Single source unsplittable flows with arc-wise lower and upper bounds

Abstract

In a digraph with a source and several destination nodes with associated demands, an unsplittable flow routes each demand along a single path from the common source to its destination. Given some flow x that is not necessarily unsplittable but satisfies all demands, it is a natural question to ask for an unsplittable flow y that does not deviate from x by too much, i.e., $$y_a\approx x_a$$ y a ≈ x a for all arcs a. Twenty years ago, in a landmark paper, Dinitz et al. (Combinatorica 19:17–41, 1999) proved that there exists an unsplittable flow y such that $$y_a\le x_a+d_{\max }$$ y a ≤ x a + d max for all arcs a, where $$d_{\max }$$ d max denotes the maximum demand value. Our first contribution is a considerably simpler one-page proof for this classical result, based upon an entirely new approach. Secondly, using a subtle variant of this approach, we obtain a new result: There is an unsplittable flow y such that $$y_a\ge x_a-d_{\max }$$ y a ≥ x a - d max for all arcs a. Finally, building upon an iterative rounding technique previously introduced by Kolliopoulos and Stein (SIAM J Comput 31:919–946, 2002) and Skutella (Math Program 91:493–514, 2002), we prove existence of an unsplittable flow that simultaneously satisfies the upper and lower bounds for the special case when demands are integer multiples of each other. For arbitrary demand values, we prove the weaker simultaneous bounds $$x_a/2-d_{\max }\le y_a\le 2x_a+d_{\max }$$ x a / 2 - d max ≤ y a ≤ 2 x a + d max for all arcs a.

Flow Time Scheduling and Prefix Beck-Fiala

“…The unsplittable flow approach used in total flow time does not seem to help in this case, which can perhaps be explained by the fact that the proof uses amortization between machines, which is unsuitable for the hard constraints in max flow time. Morell and Skutella conjecture that a strengthening of the result for unsplittable flow is possible that indeed would be sufficient for the restricted assignment case [Mor21]. However, the current status of the conjecture does not imply any meaningful guarantee.…”