Approximation Algorithms for Minimizing Congestion in Demand-Aware Networks

TL;DR

This work studies the min-congestion objective in hybrid demand-aware networks by jointly optimizing topology (static plus reconfigurable links) and routing, across four models defined by splittable vs unsplittable flows and segregated vs non-segregated routing. It delivers a comprehensive algorithmic treatment for the segregated model, including a $2$-approximation for splittable flows via LP relaxation and deterministic rounding, APX-hardness for uniform bipartite demands, and tractable single-source/destination cases; unsplittable flows admit a $O\left(\frac{\log m}{\log\log m}\right)$-approximation with matching lower bounds. For non-segregated routing, the paper proves a fundamental $\\ ext{lower bound }\Omega\left(\dfrac{c_{\max}}{c_{\min}}\right)$, with NP-hardness even for a single source/destination and uniform capacities, while identifying tractable restricted cases such as single-commodity, uniform-capacity instances. The authors validate their theoretical results with trace-driven simulations showing substantial congestion reductions over state-of-the-art baselines, underscoring practical impact for datacenter network design. Overall, the work maps a nuanced landscape of approximability across routing models and flow types in demand-aware networks, providing actionable algorithms and insights for hybrid optical-static topologies.

Abstract

Emerging reconfigurable optical communication technologies allow to enhance datacenter topologies with demand-aware links optimized towards traffic patterns. This paper studies the algorithmic problem of jointly optimizing topology and routing in such demand-aware networks to minimize congestion, along two dimensions: (1) splittable or unsplittable flows, and (2) whether routing is segregated, i.e., whether routes can or cannot combine both demand-aware and demand-oblivious (static) links. For splittable and segregated routing, we show that the problem is generally $2$-approximable, but APX-hard even for uniform demands induced by a bipartite demand graph. For unsplittable and segregated routing, we establish upper and lower bounds of $O\left(\log m/ \log\log m \right)$ and $Ω\left(\log m/ \log\log m \right)$, respectively, for polynomial-time approximation algorithms, where $m$ is the number of static links. We further reveal that under un-/splittable and non-segregated routing, even for demands of a single source (resp., destination), the problem cannot be approximated better than $Ω\left(\frac{c_{\max}}{c_{\min}} \right)$ unless P=NP, where $c_{\max}$ (resp., $c_{\min}$) denotes the maximum (resp., minimum) capacity. It remains NP-hard for uniform capacities, but is tractable for a single commodity and uniform capacities. Our trace-driven simulations show a significant reduction in network congestion compared to existing solutions.

Figures (4)

• Figure 1: Illustration of our gap-preserving reduction from the minimum vertex cover problem to the MCRN problem with $\tau=\text{SS}$. Each green node ${v_1, v_2, v_i, \ldots, v_n}$ corresponds to one vertex in the given minimum vertex cover instance. The dashed lines show a subset of reconfigurable links $\mathcal{E}$, whose two endpoints have non-zero demands in $D$, and solid lines indicate static links $E$, where there are $k$ parallel paths between $a$ and $o$ in the static network. Each bi-directed link in $\mathcal{E}\cup E$ has the same capacity in both directions, and we define capacities: $c:\vec{\mathcal{E}}\mapsto \{3\}$ and $c:\vec{E}\mapsto\{1,3\}$. The capacity value is marked on each bi-directed (static) link in the figure.
• Figure 2: Illustration of our gap-producing reduction from the min-vertex cover problem to the MCRN problem in the non-segregated model. The figure shows an instance of the MCRN, where each green node ${v_1, v_2, v_i, \ldots, v_n}$ corresponds to one vertex in the min-vertex cover instance.
• Figure 3: Algorithmic comparison of the min-congestion for four traces of exascale applications in HPC clusters, using random $k$-regular graphs for the static network.
• Figure 4: Algorithmic comparison of the min-congestion for different synthetic traces of pFabric clusters, with random $4$-regular graphs as static networks.

Theorems & Definitions (10)

• Definition 1: Min-Congestion Reconfigurable Network Problem (MCRN)
• Definition 2: Min-Congestion Multi-Commodity Flow Problem
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8