On The Closures of Monotone Algebraic Classes and Variants of the Determinant

Abstract

In this paper we prove the following two results.

• We show that for any $C \in \{\textsf {mVF}, \textsf {mVP}, \textsf {mVNP}\}$, $C = \overline{C}$. Here, $\textsf {mVF}, \textsf {mVP}$, and $\textsf {mVNP}$ are monotone variants of $\textsf {VF}, \textsf {VP}$, and $\textsf {VNP}$, respectively. For an algebraic complexity class C, $\overline{C}$ denotes the closure of C. For $\textsf {mVBP}$ a similar result was shown in Bläser et al. (in: 35th Computational Complexity Conference, CCC 2020. LIPIcs, vol 169, pp 21–12124, 2020. https://doi.org/10.4230/LIPIcs.CCC.2020.21). Here we extend their result by adapting their proof.

• We define polynomial families $\{\mathcal {P}(k)_n\}_{n \ge 0}$, such that $\{\mathcal {P}(0)_n\}_{n \ge 0}$ equals the determinant polynomial. We show that $\{\mathcal {P}(k)_n\}_{n \ge 0}$ is $\textsf {VBP}$ complete for $k=1$ and it becomes $\textsf {VNP}$ complete when $k \ge 2$. In particular, $\{\mathcal {P}(k)_n\}$ is $\mathtt {Det^{\ne k}_n(X)}$, a polynomial obtained by summing over all signed cycle covers that avoid length k cycles. We show that $\mathtt {Det^{\ne 1}_n(X)}$ is complete for $\textsf {VBP}$ and $\mathtt {Det^{\ne k}_n(X)}$ is complete for $\textsf {VNP}$ for all $k \ge 2$ over any field $\mathbb {F}$.