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Journal Article

Primal Separation for 0/1 Polytopes


Eisenbrand,  Friedrich
Discrete Optimization, MPI for Informatics, Max Planck Society;


Ventura,  Paolo
Machine Learning, MPI for Informatics, Max Planck Society;

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Eisenbrand, F., Rinaldi, G., & Ventura, P. (2003). Primal Separation for 0/1 Polytopes. Mathematical Programming / A, 95, 475-491.

Cite as: http://hdl.handle.net/11858/00-001M-0000-000F-2DCB-F
\noindent The 0/1~primal separation problem is: Given an extreme point $\bar{x}$ of a 0/1~polytope $P$ and some point $x^*$, find an inequality which is tight at $\bar{x}$, violated by $x^*$ and valid for $P$ or assert that no such inequality exists. It is known that this separation variant can be reduced to the standard separation problem for $P$. \noindent We show that 0/1~optimization and 0/1~primal separation are polynomial time equivalent. This implies that the problems 0/1~optimization, 0/1~standard separation, 0/1~augmentation, and 0/1~primal separation are polynomial time equivalent. \noindent Then we provide polynomial time primal separation procedures for matching, stable set, maximum cut, and maximum bipartite graph problems, giving evidence that these algorithms are conceptually simpler and easier to implement than their corresponding counterparts for standard separation. In particular, for perfect matching we present an algorithm for primal separation that rests only on simple max-flow computations. In contrast, the known standard separation method relies on an explicit minimum odd cut algorithm. Consequently, we obtain a very simple proof that a maximum weight perfect matching of a graph can be computed in polynomial time.