Abstract
A graph \(G = (V, E)\) is called a rectangle overlap graph if there is a bijection between V and a set \(\mathcal {R}\) of axis-parallel rectangles such that two vertices in V are adjacent if and only if the corresponding rectangles in \(\mathcal {R}\) overlap i.e. their boundaries intersect.
In this article, assuming the Unique Games Conjecture to be true we show that it is not possible to approximate the Minimum Dominating Set (MDS) problem on rectangle overlap graphs with a factor \((2-\epsilon )\) for any \(\epsilon >0\). Previously only APX hardness was known for this problem due to Erlebach and Van Leeuwen (LATIN 2008) and Damian and Pemmaraju (Inf. Process. Lett. 2006). We give an \(O(n^5)\)-time 768-approximation algorithm for the MDS problem on stabbed rectangle overlap graphs i.e. overlap graphs of rectangles intersecting a common straight line. Here n denotes the number of vertices of the input graph.
Our second result is the first constant factor approximation for MDS problem on stabbed rectangle overlap graphs which is a strict generalisation of a graphclass considered by Bandyapadhyay et al. (MFCS 2018).
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Notes
- 1.
Two rectangles R and \(R'\) are non-piercing if both \(R\setminus R'\) and \(R'\setminus R\) are connected.
References
Bandyapadhyay, S., Maheshwari, A., Mehrabi, S., Suri, S.: Approximating dominating set on intersection graphs of rectangles and L-frames. In: MFCS, pp. 37:1–37:15 (2018)
de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-77974-2
Bousquet, N., Gonçalves, D., Mertzios, G.B., Paul, C., Sau, I., Thomassé, S.: Parameterized domination in circle graphs. Theory Comput. Syst. 54(1), 45–72 (2014)
Catanzaro, D., et al.: Max point-tolerance graphs. Discret. Appl. Math. 216, 84–97 (2017)
Chepoi, V., Felsner, S.: Approximating hitting sets of axis-parallel rectangles intersecting a monotone curve. Comput. Geom. 46(9), 1036–1041 (2013)
Chlebík, M., Chlebíková, J.: Approximation hardness of dominating set problems in bounded degree graphs. Inf. Comput. 206(11), 1264–1275 (2008)
Correa, J., Feuilloley, L., Pérez-Lantero, P., Soto, J.A.: Independent and hitting sets of rectangles intersecting a diagonal line: algorithms and complexity. Discrete Comput. Geom. 53(2), 344–365 (2015)
Damian, M., Pemmaraju, S.V.: APX-hardness of domination problems in circle graphs. Inf. Process. Lett. 97(6), 231–237 (2006)
Damian-Iordache, M., Pemmaraju, S.V.: A (2+ \(\varepsilon \))-approximation scheme for minimum domination on circle graphs. J. Algorithms 42(2), 255–276 (2002)
Erlebach, T., van Leeuwen, E.J.: Domination in geometric intersection graphs. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 747–758. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78773-0_64
Govindarajan, S., Raman, R., Ray, S., Basu Roy, A.: Packing and covering with non-piercing regions, In: ESA (2016)
Katz, M.J., Mitchell, J.S.B., Nir, Y.: Orthogonal segment stabbing. Comput. Geom.: Theory Appl. 30(2), 197–205 (2005)
Keil, J.M., Mitchell, J.S.B., Pradhan, D., Vatshelle, M.: An algorithm for the maximum weight independent set problem on outerstring graphs. Comput. Geom. 60, 19–25 (2017)
Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2-\(\varepsilon \). J. Comput. Syst. Sci. 74(3), 335–349 (2008)
Mudgal, A., Pandit, S.: Covering, hitting, piercing and packing rectangles intersecting an inclined line. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, D.-Z. (eds.) COCOA 2015. LNCS, vol. 9486, pp. 126–137. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-26626-8_10
Pandit, S.: Dominating set of rectangles intersecting a straight line. In: CCCG, pp. 144–149 (2017)
Tardos, E.: A strongly polynomial algorithm to solve combinatorial linear programs. Oper. Res. 34(2), 250–256 (1986)
Acknowledgement
This research was partially funded by the IFCAM project “Applications of graph homomorphisms” (MA/IFCAM/18/39).
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Chakraborty, D., Das, S., Mukherjee, J. (2019). Dominating Set on Overlap Graphs of Rectangles Intersecting a Line. In: Du, DZ., Duan, Z., Tian, C. (eds) Computing and Combinatorics. COCOON 2019. Lecture Notes in Computer Science(), vol 11653. Springer, Cham. https://doi.org/10.1007/978-3-030-26176-4_6
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