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Abstract:
\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.
