bounds or
black-box derandomization of identity testing, the case of depth four circuits
is in a certain sense the general case. In this paper we show that smaller
depth four circuits can be obtained if we start from polynomial size arithmetic
circuits. For instance, we show that if the permanent of n*n matrices has
circuits of size polynomial in n, then it also has depth 4 circuits of size
n^O(sqrt(n)*log(n)). Our depth four circuits use integer constants of
polynomial size. These results have potential applications to lower bounds and
deterministic identity testing, in particular for sums of products of sparse
univariate polynomials. We also give an application to boolean circuit
complexity, and a simple (but suboptimal) reduction to polylogarithmic depth
for arithmetic circuits of polynomial size and polynomially bounded degree
Evaluating Matrix Circuits
The circuit evaluation problem (also known as the compressed word problem)
for finitely generated linear groups is studied. The best upper bound for this
problem is , which is shown by a reduction to polynomial
identity testing. Conversely, the compressed word problem for the linear group
is equivalent to polynomial identity testing. In
the paper, it is shown that the compressed word problem for every finitely
generated nilpotent group is in . Within
the larger class of polycyclic groups we find examples where the compressed
word problem is at least as hard as polynomial identity testing for skew
arithmetic circuits
On Computing Multilinear Polynomials Using Multi-r-ic Depth Four Circuits
International audienceIn this paper, we are interested in understanding the complexity of computing multilinear polynomials using depth four circuits in which polynomial computed at every node has a bound on the individual degree of r (referred to as multi-r-ic circuits). The goal of this study is to make progress towards proving superpolynomial lower bounds for general depth four circuits computing multilinear polynomials, by proving better and better bounds as the value of r increases. Recently, Kayal, Saha and Tavenas (Theory of Computing, 2018) showed that any depth four arithmetic circuit of bounded individual degree r computing a multilinear polynomial on n^O(1) variables and degree d = o(n), must have size at least (n/r^1.1)^{\sqrt{d/r}} when r is o(d) and is strictly less than n^1/1.1. This bound however deteriorates with increasing r. It is a natural question to ask if we can prove a bound that does not deteriorate with increasing r or a bound that holds for a larger regime of r. We here prove a lower bound which does not deteriorate with r , however for a specific instance of d = d (n) but for a wider range of r. Formally, we show that there exists an explicit polynomial on n^{O(1)} variables and degree Θ(log^2(n)) such that any depth four circuit of bounded individual degree r < n^0.2 must have size at least exp(Ω (log^2 n)). This improvement is obtained by suitably adapting the complexity measure of Kayal et al. (Theory of Computing, 2018). This adaptation of the measure is inspired by the complexity measure used by Kayal et al. (SIAM J. Computing, 2017)
Sparse multivariate polynomial interpolation in the basis of Schubert polynomials
Schubert polynomials were discovered by A. Lascoux and M. Sch\"utzenberger in
the study of cohomology rings of flag manifolds in 1980's. These polynomials
generalize Schur polynomials, and form a linear basis of multivariate
polynomials. In 2003, Lenart and Sottile introduced skew Schubert polynomials,
which generalize skew Schur polynomials, and expand in the Schubert basis with
the generalized Littlewood-Richardson coefficients.
In this paper we initiate the study of these two families of polynomials from
the perspective of computational complexity theory. We first observe that skew
Schubert polynomials, and therefore Schubert polynomials, are in \CountP
(when evaluating on non-negative integral inputs) and \VNP.
Our main result is a deterministic algorithm that computes the expansion of a
polynomial of degree in in the basis of Schubert
polynomials, assuming an oracle computing Schubert polynomials. This algorithm
runs in time polynomial in , , and the bit size of the expansion. This
generalizes, and derandomizes, the sparse interpolation algorithm of symmetric
polynomials in the Schur basis by Barvinok and Fomin (Advances in Applied
Mathematics, 18(3):271--285). In fact, our interpolation algorithm is general
enough to accommodate any linear basis satisfying certain natural properties.
Applications of the above results include a new algorithm that computes the
generalized Littlewood-Richardson coefficients.Comment: 20 pages; some typos correcte
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