Robust maximum weighted independent-set problems on interval graphs.

We study the maximum weighted independent-set problem on interval graphs with uncertainty on the vertex weights. We use the absolute robustness criterion and the min-max regret criterion to evaluate solutions. For a discrete scenario set, we find that the problem is NP-hard for each of the robustness criteria; we also provide pseudo-polynomial time algorithms when there is a constant number of scenarios and show that the problem is strongly NP-hard when the set of scenarios is unbounded. When the scenario set is a Cartesian product, we prove that the problem is equivalent to a maximum weighted independent-set problem on the same interval graph but without uncertainty for the first objective function and that the scenario set can be reduced for the second objective function.Combinatorial problems; Computational complexity; Interval graphs; Independent set;

On graph-like continua of finite length

We extend the notion of effective resistance to metric spaces that are similar to graphs but can also be similar to fractals. Combined with other basic facts proved in the paper, this lays the ground for a construction of Brownian Motion on such spaces completed in [10]