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Algorithms and Data Structures: 9th International Workshop, by Allan Borodin (auth.), Frank Dehne, Alejandro López-Ortiz,

By Allan Borodin (auth.), Frank Dehne, Alejandro López-Ortiz, Jörg-Rüdiger Sack (eds.)

This booklet constitutes the refereed lawsuits of the ninth foreign Workshop on Algorithms and knowledge constructions, WADS 2005, held in Waterloo, Canada, in August 2005.

The 37 revised complete papers provided have been conscientiously reviewed and chosen from ninety submissions. A wide number of subject matters in algorithmics and knowledge constructions is addressed together with looking out and sorting, approximation, graph and community computations, computational geometry, randomization, communications, combinatorial optimization, scheduling, routing, navigation, coding, and trend matching.

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Extra resources for Algorithms and Data Structures: 9th International Workshop, WADS 2005, Waterloo, Canada, August 15-17, 2005. Proceedings

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Let v ∈ S. We claim that the cost of the facility we locate at v is at most twice the cost of the optimal fractional facilities located at v. There are two cases: 1. N (v) ⊆ S. In this case the cost of the cheap facility we locate at v is αv ≤ 2αv (zv∗ + yv∗ ) ≤ 2(αv zv∗ + βv yv∗ ) . 2. N (v) S, that is, there is a vertex u ∈ N (v) such that u ∈ / S. Since (x∗ , y ∗ , z ∗ ) is feasible for (LP), 1 ≤ yu∗ + yv∗ + x∗u,v = yu∗ + yv∗ + min{zu∗ , zv∗ } ≤ yu∗ + yv∗ + zu∗ , and yv∗ ≥ 1 − (yu∗ + zu∗ ) > 12 .

We also provide a problem kernelization. Altogether, we thus show that Capacitated Vertex Cover—including two variants with “hard” and “soft” capacities—is fixed-parameter tractable. 3. For Maximum Partial Vertex Cover, one only wants to cover a specified number of edges (that is, not necessarily all) by at most k vertices. This 38 J. Guo, R. Niedermeier, and S. Wernicke Table 1. New parameterized complexity results for some NP-complete generalizations of Vertex Cover shown in this work. 2 + n2 W[1]-hard W[1]-hard Thm.

Then delete from G a vertex vi ∈ {v1 , v2 , . . , vk+1 } which has minimum capacity. This rule is correct because any size-k capacitated vertex cover C containing vi can be modified by replacing vi with a vertex from {v1 , v2 , . . , vk+1 } which is not in C. ˜ can be computed from G as claimed by Based on this data reduction rule, G the following two steps: 1. Use the straightforward linear-time factor-2 approximation algorithm to find a vertex cover S for G of size at most 2k (where k is the size of a minimum vertex cover for G and hence k ≤ k).

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