Berman, Piotr; Karpinski, Marek; Lingas, Andrzej Exact and approximation algorithms for geometric and capacitated set cover problems. (English) ✄ ✂  ✁ Zbl 1252.68347 Algorithmica 64, No. 2, 295-310 (2012). Summary: First, we study geometric variants of the standard set cover motivated by assignment of direc- tional antenna and shipping with deadlines, providing the first known polynomial-time exact solutions. Next, we consider the following general (non-necessarily geometric) capacitated set cover problem. There is given a set of elements with real weights and a family of sets of the elements. One can use a set if it is a subset of one of the sets in the family and the sum of the weights of its elements is at most one. The goal is to cover all the elements with the allowed sets. We show that any polynomial-time algorithm that approximates the uncapacitated version of the set cover problem with ratio r can be converted to an approximation algorithm for the capacitated version with ratio r +1.357. In particular, the composition of these two results yields a polynomial-time approximation algorithm for the problem of covering a set of customers represented by a weighted n-point set with a minimum number of antennas of variable angular range and fixed capacity with ratio 2.357. This substantially improves on the best known approximation ratio for the latter antenna problem equal to 3. Furthermore, we provide a PTAS for the dual problem where the number of sets (e.g., antennas) to use is fixed and the task is to minimize the maximum set load, in case the sets correspond to line intervals or arcs. Finally, we discuss the approximability of the generalization of the antenna problem to include several base stations for antennas, and in particular show its APX-hardness already in the uncapacitated case. MSC: 68W25 Approximation algorithms 68Q17 Computational difficulty of problems (lower bounds, completeness, dif- ficulty of approximation, etc.) 68U05 Computer graphics; computational geometry (digital and algorithmic aspects) 05B40 Combinatorial aspects of packing and covering 90C27 Combinatorial optimization Cited in 1 Document Keywords: set cover; geometric set cover; capacitated set cover; assignment of directional antenna; shipping with deadlines; approximation algorithms; polynomial-time approximation scheme; APX-hardness; exact al- gorithms; time complexity Full Text: DOI References: [1] Aronov, B., Ezra, E., Sharir, M.: Small-size {\(\epsilon\)}-nets for axis-parallel rectangles and boxes. SIAM J. Comput. 39(7), 3248–3282 (2010) · Zbl 1209.68624 · doi:10.1137/090762968 [2] Bao, L., Garcia-Luna-Aceves, J.: Transmission scheduling in ad hoc networks with directional antennas. In: Proc. ACM MOBICOM, pp. 48–58 (2002) [3] Berman, P., Kasiviswanathan, S.P., Urgaonkar, B.: Packing to angles and sectors. In: Proc. Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), pp. 171–180 (2007) [4] Berman, P., Karpinski, M.: Improved approximation lower bounds on small occurrence optimization. Electron. Colloq. Com- putat. Complex. 10(008) (2003) [5] Broden, B., Hammar, M., Nilsson, B.J.: Guarding lines and 2-link polygons is APX-hard. In: Proc. Canadian Conference on Computational Geometry (CCCG), pp. 45–48 (2001) Edited by FIZ Karlsruhe, the European Mathematical Society and the Heidelberg Academy of Sciences and Humanities © 2022 FIZ Karlsruhe GmbH Page 1