On an online traveling repairman problem with flowtimes: Worst-case and average-case analysis

We consider an online problem where a server operates on an edge-weighted graph C and an adversarial sequence of requests to vertices is released over time. Each request requires one unit of servicetime. The server is free to choose the ordering of service and intends to minimize the total flowtime of the requests. A natural class of algorithms for this problem are IGNORE algorithms. From worst-case perspective we show that IGNORE algorithms are not competitive for flowtime minimization. From an average-case point of view, we obtain a more detailed picture. In our model, the adversary may still choose the vertices of the requests arbitrarily. But the arrival times are according to a stochastic process (with some rate lambda > 0), chosen by the adversary out of a natural class of processes. The class contains the Poisson-process and (some) deterministic arrivals as special cases. We then show that there is an IGNORE algorithm that is competitive if and only if lambda not equal 1. Specifically, for lambda not equal 1, the expected competitive ratio of the algorithm is within a constant of the length of a shortest cycle that visits all vertices of G. The reason for this is that if lambda not equal 1 the requests either arrive slow enough for our algorithm or too fast even for an offline optimal algorithm. For lambda = 1 the routing-mistakes of the online algorithm accumulate just as in the worst case. As an additional result, we show how IGNORE tours are constructed optimally in polynomial time, if the underlying graph G is a line.