The Running Intersection Relaxation of the Multilinear Polytope

Abstract

The multilinear polytope of a hypergraph is the convex hull of a set of binary points satisfying a collection of multilinear equations. We introduce the running intersection inequalities, a new class of facet-defining inequalities for the multilinear polytope. Accordingly, we define a new polyhedral relaxation of the multilinear polytope, referred to as the running intersection relaxation, and identify conditions under which this relaxation is tight. Namely, we show that for kite-free beta-acyclic hypergraphs, a class that lies between gamma-acyclic and beta-acyclic hypergraphs, the running intersection relaxation coincides with the multilinear polytope and it admits a polynomial size extended formulation.

A Polyhedral Study of Lifted Multicuts

“…once the transformation is applied. We observe that this inequality resembles the running intersection inequalities introduced by Del Pia and Khajavirad (2021). In fact, there is a center, E uv , and two neighbors, E u, v(u) , E u(v),v , that satisfy the running intersection property.…”

“…Proof of Proposition 13. We observe that the structure of this proof is similar to the proof of validity for the running intersection inequalities in Del Pia and Khajavirad (2021). Let us partition…”

“…Note that ( 22) includes the running intersection inequalities. The interested reader will have noticed the similarities between our definition of the sets N k in ( 19) and the definition of N (e 0 ∩ e k ) by Del Pia and Khajavirad (2021). Moreover, our proof of the validity of the generalized intersection inequalities, Proposition 13, uses the same technique as the proof by Del Pia and Khajavirad (2021) of the validity of the running intersection inequalities.…”

“…Recent research also considers the linearization of more general multilinear forms (Del Pia and Khajavirad, 2017Khajavirad, , 2021Del Pia and Di Gregorio, 2021;Hojny et al, 2019). This line of work is motivated by the idea of avoiding the additional constraints and variables introduced in order to express the problem as a quadratic one, and to exploit the structure of the original problem.…”

Simple Odd $$\beta $$-Cycle Inequalities for Binary Polynomial Optimization

“…Recently, several classes of inequalities valid for ML(G) have been introduced, including 2-link inequalities [4], flower inequalities [7], running intersection inequalities [8], and odd β-cycle inequalities [5]. On a theoretical level, these inequalities fully describe the multilinear polytope for several hypergraph instances: flower inequalities for γ-acyclic hypergraphs, running intersection inequalities for kite-free β-acyclic hypergraphs, and flower inequalities together with odd β-cycle inequalities for cycle hypergraphs.…”