Constrained hitting set problem with intervals: Hardness, FPT and approximation algorithms

0585147 - ÚI 2025 RIV NL eng J - Journal Article
Acharyya, A. - Keikha, Vahideh - Majumdar, D. - Pandit, S.
Constrained hitting set problem with intervals: Hardness, FPT and approximation algorithms.
Theoretical Computer Science. Roč. 990, 1 April 2024 (2024), č. článku 114402. ISSN 0304-3975. E-ISSN 1879-2294
R&D Projects: GA ČR(CZ) GJ19-06792Y
Institutional support: RVO:67985807
Keywords : Constrained geometric hitting set * Computational complexity * Approximation algorithms * Parameterized complexity * Kernelization * Set cover conjecture
OECD category: Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Impact factor: 1.1, year: 2022
https://doi.org/10.1016/j.tcs.2024.114402

We study a constrained version of the Geometric Hitting Set problem where we are given a set of points, partitioned into pairwise 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, we must select all the points from that subset. 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. On the contrary, if each subset contains at most t points, where t >= 2, and the intervals are overlapping, we show that the problem becomes NP-Hard. Further, when each subset contains at most t points where t >= 3, and the intervals are disjoint, we prove that the problem is NP-Hard, and we provide two constant factor approximation algorithms for this problem. We also study the problem from the parameterized complexity perspective. If the intervals are disjoint, then 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.
Permanent Link: https://hdl.handle.net/11104/0352882