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Reports (Research Report) Year : 2006

Constructing Incremental Sequences in Graphs


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.


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Dates and versions

inria-00070361 , version 1 (19-05-2006)


  • HAL Id : inria-00070361 , version 1


Ralf Klasing, Christian Laforest, Joseph Peters, Nicolas Thibault. Constructing Incremental Sequences in Graphs. [Research Report] RR-5648, INRIA. 2006, pp.12. ⟨inria-00070361⟩
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