
Packing Disks into Disks with Optimal WorstCase Density
We provide a tight result for a fundamental problem arising from packing...
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The biclique covering number of grids
We determine the exact value of the biclique covering number for all gri...
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Outreach Strategies for Vaccine Distribution: A TwoPeriod Robust Approach
Vaccination has been proven to be the most effective method to prevent i...
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Online Circle Packing
We consider the online problem of packing circles into a square containe...
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Online unit covering in L_2
Given a set of points P in L_2, the classic Unit Covering (UC) problem a...
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RandomOrder Models
This chapter introduces the randomorder model in online algorithms. In ...
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Covering of highdimensional cubes and quantization
As the main problem, we consider covering of a ddimensional cube by n b...
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WorstCase Optimal Covering of Rectangles by Disks
We provide the solution for a fundamental problem of geometric optimization by giving a complete characterization of worstcase optimal disk coverings of rectangles: For any λ≥ 1, the critical covering area A^*(λ) is the minimum value for which any set of disks with total area at least A^*(λ) can cover a rectangle of dimensions λ× 1. We show that there is a threshold value λ_2 = √(√(7)/2  1/4)≈ 1.035797..., such that for λ<λ_2 the critical covering area A^*(λ) is A^*(λ)=3π(λ^2/16 +5/32 + 9/256λ^2), and for λ≥λ_2, the critical area is A^*(λ)=π(λ^2+2)/4; these values are tight. For the special case λ=1, i.e., for covering a unit square, the critical covering area is 195π/256≈ 2.39301.... The proof uses a careful combination of manual and automatic analysis, demonstrating the power of the employed interval arithmetic technique.
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