Universal Packet Routing with Arbitrary Bandwidths and Transit Times

Abstract

A fundamental problem in communication networks is store-andforward packet routing. In a celebrated paper Leighton, Maggs, and Rao [12] 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.

The Moser-Tardos Framework with Partial Resampling

“…As [28] showed, it suffices to control the congestion on intervals of length 2. Given our infeasible schedule, we can view each interval of length 2 as defining a new subproblem.…”

“…The congestion of this subproblem is exactly the congestion of the schedule. We quote the following result from [28]:…”

“…See Figure 1. Note that in the analysis of [28], the only thing that matters is the total congestion of an edge in each 2-frame. In deciding how to shift packets into the overflow times, we need to be careful to account for how often the edge appears as the first or second transit.…”

Constraint satisfaction, packet routing, and the lovasz local lemma

“…A seminal result proven in [20] (via iterated application of the LLL, which has become a key tool in its own right [21]) is that in fact T ≤ O(C + D) for all input instances, using constant-sized queues at the edges; both this result and its approach, have been used in much work in networks and combinatorial optimization. This argument was refined and simplified in [29,25], leading to a (constructive) bound of 23.4(C + D) [25]. By introducing several new ideas in scheduling packets along time intervals, we improve this to 7.26(C + D) (non-constructively) and 8.84(C + D) (constructively), thus approaching the simple lower bound for this fundamental problem.…”

“…Nevertheless, this yields a pathway to proving the theorem. We apply the assignment LLL to improve bounds of [1,3,8,15,16,19,20,25,31]. However, our approach here does not appear to lead to an algorithmic counterpart of our assignment LLL.…”