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.
|Keywords||Combinatorial optimization, Communication complexity, Linear programming, Quantum communication complexity, Semidefinite programming|
|Journal||Journal of the ACM|
Fiorini, S. (Samuel), Massar, S. (Serge), Pokutta, S. (Sebastian), Tiwary, H.R. (Hans Raj), & De Wolf, R. (Ronald). (2015). Exponential lower bounds for polytopes in combinatorial optimization. Journal of the ACM, 62(2). doi:10.1145/2716307