Minimum Dominating Sets of Intervals on Lines

We study the problem of computing minimum dominating sets of n intervals on lines in three cases: (1) the lines intersect at a single point, (2) all lines except one are parallel, and (3) one line with t weighted points on it and the minimum dominating set must maximize the sum of the weights of the points covered. We propose polynomial-time algorithms for the first two problems, which are special cases of the minimum dominating set problem for path graphs which is known to be NP-hard. The third problem requires identifying the structure of minimum dominating sets of intervals on a line so as to be able to select one that maximizes the weight sum of the weighted points covered. Assuming that presorting has been performed, the first problem has an O(n) -time solution, while the second and the third problems are solved by dynamic programming algorithms, requiring O(n log n) and O(n + t) time, respectively.