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We provide a numerical refutation of the developments of Fiorini et al. (2015) * for modelswith disjoint sets of descriptive variables. We also provide an insight into the meaning of the existence of a one-to-one linear map between solutions of such models.

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 ).

The weighted tree augmentation problem (WTAP) is a fundamental network design problem. We are given an undirected tree G = (V, E) with n = |V | nodes, an additional set of edges L called links and a cost vector c ∈ R L ≥1 . The goal is to choose a minimum cost subset S ⊆ L such that G = (V, E ∪ S) is 2-edgeconnected. In the unweighted case, that is, when we have c = 1 for all ∈ L, the problem is called the tree augmentation problem (TAP).Both problems are known to be APX-hard, and the best known approximation factors are 2 for WTAP by (Frederickson and JáJá, '81) and 3 2 for TAP due to (Kortsarz and Nutov, TALG '16). Adjashvili (SODA '17) recently presented an ≈ 1.96418 + ε-approximation algorithm for WTAP for the case where all link costs are bounded by a constant. This is the first approximation with a better guarantee than 2 that does not require restrictions on the structure of the tree or the links.In this paper, we improve Adjiashvili's approximation to a 3 2 + ε-approximation for WTAP under the bounded cost assumption. We achieve this by introducing a strong LP that combines 0, 1 2 -Chvátal-Gomory cuts for the standard LP for the problem with bundle constraints from Adjiashvili. We show that our LP can be solved efficiently and that it is exact for some instances that arise at the core of Adjiashvili's approach. This results in the improved performance * Martin Groß, Jochen Könemann and Laura Sanità acknowledge support from the NSERC Discovery Grant Program as well as an Early Researcher Award by the Province of Ontario. Samuel Fiorini acknowledges support from F.R.S. -FNRS (grant number 1861993, "Polyhedral combinatorics: cycle transversals, {0, 1/2}-cuts") and the ERC (Consolidator Grant 615640-ForEFront). +ε, which is asymptotically on par with the result by Kortsarz and Nutov. Our result also is the best-known LP-relative approximation algorithm for TAP.

We develop a framework for proving approximation limits of polynomial-size linear programs from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any linear program as opposed to only programs generated by hierarchies. Using our framework, we prove that O(n 1/2−ǫ )-approximations for CLIQUE require linear programs of size 2 n Ω(ǫ) . This lower bound applies to linear programs using a certain encoding of CLIQUE as a linear optimization problem. Moreover, we establish a similar result for approximations of semidefinite programs by linear programs.Our main technical ingredient is a quantitative improvement of Razborov's rectangle corruption lemma (1992) for the high error regime, which gives strong lower bounds on the nonnegative rank of shifts of the unique disjointness matrix.