Approximating the Canadian Traveller Problem with Online Randomization

Abstract

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

$$2k+1$$

2 k + 1

, while the only randomized result known so far is a lower bound of

$$k+1$$

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

$$2k+1$$

2 k + 1

by an o(1) factor. Moreover, we prove that the randomized algorithm can achieve a competitive ratio of

$$\big (1+ \frac{\sqrt{2}}{2} \big )k + \sqrt{2}$$

(

1 +

2

2

)

k +

2

in pseudo-polynomial time. The proposed techniques can also be exploited to implicitly represent multiple near-shortest s-t paths.