8.997 Topics in Combinatorial Optimization 1.1 Example: Circulation
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چکیده
1 Special cases of submodular flows We saw last time that orientation of a 2k-edge connected graph into a k-arc connected digraph and the Lucchesi and Younger Theorem were special cases of submodular flows. Other familiar problems can also be phrased as submodular flows. Let C = 2 V \ {∅, V } and let f be identically zero. Then for any U ∈ C, x(δ in (U)) − x(δ out (U) ≤ 0 (1) x(δ in (V \ U)) − x(δ out (V \ U) ≤ 0 which implies that x(δ in (U)) = x(δ out (U) for all U ⊂ V. In particular, we have that x(δ in (v)) = x(δ out (v)) for all v ∈ V. In this case, the submodular flow reduces to the circulation problem from network flows. If A and B are elements of C with nonempty intersection and A ∪ B = V , then and hence for all cases, A ∪ B and A ∩ B are in C proving that C is a crossing family. f is readily seen to be crossing submodular by checking cases. If A ⊆ S 1 , B ⊆ S 1 then the submodular inequality for f follows from submodularity of r 1. If S 1 ⊆ A, S 1 ⊆ B then by deMorgan's laws V \ (A ∪ B) = (V \ A) ∩ (V \ B) (4) V \ (A ∩ B) = (V \ A) ∪ (V \ B). Therefore, submodularity of f follows from the submodularity of r 2. Finally, if S 1 ⊆ A, B ⊆ S 1 then f (A∩B) = r 1 (A∩B) and f (A∪B) = r 2 (V \(A∪B)). Since |B| ≥ |A∩B| and |V \A| ≥ |V \(A∪B)|, the submodular inequality holds here as well.
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8.997 Topics in Combinatorial Optimization
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تاریخ انتشار 2004