Talk:Matching polytope

The proof https://en.wikipedia.org/wiki/Matching_polytope#Proof_using_the_definition_of_extreme_points does not appear to be correct. Consider a path of 4 vertices with $$x_e$$ given by $$\frac12, \frac12, 0$$ in that order along the path. Clearly this is a feasible solution but it is not possible to subtract an $$\epsilon > 0$$ from the very last edge without violating $$\mathbf 0 \le \mathbf x$$. 81.221.145.119 (talk) 17:11, 21 November 2020 (UTC) Henrik Laxhuber


 * You are right, the proof is incomplete. I removed it until I (or someone else) finds a way to fix it. --Erel Segal (talk) 21:36, 21 November 2020 (UTC)


 * A fix could be to only modify those vertices reachable from $$x_e$$ without going through an edge of integral weight. It should be possible to show that none of these edges are adjacent to an edge of weight ($$x_u$$) 1 due to $$\mathbf{A_G x \le 1}$$. The idea carries over also to the bipartite vertex cover polytope, where it should be possible to show that none of the fractional vertices are adjacent to a vertex of zero weight. --81.221.145.119 (talk) 08:20, 22 November 2020 (UTC) Henrik Laxhuber