Optimal online escape path against a certificate

TL;DR

This paper introduces a new 'certificate path' for escaping convex environments from any starting point, and develops an online strategy that guarantees a path length within a factor of about 3.32 of this certificate path, improving previous bounds.

Contribution

It proposes the certificate path as an intuitive alternative to the optimal escape path and designs an online strategy with a near-optimal competitive ratio for convex environments.

Findings

The certificate path always leads out of the environment sooner than the optimal path in convex shapes.

The online strategy guarantees a path length less than 3.318764 times the certificate path.

A lower bound of 3.313126 is established, showing near-tightness of the competitive ratio.

Abstract

More than fifty years ago, Bellman asked for the best escape path within a known forest but for an unknown starting position. This deterministic finite path is the shortest path that leads out of a given environment from any starting point. There are some worst case positions where the full path length is required. Up to now such a fixed ultimate optimal escape path for a known shape for any starting position is only known for some special convex shapes (i.e., circles, strips of a given width, fat convex bodies, some isosceles triangles). Therefore, we introduce a different, simple and intuitive escape path, the so-called certificate path. This escape path depends on the starting position s and takes the distances from s to the outer boundary of the environment into account. Due to the additional information, the certificate path always (for any position s) leaves the environment earlier than the ultimate escape path, in the above convex examples. Next we assume that fewer information is available. Neither the precise shape of the envir- onment, nor the location of the starting point is known. For a class of environments (convex shapes and shapes with kernel positions), we design an online strategy that always leaves the environment. We show that the path length for leaving the environment is always shorter than 3.318764 the length of the corresponding certificate path. We also give a lower bound of 3.313126, which shows that for the above class of environments the factor 3.318764 is (almost) tight.