Disjoint Path Allocation with Sublinear Advice

Abstract

We study the disjoint path allocation problem. In this setting, a path $P$ of length $L$ is given, and a sequence of subpaths of $P$ arrives online, one in every time step. Each such path requests a permanent connection between its two end-vertices. An online algorithm can admit or reject such a request; in the former case, none of the involved edges can be part of any other connection. We investigate how much additional binary information (called “advice”) can help to obtain a good solution. It is known that, with roughly $\log _2\log _2L$ advice bits, it can be guaranteed that a $\log _2L$-competitive solution is computed. In this paper, we prove the surprising result that, with $L ^{1-\varepsilon }$ advice bits, it is not possible to obtain a solution with a competitive ratio better than $(\delta \log _2L)/2$, where $0<\delta <\varepsilon <1$. This shows an interesting threshold behavior of the problem. A fairly good competitive ratio, namely $\log _2 L$, can be obtained with very few advice bits. However, any increase of the advice does not help any further until an almost linear number of advice bits is supplied. Then again, it is also known that linear advice allows for optimality.

Partially funded by SNF grant 200021-141089 and VEGA grant 1/0979/12.