# PHD defense: A guide book for the traveller on graphs full of blockages

Time: 14:00 -- Location: CentraleSupelec, amphi IV

summary: We study NP-hard problems dealing with graphs containing blockages.

We analyze cut problems via the parameterized complexity framework. The size p of the cut is the parameter. Given a set of sources {s_1,...,s_k} and a target t, we propose an algorithm deciding whether a cut of size at most p separates at least r sources from t. This problem is called Partial One-Target Cut. Our algorithm is FPT. We also prove that the vertex version of the problem, where the cut contains vertices instead of edges, is W[1]-hard. Our second contribution is an algorithm counting minimum (S,T)-cuts in time 2^{O(p \log p)}n^{O(1)}.

Then, we present numerous results on the competitive ratio of deterministic and randomized strategies for the Canadian Traveller Problem. We show that randomized strategies which do not use memory cannot improve the ratio 2k+1. We also provide theorems on the competitive ratio of randomized strategies in general. We study the distance competitive ratio of a group of travellers with and without communication. Eventually, we treat the competitiveness of deterministic strategies dedicated to certain families of graphs. Two strategies with a ratio below 2k+1 are proposed: one for equal-weight chordal graphs and another one for graphs such that the largest minimal (s,t)-cut is at most k.