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Abstract
The maximum weighted independent set (MWIS) problem is important since it occurs in various applications, such as facility location, selection of non-overlapping time slots, labeling of digital maps, etc. However, in real-life situations, input parameters within those models are often loosely defined or subject to change. For such reasons, this paper studies robust variants of the MWIS problem. The study is restricted to cases where the involved graph is a tree. Uncertainty of vertex weights is represented by intervals. First, it is observed that the max–min variant of the problem can be solved in linear time. Next, as the most important original contribution, it is proved that the min–max regret variant is NP-hard. Finally, two mutually related approximation algorithms for the min–max regret variant are proposed. The first of them is already known, but adjusted to the considered situation, while the second one is completely new. Both algorithms are analyzed and evaluated experimentally.
Highlights
Symmetry 2021, 13, 2259. https://Our introduction starts with some well-known definitions, which can be found in many textbooks, e.g., [1,2]
In order to find the solution of our maximum weighted independent set (MWIS) problem instance, one can restrict to nontrivial independent sets
We have studied two robust variants of the MWIS problem, i.e., the max
Summary
Our introduction starts with some well-known definitions, which can be found in many textbooks, e.g., [1,2]. The second criterion is called min–max regret [3] or robust deviation [7]—it selects the independent set whose maximal deviation of weight from the conventional optimum, measured over all scenarios, is as small as possible. We have been inspired by [8,12], where an analogous complexity analysis of robust MWIS problem variants has been done for so-called interval graphs rather than for trees. This work is in a close relationship with our previous paper [11], in which the same robust MWIS problem variants on trees have been considered under discrete uncertainty representation.
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