Download e-book for kindle: Algorithms and Computation: 18th International Symposium, by Pankaj K. Agarwal (auth.), Takeshi Tokuyama (eds.)

By Pankaj K. Agarwal (auth.), Takeshi Tokuyama (eds.)

ISBN-10: 3540771182

ISBN-13: 9783540771180

This ebook constitutes the refereed complaints of the 18th overseas Symposium on Algorithms and Computation, ISAAC 2007, held in Sendai, Japan, in December 2007.

The seventy seven revised complete papers awarded including 2 invited talks have been conscientiously reviewed and chosen from 220 submissions. The papers are geared up in topical sections on graph algorithms, computational geometry, complexity, graph drawing, dispensed algorithms, optimization, information constitution, online game thought, database purposes, on-line algorithms, I/O algorithms, networks, geometric functions, and string.

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Additional info for Algorithms and Computation: 18th International Symposium, ISAAC 2007, Sendai, Japan, December 17-19, 2007. Proceedings

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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 .

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Algorithms and Computation: 18th International Symposium, ISAAC 2007, Sendai, Japan, December 17-19, 2007. Proceedings by Pankaj K. Agarwal (auth.), Takeshi Tokuyama (eds.)


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