Monotone Complexity of Spanning Tree Polynomial Re-visited

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.