Sebastian Pokutta

Exponential Lower Bounds for Polytopes in Combinatorial Optimization

We solve a 20-year old problem posed by Yannakakis and prove that no polynomial-size linear program (LP) exists whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope and the stable set polytope. These results were discovered through a new connection that we make between one-way quantum communication protocols and semidefinite programming reformulations of LPs.

Approximation Limits of Linear Programs (Beyond Hierarchies)

We develop a framework for proving approximation limits of polynomial size linear programs (LPs) from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any LP as opposed to only programs generated by hierarchies. Using our framework, we prove that O(n1/2-ϵ)-approximations for CLIQUE require LPs of size 2nΩ(ϵ). This lower bound applies to LPs using a certain encoding of CLIQUE as a linear optimization problem. Moreover, we establish a similar result for approximations of semidefinite programs by LPs. Our main technical ingredient is a quantitative improvement of Razborov’s [38] rectangle corruption lemma for the high error regime, which gives strong lower bounds on the nonnegative rank of shifts of the unique disjointness matrix.

Strict linear prices in non-convex European day-ahead electricity markets

The European power grid can be divided into several market areas where the price of electricity is determined in a day-ahead auction. Market participants can provide continuous hourly bid curves and combinatorial bids with associated quantities given the prices. The goal of our auction is to maximize the economic surplus of all participants subject to quantity constraints and price constraints. The price constraints ensure that no one incurs a loss. Only traders who submitted a combinatorial bid might miss a not-realized profit. The resulting problem is a large-scale mathematical program with equilibrium constraints (MPEC) and binary variables that cannot be solved efficiently by standard solvers. We present an exact algorithm and a fast heuristic for this type of problem. Both algorithms decompose the MPEC into a master problem (a mixed-integer quadratic program) and pricing subproblems (linear programs). The modelling technique and the algorithms are applicable to a wide variety of combinatorial auctions that are based on mixed-integer programs.

On the Existence of 0/1 Polytopes with High Semidefinite Extension Complexity

Rothvoss [1] showed that there exists a 0/1 polytope (a polytope whose vertices are in {0,1} n ) such that any higher-dimensional polytope projecting to it must have 2Ω(n) facets, i.e., its linear extension complexity is exponential. The question whether there exists a 0/1 polytope with high PSD extension complexity was left open. We answer this question in the affirmative by showing that there is a 0/1 polytope such that any spectrahedron projecting to it must be the intersection of a semidefinite cone of dimension 2Ω(n) and an affine space. Our proof relies on a new technique to rescale semidefinite factorizations.

A note on the extension complexity of the knapsack polytope

We show that there are 0-1 and unbounded knapsack polytopes with super-polynomial extension complexity. More specifically, for each

The matching polytope does not admit fully-polynomial size relaxation schemes

n \in N

Lower bounds for the Chvátal-Gomory rank in the 0/1 cube

we exhibit 0-1 and unbounded knapsack polytopes in dimension

Approximation and online algorithms for multidimensional bin packing: A survey

n

Cutting-planes for weakly-coupled 0/1 second order cone programs

with extension complexity

On the rank of cutting-plane proof systems

\Omega(2^{\sqrt{n}})