The Competitiveness of Randomized Strategies for Canadians via Systems of Linear Inequalities
The Canadian Traveller Problem is a PSPACE-complete optimization problem where a traveller traverses an undirected weighted graph G from source s to target t where some edges \(E_*\) are blocked. At the beginning, the traveller does not know which edges are blocked. He discovers them when arriving at one of their endpoints. The objective is to minimize the distance traversed to reach t.
Westphal proved that no randomized strategy has a competitive ratio smaller than \(\left| E_* \right| +1\). We show, using linear algebra techniques, that this bound cannot be attained, especially on a specific class of graphs: apex trees. Indeed, no randomized strategy can be \(\left( \left| E_* \right| +1\right) \)-competitive, even on apex trees with only three simple \((s,t)\)-paths.
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