A Combinatorial Metrical Task System Problem under the Uniform Metric

We consider a variant of the metrical task system (MTS) problem under the uniform metric, where each decision corresponds to some combinatorial object in a fixed set (e.g., the set of all s-t paths of a fixed graph). Typical algorithms such as Marking algorithm are not known to solve this problem efficiently and straightforward implementations takes exponential time for many classes of combinatorial sets. We propose a modification of Marking algorithm, which we call Weighted Marking algorithm. We show that Weighted Marking algorithm still keeps O(logn) competitive ratio for the standard MTS problem with n states. On the other hand, combining with known sampling techniques for combinatorial sets, Weighted Marking algorithm works efficiently for various classes of combinatorial sets