Low congestion online routing and an improved mistake bound for online   prediction of graph labeling
  In this paper, we show a connection between a certain online low-congestion routing problem and an online prediction of graph labeling. More specifically, we prove that if there exists a routing scheme that guarantees a congestion of $\alpha$ on any edge, there exists an online prediction algorithm with mistake bound $\alpha$ times the cut size, which is the size of the cut induced by the label partitioning of graph vertices. With previous known bound of $O(\log n)$ for $\alpha$ for the routing problem on trees with $n$ vertices, we obtain an improved prediction algorithm for graphs with high effective resistance.   In contrast to previous approaches that move the graph problem into problems in vector space using graph Laplacian and rely on the analysis of the perceptron algorithm, our proof are purely combinatorial. Further more, our approach directly generalizes to the case where labels are not binary. 
Jittat Fakcharoenphol, Boonserm Kijsirikul