Complexity of linear relaxations in integer programming

Semidefinite Relaxations for Integer Programming

We survey some recent developments in the area of semidefinite optimization applied to integer programming. After recalling some generic modeling techniques to obtain semidefinite relaxations for NP-hard problems, we look at the theoretical power of semidefinite optimization in the context of the Max-Cut and the Coloring Problem. In the second part, we consider algorithmic questions related to ...

Stronger Linear Programming Relaxations

Linear programming relaxations of maxcut

It is well-known that the integrality gap of the usual linear programming relaxation for Maxcut is 2 − ǫ. For general graphs, we prove that for any ǫ and any fixed boundk, adding linear constraints of support bounded by k does not reduce the gap below 2−ǫ. We generalize this to prove that for any ǫ and any fixed bound k, strengthening the usual linear programming relaxation by doing k rounds of...

Complexity scaling of mixed-integer linear programming decoding

In this talk we discuss a hybrid belief-propagation (BP) / mixed-integer linear programming (MI-LP) decoder. The failure of a first-stage BP decoding attempt triggers a second-stage MI-LP decoder. The MI-LP decoder was presented at ISIT 2007 (Draper, Yedidia, and Wang) where it was shown to achieve the optimum maximum-likelihood (ML) decoding performance on a (155, 64) LDPC code introduced by T...

Linear Programming and Integer Linear Programming

5 Algorithms 8 5.1 Fourier-Motzkin Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5.2 The Simplex Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5.3 Seidel’s Randomized Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5.4 The Ellipsoid Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5.5 Using the Ellipsoid...