Transparency Beyond VNP in the Monotone Setting
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Recently Hrubes and Yehudayoff (2021) showed a connection between the monotone algebraic circuit complexity of transparent polynomials and a geometric complexity measure of their Newton polytope. They then used this connection to prove lower bounds against monotone VP (mVP). We extend their work by showing that their technique can be used to prove lower bounds against classes that are seemingly more powerful than monotone VNP (mVNP). In the process, we define a natural monotone analogue of VPSPACE – a well-studied class in the non-monotone setting (Poizat 2008, Koiran-Perifel 2009, Malod 2011, Mahajan-Rao 2013) – and prove an exponential separation between the computational powers of this class and mVNP. To show this separation, we define a new polynomial family with an interesting combinatorial structure which we use heavily in our lower bound. Both the polynomial and the combinatorial nature of our proof might be of independent interest.
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