eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
16:1
16:14
10.4230/LIPIcs.FSTTCS.2022.16
article
On the VNP-Hardness of Some Monomial Symmetric Polynomials
Curticapean, Radu
1
2
https://orcid.org/0000-0001-7201-9905
Limaye, Nutan
1
2
https://orcid.org/0000-0002-0238-1674
Srinivasan, Srikanth
3
4
https://orcid.org/0000-0001-6491-124X
IT University of Copenhagen, Denmark
Basic Algorithms Research Copenhagen, Denmark
Department of Computer Science, Aarhus University, Denmark
On leave from Department of Mathematics, IIT Bombay, India
A polynomial P ∈ 𝔽[x_1,…,x_n] is said to be symmetric if it is invariant under any permutation of its input variables. The study of symmetric polynomials is a classical topic in mathematics, specifically in algebraic combinatorics and representation theory. More recently, they have been studied in several works in computer science, especially in algebraic complexity theory.
In this paper, we prove the computational hardness of one of the most basic kinds of symmetric polynomials: the monomial symmetric polynomials, which are obtained by summing all distinct permutations of a single monomial. This family of symmetric functions is a natural basis for the space of symmetric polynomials (over any field), and generalizes many well-studied families such as the elementary symmetric polynomials and the power-sum symmetric polynomials.
We show that certain families of monomial symmetric polynomials are VNP-complete with respect to oracle reductions. This stands in stark contrast to the case of elementary and power symmetric polynomials, both of which have constant-depth circuits of polynomial size.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol250-fsttcs2022/LIPIcs.FSTTCS.2022.16/LIPIcs.FSTTCS.2022.16.pdf
algebraic complexity
symmetric polynomial
permanent
Sidon set