# The Canadian Tour Operator Problem on paths: tight bounds and resource augmentation

- 150 Downloads

## Abstract

In the prize-collecting travelling salesman problem, we are given a weighted graph \(G=(V,E)\) with edge weights \(\ell :E\rightarrow \mathbb {R}_+\), a special vertex \(r\in V\), penalties \(\pi :V\rightarrow \mathbb {R}_+\) and the goal is to find a closed tour \(T\) such that \(r\in V(T)\) and such that the cost \(\ell (T)+\pi (V\setminus V(T))\), which is the sum of the edges in the tour and the cost of the vertices not spanned by \(T\), is minimized. We consider an online variant of the prize-collecting travelling salesman problem related to graph exploration. In the *Canadian Tour Operator Problem* the task is to find a closed route for a tourist bus in a given network \(G=(V,E)\) in which some edges are blocked by avalanches. An online algorithm learns from a blocked edge only when reaching one of its endpoints. The bus operator has the option to avoid visiting each node \(v\in V\) by paying a refund of \(\pi (v)\) to the tourists. The goal consists of minimizing the sum of the travel costs and the refunds. We study the problem on a simple (weighted) path and prove tight bounds on the competitiveness of deterministic algorithms. Specifically, we give an algorithm with competitive ratio equal to the golden ratio \(\phi =(1+\sqrt{5})/2\). We also study the effect of resource augmentation, where the online algorithm either pays a discounted cost for traversing edges or for the penalties.

## Keywords

Online computation Competitive analysis Resource augmentation Graph exploration## References

- Ausiello G, Bonifaci V, Laura L (2008) The online prize-collecting traveling salesman problem. Inform Process Lett 107(6):199–204MathSciNetCrossRefMATHGoogle Scholar
- Balas E (1989) The prize collecting traveling salesman problem. Networks 19(6):621–636MathSciNetCrossRefMATHGoogle Scholar
- Bender M, Westphal S (2013) An optimal randomized online algorithm for the k-Canadian Traveller Problem on node-disjoint paths. J Comb Optim. doi: 10.1007/s10878-013-9634-8
- Borodin A, El-Yaniv R (1998) Online computation and competitive analysis, vol 53. Cambridge University Press, New YorkMATHGoogle Scholar
- Fiat A, Woeginger GJ (eds) (1998) Online algorithms: the state of the art. Lecture Notes in Computer Science, vol 1442. Springer, BerlinGoogle Scholar
- Jaillet P, Wagner MR (2008) Generalized online routing: new competitive ratios, resource augmentation, and asymptotic analyses. Oper Res 56(3):745–757MathSciNetCrossRefMATHGoogle Scholar
- Kalyanasundaram B, Pruhs K (1995) Speed is as powerful as clairvoyance. In: Proceedings of the 36th annual symposium on foundations of computer science, pp 214–221Google Scholar
- Megow N, Mehlhorn K, Schweitzer P (2011) Online graph exploration: new results on old and new algorithms. In: Proceedings of the 38th international colloquium on automata, languages and programming. Lecture Notes in Computer Science, vol 6756. Springer, Berlin, pp 478–489Google Scholar
- Papadimitriou CH, Yannakakis M (1989) Shortest paths without a map. In: Proceedings of the 16th international colloquium on automata, languages and programming. Springer, Berlin, pp 610–620Google Scholar
- Phillips CA, Stein C, Torng E, Wein J (1997) Optimal time-critical scheduling via resource augmentation. In: Proceedings of the twenty-ninth annual ACM symposium on theory of computing, ACM, pp 140–149Google Scholar
- Westphal S (2008) A note on the k-Canadian Traveller Problem. Inform Process Lett 106(3):87–89MathSciNetCrossRefMATHGoogle Scholar
- Yao ACC (1977) Probabilistic computations: towards a unified measure of complexity. In: Proceedings of the 18th annual IEEE symposium on the foundations of computer science, pp 222–227Google Scholar