Extended Formulation Lower Bounds via Hypergraph Coloring?

Exploring the power of linear programming for combinatorial optimization problems has been recently receiving renewed attention after a series of breakthrough impossibility results. From an algorithmic perspective, the related questions concern whether there are compact formulations even for problems that are known to admit polynomial-time algorithms. We propose a framework for proving lower bounds on the size of extended formulations. We do so by introducing a specific type of extended relaxations that we call product relaxations and is motivated by the study of the Sherali-Adams (SA) hierarchy. Then we show that for every approximate relaxation of a polytope P, there is a product relaxation that has the same size and is at least as strong. We provide a methodology for proving lower bounds on the size of approximate product relaxations by lower bounding the chromatic number of an underlying hypergraph, whose vertices correspond to gap-inducing vectors. We extend the definition of product relaxations and our methodology to mixed integer sets. However in this case we are able to show that mixed product relaxations are at least as powerful as a special family of extended formulations. As an application of our method we show an exponential lower bound on the size of approximate mixed product formulations for the metric capacitated facility location problem, a problem which seems to be intractable for linear programming as far as constant-gap compact formulations are concerned

On the extension complexity of combinatorial polytopes

In this paper we extend recent results of Fiorini et al. on the extension complexity of the cut polytope and related polyhedra. We first describe a lifting argument to show exponential extension complexity for a number of NP-complete problems including subset-sum and three dimensional matching. We then obtain a relationship between the extension complexity of the cut polytope of a graph and that of its graph minors. Using this we are able to show exponential extension complexity for the cut polytope of a large number of graphs, including those used in quantum information and suspensions of cubic planar graphs.Comment: 15 pages, 3 figures, 2 table

Lifting Linear Extension Complexity Bounds to the Mixed-Integer Setting

Mixed-integer mathematical programs are among the most commonly used models for a wide set of problems in Operations Research and related fields. However, there is still very little known about what can be expressed by small mixed-integer programs. In particular, prior to this work, it was open whether some classical problems, like the minimum odd-cut problem, can be expressed by a compact mixed-integer program with few (even constantly many) integer variables. This is in stark contrast to linear formulations, where recent breakthroughs in the field of extended formulations have shown that many polytopes associated to classical combinatorial optimization problems do not even admit approximate extended formulations of sub-exponential size. We provide a general framework for lifting inapproximability results of extended formulations to the setting of mixed-integer extended formulations, and obtain almost tight lower bounds on the number of integer variables needed to describe a variety of classical combinatorial optimization problems. Among the implications we obtain, we show that any mixed-integer extended formulation of sub-exponential size for the matching polytope, cut polytope, traveling salesman polytope or dominant of the odd-cut polytope, needs $\Omega(n/\log n)$ many integer variables, where $n$ is the number of vertices of the underlying graph. Conversely, the above-mentioned polyhedra admit polynomial-size mixed-integer formulations with only $O(n)$ or $O(n \log n)$ (for the traveling salesman polytope) many integer variables. Our results build upon a new decomposition technique that, for any convex set $C$, allows for approximating any mixed-integer description of $C$ by the intersection of $C$ with the union of a small number of affine subspaces.Comment: A conference version of this paper will be presented at SODA 201

An extension of disjunctive programming and its impact for compact tree formulations

In the 1970's, Balas introduced the concept of disjunctive programming, which is optimization over unions of polyhedra. One main result of his theory is that, given linear descriptions for each of the polyhedra to be taken in the union, one can easily derive an extended formulation of the convex hull of the union of these polyhedra. In this paper, we give a generalization of this result by extending the polyhedral structure of the variables coupling the polyhedra taken in the union. Using this generalized concept, we derive polynomial size linear programming formulations (compact formulations) for a well-known spanning tree approximation of Steiner trees, for Gomory-Hu trees, and, as a consequence, of the minimum $T$-cut problem (but not for the associated $T$-cut polyhedron). Recently, Kaibel and Loos (2010) introduced a more involved framework called {\em polyhedral branching systems} to derive extended formulations. The most parts of our model can be expressed in terms of their framework. The value of our model can be seen in the fact that it completes their framework by an interesting algorithmic aspect.Comment: 17 page