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., $y_a\approx x_a$ for all arcs a. Twenty years ago, in a landmark paper, Dinitz, Garg, and Goemans [3] proved that there is an unsplittable flow y such that $y_a\le x_a+d_{\max }$ for all arcs a, where $d_{\max }$ 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 $y_a\ge x_a-d_{\max }$ for all arcs a. Finally, building upon an iterative rounding technique previously introduced by Kolliopoulos and Stein [10] and Skutella [15], 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 slightly weaker simultaneous bounds $x_a/2-d_{\max }\le y_a\le 2x_a+d_{\max }$ for all arcs a.

Partially supported by DFG Priority Programme 1736 (grant SK 58/10-2).

Appendices

A Illustration of UBP and LBP Augmentation Steps

Fig. 2.

For a given unsplittable flow y, let v be UBP-reachable w.r.t. y along a path Q (illustrated dashed on the left) and let i be a commodity such that v lies on path $P_i^y$. A UBP augmentation step reroutes commodity i from s to v along path Q. The resulting unsplittable flow $y'$ is illustrated on the right.

Fig. 3.

For a given unsplittable flow y, let v be LBP-reachable for some commodity i w.r.t. y and Q be an arbitrary s-v-path (illustrated dashed on the left). An LBP augmentation step reroutes commodity i from s to v along path Q. The resulting unsplittable flow $y'$ is illustrated on the right.

B Counterexample

In Fig. 4 we give an example showing that the algorithm of Dinitz, Garg, and Goemans [3] (that was designed for the problem with arc-wise upper bounds) is not adapted for handling arc-wise lower bounds. Their violation can be arbitrarily large, as we can see by adding commodities and expanding the graph in Fig. 4. Consequently, even though the two problems regarding arc-wise upper and lower bounds seem to be similar in spirit, we need completely new tools in order to solve the above mentioned problem.

Fig. 4.

Let the digraph have three commodities with demand 1, as illustrated on the left, and let the fractional flow x be given as follows: $x_{\text {solid}}=1$, $x_{\text {dashed}}=1-\epsilon$, $x_{\text {dotted}}=\epsilon$, and $x_{\text {red}}=3-3 \epsilon$ for some $\epsilon >0$. The algorithm in [3] first moves one of the demands to the source node along its corresponding dotted arc, hence decreasing flow on the red arc by $1-\epsilon$. The remaining instance with two commodities and $x'_{\text {red}}=2-2 \epsilon$ is illustrated on the right. Any further augmentation step decreases flow on the red arc by $1-\epsilon$, hence violating the desired lower bound on the red arc. (Color figure online)

C Proof of Lemma 4

In order to prove the correctness of Algorithm 1 and the lower (resp. upper) bound, we use the following well-known results on splittable flows, also known as the cut condition; see, e.g., [1]: Given an arc-capacitated directed graph $D=(V,A)$ with a source node s and k commodities with corresponding destination nodes $t_1, \ldots , t_k$ and demands $d_1, \ldots , d_k \in \mathbb {R}_{>0}$, there is a feasible flow satisfying all demands if and only if, for any subset $T \subset V \setminus \{s\}$, the sum of capacities of arcs in the directed cut $(V\setminus T, T)$ is at least d(T).

By definition, $\bar{d_i} \le d_i < 2 \bar{d_i}$ for all $i=1,\dots ,k$. We thus get $d_i -\bar{d_i} < \frac{d_i}{2}$ for all $i=1,\dots ,k$. We first show that there is a flow $\bar{x}$ satisfying demands $\bar{d_1}, \ldots , \bar{d_k}$ such that $\bar{x}_a \ge \frac{x_a}{2}$ for all $a \in A$. By the cut condition for flow x, it holds that

$$\begin{aligned} \sum _{a \in \delta ^{\text {out}}(T)} x_a \ge d(T)\qquad \text {for each} \ T \subseteq V \setminus \{s\}. \end{aligned}$$

Therefore,

$$\begin{aligned} \sum _{a \in \delta ^{\text {out}}(T)} \frac{x_a}{2} \ge \frac{d(T)}{2} > d(T)-\bar{d}(T). \end{aligned}$$

By the cut condition, there is a flow satisfying demands $d_i -\bar{d_i}$ for $i=1,\dots ,k$ and obeying arc capacities $\frac{x_a}{2}$ for all $a\in A$. Subtract this flow from the original flow x to get a flow $\bar{x}$ satisfying demands $\bar{d_1}, \ldots , \bar{d_k}$. The flow $\bar{x}$ actually satisfies

$$\begin{aligned} \bar{x}_a \ge x_a -\frac{x_a}{2}= \frac{x_a}{2} \quad \text {for all } a \in A. \end{aligned}$$

Applying Theorem 4 to $\bar{x}$ leads to an unsplittable flow $\bar{y}$ such that

$$\begin{aligned} \bar{x}_a -\bar{d}_{\max } \; \le \; \bar{y}_a \; \le \; \bar{x}_a +\bar{d}_{\max } \quad \text {for all } a \in A. \end{aligned}$$

By construction of flow y, we obtain the lower bound

$$\begin{aligned} y_a \; \ge \; \bar{y}_a \; \ge \; \bar{x}_a -\bar{d}_{\max } \; \ge \; \frac{x_a}{2} - d_{\max } \quad \text {for all } a \in A. \end{aligned}$$

In order to prove the upper bound, let $d_{i_a}$ be a commodity with maximal demand that is routed across an arc $a \in A$. Notice that $\bar{y}$ even satisfies a slightly stronger upper bound (also see [15]): we have $\bar{y}_a \le \bar{x}_a + \bar{d}_{i_a}$ for each arc a. Therefore,

$$\begin{aligned} y_a = \sum _{i: a \in P_i} d_i \; \le \; d_{i_a} + 2 \sum _{\begin{array}{c} i: a \in P_i, \\ i \ne i_a \end{array}} \bar{d_i} \; \le \; d_{i_a} + 2 \bar{x}_a \; \le \; 2 x_a + d_{\max }. \end{aligned}$$

This concludes the proof.   $\square$