@article{Klasing_Laforest_Peters_Thibault_2006, title={Constructing Incremental Sequences in Graphs}, volume={1}, url={https://journals.lib.unb.ca/index.php/AOR/article/view/666}, abstractNote={<p>Given a weighted graph <img src="/static_content/AOR/vol1(2)1-7.gif" alt="" />, we investigate the problem of constructing a sequence of <img src="/static_content/AOR/vol1(2)1-6.gif" alt="" /> subsets of vertices <img src="/static_content/AOR/vol1(2)1-1.gif" alt="" /> (called groups) with small diameters, where the diameter of a group is calculated using distances in G. The constraint on these <em>n</em> groups is that they must be incremental: <img src="/static_content/AOR/vol1(2)1-2.gif" alt="" />. The cost of a sequence is the maximum ratio between the diameter of each group Mi and the diameter of a group <img src="/static_content/AOR/vol1(2)1-4.gif" alt="" /> with <em>I</em> vertices and minimum diameter:</p> <p><img src="/static_content/AOR/vol1(2)1-3.gif" alt="" />.</p> <p>This quantity captures the impact of the incremental constraint on the diameters of the groups in a sequence. We give general bounds on the value of this ratio and we prove that the problem of constructing an optimal incremental sequence cannot be solved approximately in polynomial time with an approximation ratio less than 2 unless P = NP. Finally, we give a 4-approximation algorithm and we show that the analysis of our algorithm is tight.</p>}, number={2}, journal={Algorithmic Operations Research}, author={Klasing, Ralf and Laforest, Christian and Peters, Joseph and Thibault, Nicolas}, year={2006}, month={Jun.} }