# Download e-book for iPad: Computational geometry for design and manufacture by I. D. Faux, Michael J. Pratt

By I. D. Faux, Michael J. Pratt

ISBN-10: 0470270691

ISBN-13: 9780470270691

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Claim. Every set in W0 − {W1 , W2 , W3 } is disjoint with W1 ∩ W2 ∩ W3 . Lemma 10. Let Wi , Wj be two minimal deﬁcient sets wrt. vi and vj , respectively, such that Wi ∩Wj = ∅, |NG (Wi ∪Wj )| ≤ 2, d(vi ) = 4, and {vi , vj }∩(Wi ∩Wj ) = ∅. Then for any feasible solution S to 4LVSLP, we have |S ∩ (Wi ∪ Wj )| ≥ 2. Proof. By Lemma 2, we have S ∩ (Wi ∪ Wj ) = ∅; let s ∈ S ∩ (Wi ∪ Wj ). If s = vi , then |NG (Wi ∪ Wj − {s})| ≤ |NG (Wi ∪ Wj )| + 1 ≤ 3 < d(vi ). Hence, in this case, again by Lemma 2, we have S ∩ (Wi ∪ Wj − {s}) = ∅ and |S ∩ (Wi ∪ Wj )| ≥ 2.

We can observe the following property on extreme subsets of a set function (the proof is omitted for space reasons). 22 H. Nagamochi Lemma 1. Let f be a set function on a ﬁnite set V , and X (f ) be the family of extreme subsets of f . (i) If f is intersecting posi-modular or symmetric and crossing submodular, then X (f ) is laminar. (ii) There is a crossing posi-modular set function f such that X (f ) is not laminar. (iii) There is an asymmetric and fully submodular set function f such that X (f ) is not laminar.

Corollary 2. Let f be a set function f on V with n = |V | ≥ 2. If f is symmetric and crossing submodular or intersecting submodular and posi-modular, then a ﬂat pair of f can be found in O(n2 Tf ) time. Proof. If f is symmetric and crossing submodular, then we compute an MD ordering π of f in O(n2 Tf ) time and choose the pair of the last two elements in π, which is ﬂat by Theorem 5. Consider the case where f is intersecting submodular and posi-modular, where we assume f (∅) = f (V ) = −∞ as it does not lose the intersecting submodularity and posi-modularity of f .