Abstract
Set cover is one of the most studied optimization problems in Computer Science. In this paper, we target two interesting variations of this problem in a geometric setting: (i) maximum disjoint coverage (MDC), and (ii) maximum independent coverage (MIC) problems. In both problems, the input consists of a set P of points and a set O of geometric objects in the plane. The objective is to maximize the number of points covered by a set \(O'\) of selected objects from O. In the MDC problem we restrict the objects in \(O'\) are pairwise disjoint (non-intersecting). Whereas, in the MIC problem any pair of objects in \(O'\) should not share a point from P (however, they may intersect each other). We consider various geometric objects as covering objects such as axis-parallel infinite lines, axis-parallel line segments, unit disks, axis-parallel unit squares, and intervals on a real line. For the covering objects axis-parallel infinite lines, we show that both MDC and MIC problems admit polynomial time algorithms. In addition to that, we give polynomial time algorithms for both MDC and MIC problems with intervals on the real line. On the other hand, we prove that the MIC problem is \({\mathsf {NP}}\)-complete when the objects are horizontal infinite lines and vertical segments. We also prove that both MDC and MIC problems are \({\mathsf {NP}}\)-complete for axis-parallel unit segments in the plane. For unit disks and axis-parallel unit squares, we prove that both these problems are \({\mathsf {NP}}\)-complete. Further, we present \({\mathsf {PTAS}}\) es for the MDC problem for unit disks as well as unit squares using Hochbaum and Maass’s “shifting strategy”. For unit squares, we design a \({\mathsf {PTAS}}\) for the MIC problem using Chan and Hu’s “mod-one transformation” technique.
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Notes
Let G(V, E) be a bipartite graph. Finding a minimum weight vertex cover \(V^*\subset V\) in G can be solved by a minimum cut computation or a maximum flow computation in a related graph. Then the maximum weight independent set of G is \(V{\setminus } V^*\).
References
Adamaszek A, Wiese A (2013) Approximation schemes for maximum weight independent set of rectangles. In: 2013 IEEE 54th annual symposium on foundations of computer science, pp 400–409
Ahuja RK, Magnanti TL, Orlin JB (1993) Network flows: theory, algorithms, and applications. Prentice-Hall Inc., Upper Saddle River
Chan TM, Grant E (2014) Exact algorithms and APX-hardness results for geometric packing and covering problems. Comput Geom 47(2):112–124
Chan TM, Har-Peled S (2012) Approximation algorithms for maximum independent set of pseudo-disks. Discrete Comput Geom 48(2):373–392
Chan TM, Hu N (2015) Geometric red-blue set cover for unit squares and related problems. Comput Geom 48(5):380–385
Clark BN, Colbourn CJ, Johnson DS (1990) Unit disk graphs. Discrete Math 86(1):165–177
Dhar AK, Madireddy RR, Pandit S, Singh J (2019) Maximum independent and disjoint coverage. In: Theory and applications of models of computation (TAMC), pp. 134–153
Erlebach T, Jansen K, Seidel E (2005) Polynomial-time approximation schemes for geometric intersection graphs. SIAM J Comput 34(6):1302–1323
Erlebach T, van Leeuwen EJ (2008) Approximating geometric coverage problems. In: ACM-SIAM symposium on discrete algorithms (SODA), pp 1267–1276
Feige U (1998) A threshold of \(\ln n\) for approximating set cover. J ACM 45(4):634–652
Hochbaum DS, Maass W (1985) Approximation schemes for covering and packing problems in image processing and VLSI. J ACM 32(1):130–136
Ito T, ichi Nakano S, Okamoto Y, Otachi Y, Uehara R, Uno T, Uno Y (2014) A 4.31-approximation for the geometric unique coverage problem on unit disks. Theor Comput Sci 544:14–31
Ito T, ichi Nakano S, Okamoto Y, Otachi Y, Uehara R, Uno T, Uno Y (2016) A polynomial-time approximation scheme for the geometric unique coverage problem on unit squares. Comput Geom 51:25–39
Kleinberg J, Tardos E (2006) Algorithm design. Addison Wesley, Boston
Madireddy RR, Mudgal A, Pandit S (2018) Hardness results and approximation schemes for discrete packing and domination problems. In: Conference on combinatorial optimization and applications (COCOA), pp 216–226
Mehrabi S (2016) Geometric unique set cover on unit disks and unit squares. In: Canadian conference on computational geometry (CCCG), pp 195–200
Mulzer W, Rote G (2008) Minimum-weight triangulation is NP-hard. J ACM 55(2):11:1–11:29
Nandy SC, Pandit S, Roy S (2017) Covering points: minimizing the maximum depth. In: Canadian conference on computational geometry (CCCG), pp 37–42
Nieberg T, Hurink J, Kern W (2005) A robust PTAS for maximum weight independent sets in unit disk graphs. In: Graph-theoretic concepts in computer science, pp 214–221
Schaefer TJ (1978) The complexity of satisfiability problems. In: ACM symposium on theory of computing (STOC), pp 216–226
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A preliminary version of this paper appeared in the 15th Annual International Conference on Theory and Applications of Models of Computation (TAMC) 2019 (Dhar et al. 2019)
This work was done while S. Pandit was affiliated with the Stony Brook University, Stony Brook, NY, USA and was partially supported by the Indo-US Science and Technology Forum (IUSSTF) under the SERB Indo-US Postdoctoral Fellowship scheme with Grant Number 2017/94, Department of Science and Technology, Government of India.
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Dhar, A.K., Madireddy, R.R., Pandit, S. et al. Maximum independent and disjoint coverage. J Comb Optim 39, 1017–1037 (2020). https://doi.org/10.1007/s10878-020-00536-w
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DOI: https://doi.org/10.1007/s10878-020-00536-w