Combinatorial bounds on nonnegative rank and extended formulations

Abstract

An extended formulation of a polytope P is a system of linear inequalities and equations that describe some polyhedron which can be projected onto P . Extended formulations of small size (i.e., number of inequalities) are of interest, as they allow to model corresponding optimization problems as linear programs of small sizes. In this paper, we describe several aspects and new results on the main known approach to establish lower bounds on the sizes of extended formulations, which is to bound from below the number of rectangles needed to cover the support of a slack matrix of the polytope. Our main goals are to shed some light on the question how this combinatorial rectangle covering bound compares to other bounds known from the literature, and to obtain a better idea of the power as well as of the limitations of this bound. In particular, we provide geometric interpretations (and a slight sharpening) of Yannakakis ' [35] result on the relation between minimal sizes of extended formulations and the nonnegative rank of slack matrices, and we describe the fooling set bound on the nonnegative rank (due to Dietzfelbinger et al. [7]) as the clique number of a certain graph. Among other results, we prove that both the cube as well as the Birkhoff polytope do not admit extended formulations with fewer inequalities than these polytopes have facets, and we show that every extended formulation of a d-dimensional neighborly polytope with Ω(d 2 ) vertices has size Ω(d 2 ).

Smallest compact formulation for the permutahedron

“…We also show that any extended formulation with O(n log n) inequalities requires Θ(n log n) dimensions. The only other tight results we are aware of show that both the hypercube and the Birkhoff polytope admit no smaller extended formulations than themselves, see Fiorini et al [4].…”

Sparse sums of squares on finite abelian groups and improved semidefinite lifts

“…, a lower bound from [FKPT13] concerning neighborly polytopes shows that any linear programming lift of T C(d 2 , 2d) must have size at least Ω(d 2 ), whereas our semidefinite programming lift in this case has size…”

Characteristics of Complexity: Clique Number of a Polytope Graph and Rectangle Covering Number

“…Во всяком случае, список задач, для которых этот факт установлен, весьма внушителен [11]. В частности, для многогранника паросочетаний MATCH n число прямоугольного покрытия больше, чем n 2 , но меньше, чем n 4 [14]. (Напомним, что лучшие из известных алгоритмов решают задачу о паросочетаниях за O(n 3 ) опера-ций.)…”

“…Более того, за счет малых смещений мы всегда можем добиться того, что все вершины окажутся в общем положении 7 , соответственно, сам новый многогранник станет симплициальным. Остается вос-пользоваться тем, что число прямоугольного покрытия для симплициального мно-гогранника равно O(d 2 log 2 K), где d размерность многогранника, а K число его вершин [14]. То есть в нашем случае это число окажется равным O(n 3 (n + 1) 2 ).…”