Integrality of Linearizations of Polynomials over Binary Variables using Additional Monomials

Polynomial optimization problems over binary variables can be expressed as integer programs using a linearization with extra monomials in addition to those arising in the given polynomial. We characterize when such a linearization yields an integral relaxation polytope, generalizing work by Del Pia and Khajavirad (SIAM Journal on Optimization, 2018) and Buchheim, Crama and Rodr\'iguez-Heck (European Journal of Operations Research, 2019). We also present an algorithm that finds these extra monomials for a given polynomial to yield an integral relaxation polytope or determines that no such set of extra monomials exists. In the former case, our approach yields an algorithm to solve the given polynomial optimization problem as a compact LP, and we complement this with a purely combinatorial algorithm.Comment: 27 pages, 11 figure

A new separation algorithm for the Boolean quadric and cut polytopes

A separation algorithm is a procedure for generating cutting planes. Up to now, only a few polynomial-time separation algorithms were known for the Boolean quadric and cut polytopes. These polytopes arise in connection with zero-one quadratic programming and the maxcut problem, respectively. We present a new algorithm, which separates over a class of valid inequalities that includes all odd bicycle wheel inequalities and (2p + 1, 2)-circulant inequalities. It exploits, in a non-trivial way, three known results in the literature: one on the separation of {0,1/2}-cuts, one on the symmetries of the polytopes in question, and one on an affine mapping between the polytopes