Monotone Complexity of Spanning Tree Polynomial Re-visited

TL;DR

This paper establishes new exponential lower bounds on the monotone arithmetic circuit complexity of the spanning tree polynomial, advancing understanding of algebraic complexity and circuit lower bounds.

Contribution

It proves the first strongly exponential monotone lower bounds for spanning tree polynomials over expander graphs and introduces epsilon-sensitive bounds within VP.

Findings

Monotone complexity of spanning tree polynomials over expander graphs is exponential.

Epsilon-sensitive lower bounds are established for specific polynomials in VP.

A communication complexity result related to spanning trees with low discrepancy is also presented.

Abstract

We prove two results that shed new light on the monotone complexity of the spanning tree polynomial, a classic polynomial in algebraic complexity and beyond. First, we show that the spanning tree polynomials having $n$ variables and defined over constant-degree expander graphs, have monotone arithmetic complexity $2^{\Omega(n)}$. This yields the first strongly exponential lower bound on the monotone arithmetic circuit complexity for a polynomial in VP. Before this result, strongly exponential size monotone lower bounds were known only for explicit polynomials in VNP (Gashkov-Sergeev'12, Raz-Yehudayoff'11, Srinivasan'20, Cavalar-Kumar-Rossman'20, Hrubes-Yehudayoff'21). Recently, Hrubes'20 initiated a program to prove lower bounds against general arithmetic circuits by proving $\epsilon$-sensitive lower bounds for monotone arithmetic circuits for a specific range of values for $\epsilon \in (0,1)$. We consider the spanning tree polynomial $ST_{n}$ defined over the complete graph on $n$ vertices and show that the polynomials $F_{n-1,n} - \epsilon \cdot ST_{n}$ and $F_{n-1,n} + \epsilon \cdot ST_{n}$ defined over $n^2$ variables, have monotone circuit complexity $2^{\Omega(n)}$ if $\epsilon \geq 2^{-\Omega(n)}$ and $F_{n-1,n} = \prod_{i=2}^n (x_{i,1} +\cdots + x_{i,n})$ is the complete set-multilinear polynomial. This provides the first $\epsilon$-sensitive exponential lower bound for a family of polynomials inside VP. En-route, we consider a problem in 2-party, best partition communication complexity of deciding whether two sets of oriented edges distributed among Alice and Bob form a spanning tree or not. We prove that there exists a fixed distribution, under which the problem has low discrepancy with respect to every nearly-balanced partition. This result could be of interest beyond algebraic complexity.