Abstract
According to a classical result of Grünbaum, the transversal number τ(ℱ) of any family ℱ of pairwise-intersecting translates or homothets of a convex body C in ℝd is bounded by a function of d. Denote by α(C) (resp. β(C)) the supremum of the ratio of the transversal number τ(ℱ) to the packing number ν(ℱ) over all finite families ℱ of translates (resp. homothets) of a convex body C in ℝd. Kim et al. recently showed that α(C) is bounded by a function of d for any convex body C in ℝd, and gave the first bounds on α(C) for convex bodies C in ℝd and on β(C) for convex bodies C in the plane.
Here we show that β(C) is also bounded by a function of d for any convex body C in ℝd, and present new or improved bounds on both α(C) and β(C) for various convex bodies C in ℝd for all dimensions d. Our techniques explore interesting inequalities linking the covering and packing densities of a convex body. Our methods for obtaining upper bounds are constructive and lead to efficient constant-factor approximation algorithms for finding a minimum-cardinality point set that pierces a set of translates or homothets of a convex body.
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A preliminary version of this paper appeared in the Proceedings of the 17th Annual European Symposium on Algorithms (ESA 2009), LNCS, vol. 5757, pp. 131–142.
A. Dumitrescu was supported in part by NSF CAREER grant CCF-0444188.
M. Jiang was supported in part by NSF grant DBI-0743670.
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Dumitrescu, A., Jiang, M. Piercing Translates and Homothets of a Convex Body. Algorithmica 61, 94–115 (2011). https://doi.org/10.1007/s00453-010-9410-4
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DOI: https://doi.org/10.1007/s00453-010-9410-4