It is simple to find a major e that maintains the domain constraints: it truly is both the smallest difference between an odd variable and 0, or maybe the smallest difference between a fair variable and 1. Now, We've 1 much less fractional variable, so k(x) decreases by one. The objective price stays a similar, since in any other case we could boost it by choosing e to be constructive or negative, in contradiction to the assumption that it's maximal.
To guarantee which the variables indeed depict an ideal matching, we add constraints stating that each vertex is adjacent to accurately 1 edge while in the matching, i.
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