On linear programming formulations for the TSP polytope
Next week I am planning to give a talk on our recent paper which is joint work with Samuel Fiorini, Serge Massar, Hans Raj Tiwary, and Ronald de Wolf where we consider linear and semidefinite extended formulations and we prove that any linear programming formulation of the traveling salesman polytope has super-polynomial size (independent of P-vs-NP). From the abstract:
We solve a 20-year old problem posed by M. Yannakakis and prove that there exists no polynomial-size linear program (LP) 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 maximum cut problem and the stable set problem. These results follow from a new connection that we make between one-way quantum communication protocols and semidefinite programming reformulations of LPs.
In 1986/87 Ted Swart (University of Guelph) wrote a number of papers (some of them had the title: “P=NP”) that gave linear programming formulations of polynomial size for the Hamiltonian cycle problem. Since linear programming is polynomially solvable and Hamiltonian cycle is NP-hard, Swart deduced that P=NP.
In 1988, Mihalis Yannakakis closed the discussion with his paper “Expressing combinatorial optimization problems by linear programs” (Proceedings of STOC 1988, pp. 223-228). Yannakakis proved that expressing the traveling salesman problem by a symmetric linear program (as in Swart’s approach) requires exponential size. The journal version of this paper has been published in Journal of Computer and System Sciences 43, 1991, pp. 441-466.
In his paper, Yannakakis posed the question whether one can show such a lower bound unconditionally, i.e., without the symmetry assumption and Yannakakis conjectured that symmetry ‘should not help much’. This sounded reasonable however no proof was known. In 2010, to the surprise of many, Kaibel, Pashkovich, and Theis proved that there exist polytopes whose symmetric extension complexity (the number of facets of the polytope) is super-polynomial, whereas there exists asymmetric extended formulations that use only polynomially many inequalities; i.e., symmetry does matter. On top of that, the considered polytopes were closely related to the matching polytope (used by Yannakakis to establish the TSP result) which rekindled the discussion on the (unconditional) extension complexity of the travelling salesman polytope and Kaibel asked whether 0/1-polytopes have extended formulations with a polynomial number of inequalities in general or if there exist 0/1-polytopes that need a super-polynomial number of facets in any extension. Beware! This is not in contradiction or related to the P-vs.-NP question as we only talk about the number of inequalities and not the encoding length of the coefficients. This was settled by Rothvoss in 2011 by a very nice counting argument: there are 0/1-polytopes that need a super-polynomial number of inequalities in any extension.
To make the following slightly more formal, let be a polytope (of some dimension). Then an extended formulation for is another polytope such that there exists a linear projection with . The size of an extension is now the number of facets of and the extension complexity of (denoted by: ) is the minimum such that there exists an extension of size . We are interested in where is the travelling salesman polytope. Our proof heavily relies on a connection between the extension complexity of a polytope and communication complexity (the basic connection was made by Yannakakis and it was later extended by Faenza, Fiorini, Grappe, and Tiwary and Fiorini, Kaibel, Pashkovich, Theis in 2011). In fact, suppose that we have an inner and outer description of our polytope , say . Then we can define the slack matrix as , i.e., the slack of the vertices with respect to the inequalities. The extension complexity of a polytope is now equal to the nonnegative rank of which is essentially equivalent to determining the best protocol to compute the entries of in expectation (Alice gets a row index and Bob a column index). The latter can be bounded from below by the non-deterministic communication complexity and we use a certain matrix which has large non-deterministic communication complexity (see de Wolf 2003). This matrix is special as it constitutes the slack for some valid inequalities for the correlation polytope which eventually leads to a exponential lower bound for the extension complexity of the correlation polytope. The latter is affine isomorphic to the cut polytope. Then via a reduction-like mechanism similar lower bounds are established for the stable set polytope () for a certain family of graphs and then we use the fact that the TSP polytope contains any stable set polytope as a face (Yannakakis) for suitably chosen parameters and we obtain .
Here are the slides: