Abstract
We introduce the notion of monotone linear programming circuits (MLP circuits), a model of computation for partial Boolean functions. Using this model, we prove the following results. 1 (1) MLP circuits are superpolynomially stronger than monotone Boolean circuits. (2) MLP circuits are exponentially stronger than monotone span programs over the reals. (3) MLP circuits can be used to provide monotone feasibility interpolation theorems for Lovász-Schrijver proof systems and for mixed Lovász-Schrijver proof systems. (4) The Lovász-Schrijver proof system cannot be polynomially simulated by the cutting planes proof system. Finally, we establish connections between the problem of proving lower bounds for the size of MLP circuits and the field of extension complexity of polytopes.
Highlights
Superpolynomial lower bounds on the size of Boolean circuits computing explicit Boolean functions have only been proved for circuits from some specific families of circuits
Size lower bounds for these systems have been proved only with respect to tree-like proofs [21], and it seems reasonable that a monotone interpolation theorem for this system may be a first step towards proving size lower bounds for general LS proof systems. Towards this goal we show that MLP circuits which are constituted by strong MLP gates can be used to provide a monotone feasible interpolation theorem for LS proof systems
In this work we introduced several models of computation based on the notion of monotone linear programs
Summary
Superpolynomial lower bounds on the size of Boolean circuits computing explicit Boolean functions have only been proved for circuits from some specific families of circuits. Our result solves one direction of this mutual relation by showing that for some tautologies, LS proofs can be superpolynomially more concise than cutting-planes proofs Using this interpolation theorem, and a size lower bound for monotone real circuits due to Fu [10], we can show that MLP-circuits cannot be polynomially simulated by monotone real circuits (Theorem 19). The theorem gives an example of a monotone function whose set of ones requires exponentially large extended formulation, but whose minterms can be separated from a large subset of maxterms by a polynomial size weak MLP gate. In this extended abstract all proofs are omitted
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