Monotone Complexity of Spanning Tree Polynomial Re-visited
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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^Ω(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 ϵ-sensitive lower bounds for monotone arithmetic circuits for a specific range of values for ϵ∈ (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 - ϵ· ST_n and F_n-1,n + ϵ· ST_n defined over n^2 variables, have monotone circuit complexity 2^Ω(n) if ϵ≥ 2^-Ω(n) and F_n-1,n = ∏_i=2^n (x_i,1 +⋯ + x_i,n) is the complete set-multilinear polynomial. This provides the first ϵ-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.
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