BREAKING THE RECTANGLE BOUND BARRIER AGAINST FORMULA SIZE LOWER BOUNDS

Abstract

Karchmer, Kushilevitz and Nisan formulated the formula size problem as an integer programming problem called the rectangle bound and introduced a technique called the LP bound, which gives a formula size lower bound by showing a feasible solution of the dual problem of its LP-relaxation. As extensions of the LP bound, we introduce novel general techniques proving formula size lower bounds, named a quasi-additive bound and the Sherali-Adams bound. While the Sherali-Adams bound is potentially strong enough to give a lower bound matching to the rectangle bound, we prove that the quasi-additive bound can surpass the rectangle bound. We also reveal that the quasi-additive bound is potentially strong enough to prove the matching formula size lower bound.