Minimum Congestion Routing of Unsplittable Flows in Data-Center Networks

TL;DR

This work studies minimum congestion routing for unsplittable flows in Clos networks, where flow demands are constrained by link capacities. The authors introduce a two-phase offline routing algorithm that bridges two classical heuristics to beat the longstanding barrier of congestion $2$, achieving a guaranteed congestion of $\frac{9}{5}$ times the optimum and an overall $\frac{9}{5}$-approximation. They establish fundamental limits by proving a $\tfrac{3}{2}$-hardness barrier for both congestion and approximation and show online algorithms cannot beat a factor of $2$, highlighting a clear separation between offline and online settings. The results significantly advance the understanding of Clos-network routing with unsplittable flows and guide future work on partial splittability and online scheduling in data-center fabrics.

Abstract

Millions of flows are routed concurrently through a modern data-center. These networks are often built as Clos topologies, and flow demands are constrained only by the link capacities at the ingress and egress points. The minimum congestion routing problem seeks to route a set of flows through a data center while minimizing the maximum flow demand on any link. This is easily achieved by splitting flow demands along all available paths. However, arbitrary flow splitting is unrealistic. Instead, network operators rely on heuristics for routing unsplittable flows, the best of which results in a worst-case congestion of $2$ (twice the uniform link capacities). But is $2$ the lowest possible congestion? If not, can an efficient routing algorithm attain congestion below $2$? Guided by these questions, we investigate the minimum congestion routing problem in Clos networks with unsplittable flows. First, we show that for some sets of flows the minimum congestion is at least $\nicefrac{3}{2}$, and that it is $NP$-hard to approximate a minimum congestion routing by a factor less than $\nicefrac{3}{2}$. Second, addressing the motivating questions directly, we present a polynomial-time algorithm that guarantees a congestion of at most $\nicefrac{9}{5}$ for any set of flows, while also providing a $\nicefrac{9}{5}$ approximation of a minimum congestion routing. Last, shifting to the online setting, we demonstrate that no online algorithm (even randomized) can approximate a minimum congestion routing by a factor less than $2$, providing a strict separation between the online and the offline setting.

Figures (11)

• Figure 1: A set of flows without a routing.
• Figure 2: A routing with minimum congestion.
• Figure 4: Worst-case flows for the Melen-Turner algorithm. Figure \ref{['fig:worst-case-melen-turner-algorithm-1']} shows a set of flows in a Clos network with $N$ middle switches composed of two types of flows: one type 1 flow with demand $1$ (in blue) and $(N-1)/\epsilon$ type 2 flows each with demand $\epsilon$ (in orange), for some small $\epsilon > 0$ (only the first input and output switches are shown). The minimum congestion routing has congestion $1$, with the type 1 flow assigned to some middle switch, and $1/\epsilon$ type 2 flows assigned to each of the remaining $N-1$. Figure \ref{['fig:worst-case-melen-turner-algorithm-2']} shows the mapping of the flows to the new Clos network, where each ToR switch is divided into $x$ copies, with $x \coloneqq 1/N + 1/\epsilon - 1/(N \times \epsilon)$. Figure \ref{['fig:worst-case-melen-turner-algorithm-3']} shows a link-disjoint routing for the flows in the new network, and Figure \ref{['fig:worst-case-melen-turner-algorithm-4']} shows the corresponding routing in the original one. The routing returned by the algorithm has congestion $2 - \epsilon - (1-\epsilon)/N$. Consequently, the worst-case congestion and the approximation factor of the algorithm approach $2$ as $N$ grows large.
• Figure 5: The cross gadget underlying Theorems \ref{['thm:intro-offline_limit_congestion']} and \ref{['thm:intro-offline_limit_approximation']} in a Clos network with $3$ middle switches.
• Figure 6: Sequence $X$.

Theorems & Definitions (37)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Corollary 2.4
• Theorem 2.5
• Corollary 2.6
• Theorem 3.1: Hwang_1983Lovasz_2009
• Corollary 3.2
• Theorem 4.1
• Lemma 4.2