# Approximating the Canadian Traveller Problem with Online Randomization

## Abstract

In this paper, we study online algorithms for the Canadian Traveller Problem defined by Papadimitriou and Yannakakis in 1991. This problem involves a traveller who knows the entire road network in advance, and wishes to travel as quickly as possible from a source vertex s to a destination vertex t, but discovers online that some roads are blocked (e.g., by snow) once reaching them. Achieving a bounded competitive ratio for the problem is PSPACE-complete. Furthermore, if at most k roads can be blocked, the optimal competitive ratio for a deterministic online algorithm is $$2k+1$$, while the only randomized result known so far is a lower bound of $$k+1$$. We show, for the first time, that a polynomial time randomized algorithm can outperform the best deterministic algorithms when there are at least two blockages, and surpass the lower bound of $$2k+1$$ by an o(1) factor. Moreover, we prove that the randomized algorithm can achieve a competitive ratio of $$\big (1+ \frac{\sqrt{2}}{2} \big )k + \sqrt{2}$$ in pseudo-polynomial time. The proposed techniques can also be exploited to implicitly represent multiple near-shortest s-t paths.

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## Notes

1. 1.

Due to space limit, the value of each $$D_E(\overrightarrow{(u,v)})$$ is shown: $$D_E(\overrightarrow{(s,v_1)})=\{ 14,15,16 \}$$, $$D_E(\overrightarrow{(s,v_2)})=\{ 14,15,16 \}$$, $$D_E(\overrightarrow{(s,v_3)})=\{ 15,18,19 \}$$; $$D_E(\overrightarrow{(v_1,s)})=\{ 19,20,21 \}$$, $$D_E(\overrightarrow{(v_1,v_2)})=\{ 11,12,13 \}$$, $$D_E(\overrightarrow{(v_1,v_4)})=\{ 9,10,11 \}$$, $$D_E(\overrightarrow{(v_1,v_5)})=\{ 9,10,11 \}$$; $$D_E(\overrightarrow{(v_2,s)})=\{ 18,19,20 \}$$, $$D_E(\overrightarrow{(v_2,v_1)})=\{ 10,11,12 \}$$, $$D_E(\overrightarrow{(v_2,v_3)})=\{ 11,14,15 \}$$, $$D_E(\overrightarrow{(v_2,v_5)})=\{ 12,13,14 \}$$, $$D_E(\overrightarrow{(v_2,v_6)})=\{ 10,13,14 \}$$; $$D_E(\overrightarrow{(v_3,s)})=\{ 20,21,22 \}$$, $$D_E(\overrightarrow{(v_3,v_2)})=\{ 12,13,14 \}$$, $$D_E(\overrightarrow{(v_3,v_6)})=\{ 9,12,13 \}$$; $$D_E(\overrightarrow{(v_4,v_1)})=\{ 13,14,15 \}$$, $$D_E(\overrightarrow{(v_4,v_5)})=\{ 5,6,7 \}$$, $$D_E(\overrightarrow{(v_4,t)})=\{ 6,12,14 \}$$; $$D_E(\overrightarrow{(v_5,v_1)})=\{ 14,15,16 \}$$, $$D_E(\overrightarrow{(v_5,v_2)})=\{ 18,19,20\}$$, $$D_E(\overrightarrow{(v_5,v_4)})=\{ 6,7,8 \}$$, $$D_E(\overrightarrow{(v_5,v_6)})=\{ 5,8,9 \}$$, $$D_E(\overrightarrow{(v_5,t)})=\{ 4,10,12 \}$$; $$D_E(\overrightarrow{(v_6,v_2)})=\{ 17,18,19 \}$$, $$D_E(\overrightarrow{(v_6,v_3)})=\{ 15,18,19 \}$$, $$D_E(\overrightarrow{(v_6,v_5)})=\{ 6,7,8 \}$$, $$D_E(\overrightarrow{(v_6,t)})=\{ 3,9,11 \}$$; $$D_E(\overrightarrow{(t,v_4)})=\{ 11,12,13 \}$$, $$D_E(\overrightarrow{(t,v_5)})=\{ 8,9,10 \}$$, $$D_E(\overrightarrow{(t,v_6)})=\{ 6,9,10 \}$$. Note that for a directed edge (uv), $$D_E(\overrightarrow{(u,v)}) = \{ \ell + d(u,v) \mid \ell \in D_V(v)\}$$.

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## Acknowledgements

The authors would like to thank anonymous referees for their helpful comments, as well as Dr. Fan Chung with UC San Diego, Dr. Kazuo Iwama with Kyoto University and Dr. Wing-Kai Hon with National Tsing Hua University for discussions on this work.

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Correspondence to Chung-Shou Liao.

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An extended abstract of this paper appeared in the proceedings of the 41st International Colloquium on Automata, Languages, and Programming (ICALP 2014). This work was partially supported by MOST Taiwan under Grants MOST105-2628-E-007-010-MY3, MOST105-2221-E-007-085-MY3, and JSPS KAKENHI 23240002.

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Demaine, E.D., Huang, Y., Liao, CS. et al. Approximating the Canadian Traveller Problem with Online Randomization. Algorithmica 83, 1524–1543 (2021). https://doi.org/10.1007/s00453-020-00792-6

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