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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References
Alon, N.: Piercing d-intervals. Discrete Comput. Geom. 19, 333–334 (1998)
Alon, N., Kleitman, D.J.: Piercing convex sets and the Hadwiger-Debrunner (p,q)-problem. Adv. Math. 96, 103–112 (1992)
Bereg, S., Dumitrescu, A., Jiang, M.: Maximum area independent sets in disk intersection graphs. Int. J. Comput. Geom. Appl. 20, 105–118 (2010). doi: 10.1142/S0218195910003220
Bereg, S., Dumitrescu, A., Jiang, M.: On covering problems of Rado. Algorithmica 57, 538–561 (2009). doi:10.1007/s00453-009-9298-z
Braß, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer, New York (2005)
Carmi, P., Katz, M.J., Lev-Tov, N.: Polynomial-time approximation schemes for piercing and covering with applications in wireless networks. Comput. Geom., Theory Appl. 39, 209–218 (2008)
Chakerian, G.D., Stein, S.K.: Some intersection properties of convex bodies. Proc. Am. Math. Soc. 18, 109–112 (1967)
Chan, T.: Polynomial-time approximation schemes for packing and piercing fat objects. J. Algorithms 46, 178–189 (2003)
Danzer, L.: Zur Lösung des Gallaischen Problems über Kreisscheiben in der Euklidischen Ebene. Stud. Sci. Math. Hung. 21, 111–134 (1986)
Danzer, L., Grünbaum, B., Klee, V.: Helly’s theorem and its relatives. In: Proceedings of Symposia in Pure Mathematics, vol. 7, pp. 101–181. American Mathematical Society, Providence (1963)
Fon-Der-Flaass, D.G., Kostochka, A.V.: Covering boxes by points. Discrete Math. 120, 269–275 (1993)
Fowler, R.J., Paterson, M.S., Tanimoto, S.L.: Optimal packing and covering in the plane are NP-complete. Inf. Process. Lett. 12, 133–137 (1981)
Grünbaum, B.: On intersections of similar sets. Port. Math. 18, 155–164 (1959)
Gyárfás, A., Lehel, J.: Covering and coloring problems for relatives of intervals. Discrete Math. 55, 167–180 (1985)
Kabatjanskiĭ, G.A., Levenšteĭn, V.I.: Bounds for packings on a sphere and in space. Probl. Pered. Inf. 14, 3–25 (1978) (in Russian). English translation: Probl. Inf. Transmiss. 14, 1–17 (1978)
Kaiser, T.: Transversals of d-intervals. Discrete Comput. Geom. 18, 195–203 (1997)
Karasev, R.N.: Transversals for families of translates of a two-dimensional convex compact set. Discrete Comput. Geom. 24, 345–353 (2000)
Károlyi, G.: On point covers of parallel rectangles. Period. Math. Hung. 23, 105–107 (1991)
Károlyi, G., Tardos, G.: On point covers of multiple intervals and axis-parallel rectangles. Combinatorica 16, 213–222 (1996)
Kim, S.-J., Nakprasit, K., Pelsmajer, M.J., Skokan, J.: Transversal numbers of translates of a convex body. Discrete Math. 306, 2166–2173 (2006)
Kuperberg, W.: An inequality linking packing and covering densities of plane convex bodies. Geom. Dedic. 23, 59–66 (1987)
Kynčl, J., Tancer, M.: The maximum piercing number for some classes of convex sets with (4,3)-property. Electron. J. Comb. 15, #R27 (2008)
Matoušek, J.: Lower bounds on the transversal numbers of d-intervals. Discrete Comput. Geom. 26, 283–287 (2001)
Naszódi, M., Taschuk, S.: On the transversal number and VC-dimension of families of positive homothets of a convex body. Discrete Math. 310, 77–82 (2010)
Nielsen, F.: On point covers of c-oriented polygons. Theoret. Comput. Sci. 263, 17–29 (2001)
Pach, J., Agarwal, P.K.: Combinatorial Geometry. Wiley, New York (1995)
Rado, R.: Some covering theorems (I). Proc. Lond. Math. Soc. 51, 232–264 (1949)
Rogers, C.A.: A note on coverings. Mathematika 4, 1–6 (1957)
Schmidt, W.M.: On the Minkowski-Hlawka theorem. Ill. J. Math. 7, 18–23 (1963)
Smith, E.: An improvement of an inequality linking packing and covering densities in 3-space. Geom. Dedic. 117, 11–18 (2006)
Tardos, G.: Transversals of 2-intervals, a topological approach. Combinatorica 15, 123–134 (1995)
Wenger, R.: Helly-type theorems and geometric transversals. In: Handbook of Discrete and Computational Geometry, 2nd edn., pp. 73–96. CRC Press, Boca Raton (2004)
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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