A method for deriving lower bounds for the complexity of monotone arithmetic circuits computing real polynomials

Lower bounds for monotone counting circuits

“…Schnorr [17] has shown that C(∆ n ) = Θ(n 3/2 ); this also follows from the lower bound of Gashkov and Sergeev [7] mentioned above, because the polynomial is (1, 1)-sparse: any triangle is uniquely determined by any two of its edges.…”

“…The proofs of Theorems 8 and 9 extend to C 0/1 (f ) the arguments used in [6,7,13] to lower-bound C(f ). The main difficulty with the extension (stipulated by the idempotence axiom x 2 = x) is that, unlike the measure µ(f ) = |mon(f )| (used to lower-bound C(f )), the measures |sup(f )| and |min(f )| are no more "monotone " in the sense that µ(f ) µ(f g).…”

“…There is yet another general lower-bounds criterion for monotone arithmetic circuits, due to Gashkov [6], and Gashkov and Sergeev [7]. They call a polynomial f (k, l)-sparse, if…”

“…In particular, if the target polynomial f is multilinear (no variable has degree larger than 1, then the circuit itself must be multilinear: the polynomials produced at inputs of each product gate must depend on disjoint sets of variables. This limitation is essentially exploited in all lower bounds for monotone arithmetic circuits, including [17,19,10,27,22,6,24,7].…”

Lower Bounds for Tropical Circuits and Dynamic Programs

“…Proof. Our argument is a mix of ideas of Gashkov and Sergeev [10], and of Pippenger [32]. Take a minimal circuit F producing f ; hence, F = f is (k, l)-free.…”