Constructing Incremental Sequences in Graphs
Ralf Klasing, Christian Laforest, Joseph G. Peters, Nicolas Rudolph Thibault, Ralf Klasing, Christian Laforest, Joseph G. Peters, Nicolas Thibault, Constructing Incremental, Ralf Klasing, Christian Laforest, Joseph Peters, Nicolas Thibault · HAL (Le Centre pour la Communication Scientifique Directe) · 2006
Given a weighted graph $G=(V,E,w)$, we investigate the problem of constructing a sequence of $n=|V|$ subsets of vertices $M_1,...,M_n$ (called groups) with small diameters, where the diameter of a group is calculated using distances in $G$. The constraint on these $n$ groups is that they must be incremental: $M_1\subsetM_2 \subset...\subsetM_n=V$. The cost of a sequence is the maximum ratio between the diameter of each group $M_i$ and the diameter of a group $N_i^*$ with $i$ vertices and minimum diameter: $\max_2 \leqi \leqn \left{ \fracD(M_i)D(N_i^*) \right}$. 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.