Better Bounds for Online Line Chasing

Abstract: We study online competitive algorithms for the \emph{line chasing problem} in Euclidean spaces $\reals^d$, where the input consists of an initial point $P_0$ and a sequence of lines $X_1,X_2,...,X_m$, revealed one at a time. At each step $t$, when the line $X_t$ is revealed, the algorithm must determine a point $P_t\in X_t$. An online algorithm is called $c$-competitive if for any input sequence the path $P_0, P_1,...,P_m$ it computes has length at most $c$ times the optimum path. The line chasing problem is a variant of a more general convex body chasing problem, where the sets $X_t$ are arbitrary convex sets. To date, the best competitive ratio for the line chasing problem was $28.1$, even in the plane. We significantly improve this bound, by providing a~$3$-competitive algorithm for any dimension $d$. We also improve the lower bound on the competitive ratio, from $1.412$ to $1.5358$.