Abstract
Given a line segment arrangement, defined as the geometric structure induced by a set of n line segments in a plane, we study the complexity of two covering problems—Cell Cover for Segments and Guarding a Set of Segments. In Cell Cover for Segments (CCS) problem, the input is a line segment arrangement and our aim is to cover all the segments with minimum number of cells. We prove that the decision version of CCS is NP-complete. Guarding a Set of Segments (GSS) Problem asks to cover all the line segments with minimum number of points in the arrangement. The decision version of GSS problem is known to be NP-complete [4]. Given k as the solution size, we prove that the GSS problem admits a kernel of \(\mathcal {O}(k^{2})\) line segments and provide an \(\mathcal {O^*}(k^{2k})\) FPT algorithm for solving GSS. We model the arrangement as its underlying planar graph which allows us to define a structural parameter called face density (d) of the arrangement, and propose an \(\mathcal {O^*}(d^{k})\) FPT algorithm for the GSS problem.
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Rema, M., Subashini, R., Methirumangalath, S., Rajan, V. (2023). Classical and Parameterized Complexity of Line Segment Covering Problems in Arrangement. In: Bhateja, V., Carroll, F., Tavares, J.M.R.S., Sengar, S.S., Peer, P. (eds) Intelligent Data Engineering and Analytics. FICTA 2023. Smart Innovation, Systems and Technologies, vol 371. Springer, Singapore. https://doi.org/10.1007/978-981-99-6706-3_29
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