Abstract
We study a constrained version of the Geometric Hitting Set problem where we are given a set of points, partitioned into disjoint subsets, and a set of intervals. The objective is to hit all the intervals with a minimum number of points such that if we select a point from a subset then we must select all the points from that subset. In general, when the intervals are disjoint, we prove that the problem is in FPT, when parameterized by the size of the solution. We also complement this result by giving a lower bound in the size of the kernel for disjoint intervals, and we also provide a polynomial kernel when the size of all subsets is bounded by a constant.
Next, we consider two special cases of the problem where each subset can have at most 2 and 3 points. If each subset contains at most 2 points and the intervals are disjoint, we show that the problem admits a polynomial-time algorithm. However, when each subset contains at most 3 points and intervals are disjoint, we prove that the problem is \(\mathsf {NP\text {-}Hard}\) and we provide two constant factor approximations for the problem.
A. Acharyya and V. Keikha—The author is supported by the Czech Science Foundation, grant number GJ19-06792Y, and by institutional support RVO: 67985807.
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Notes
- 1.
The edge cover problem defined on a graph finds the set of edges of a minimum size such that every vertex of the graph is incident to at least one edge of the set.
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Acharyya, A., Keikha, V., Majumdar, D., Pandit, S. (2021). Constrained Hitting Set Problem with Intervals. In: Chen, CY., Hon, WK., Hung, LJ., Lee, CW. (eds) Computing and Combinatorics. COCOON 2021. Lecture Notes in Computer Science(), vol 13025. Springer, Cham. https://doi.org/10.1007/978-3-030-89543-3_50
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