Universal packet routing with arbitrary bandwidths and transit times

A fundamental problem in communication networks is store-and-forward packet routing. In a celebrated paper Leighton, Maggs, and Rao proved that the length of an optimal schedule is linear in the trivial lower bounds congestion and dilation. However, there has been no improvement on the actual bounds in more than 10 years. Also, commonly the problem is studied only in the setting of unit bandwidths and unit transit times. In this paper, we prove bounds on the length of optimal schedules for packet routing in the setting of arbitrary bandwidths and arbitrary transit times. Our results generalize the existing work to a much broader class of instances and also improve the known bounds significantly. For the case of unit transit times and bandwidths, we improve the best known bound of 39(C+D) to 23.4(C+D), where C and D denote the congestion and dilation, respectively. If every link in the network has a certain minimum transit time or capacity we improve this bounds to up to 4.25(C+D). Key to our results is a framework which employs tight bounds for instances where each packet travels along only a small number of edges. Further insights for such instances would reduce our constants even more

Scheduling Bidirectional Traffic on a Path

We study the fundamental problem of scheduling bidirectional traffic along a path composed of multiple segments. The main feature of the problem is that jobs traveling in the same direction can be scheduled in quick succession on a segment, while jobs in opposing directions cannot cross a segment at the same time. We show that this tradeoff makes the problem significantly harder than the related flow shop problem, by proving that it is NP-hard even for identical jobs. We complement this result with a PTAS for a single segment and non-identical jobs. If we allow some pairs of jobs traveling in different directions to cross a segment concurrently, the problem becomes APX-hard even on a single segment and with identical jobs. We give polynomial algorithms for the setting with restricted compatibilities between jobs on a single and any constant number of segments, respectively

The Moser-Tardos Framework with Partial Resampling

The resampling algorithm of Moser \& Tardos is a powerful approach to develop constructive versions of the Lov\'{a}sz Local Lemma (LLL). We generalize this to partial resampling: when a bad event holds, we resample an appropriately-random subset of the variables that define this event, rather than the entire set as in Moser & Tardos. This is particularly useful when the bad events are determined by sums of random variables. This leads to several improved algorithmic applications in scheduling, graph transversals, packet routing etc. For instance, we settle a conjecture of Szab\'{o} & Tardos (2006) on graph transversals asymptotically, and obtain improved approximation ratios for a packet routing problem of Leighton, Maggs, & Rao (1994)