Optimal Electrical Oblivious Routing on Expanders

TL;DR

The paper investigates how well electrical flow routing serves as an oblivious routing scheme on Φ-expander graphs. By deriving tight upper and lower bounds across ℓ_p norms, it shows that electrical routing achieves ρ_p = O(log m/Φ^{1−2/p}) (with constants and localization assumptions shaping the precise dependence) and, in particular, proves ρ_1 and ρ_∞ are O(log m/Φ), while ρ_2 ties to ℓ2-localization. The results rely on a voltage-cut analysis for ℓ∞ and ℓ1 bounds, a Riesz–Thorin interpolation for all p, and a lower-bound gadget based on expander construction and effective resistance to establish near-tightness. The findings have implications for fast approximate maximum flow algorithms on expanders and broaden the understanding of localization and norm-interpolation in oblivious routing.

Abstract

In this paper, we investigate the question of whether the electrical flow routing is a good oblivious routing scheme on an $m$-edge graph $G = (V, E)$ that is a $Φ$-expander, i.e. where $\lvert \partial S \rvert \geq Φ\cdot \mathrm{vol}(S)$ for every $S \subseteq V, \mathrm{vol}(S) \leq \mathrm{vol}(V)/2$. Beyond its simplicity and structural importance, this question is well-motivated by the current state-of-the-art of fast algorithms for $\ell_{\infty}$ oblivious routings that reduce to the expander-case which is in turn solved by electrical flow routing. Our main result proves that the electrical routing is an $O(Φ^{-1} \log m)$-competitive oblivious routing in the $\ell_1$- and $\ell_\infty$-norms. We further observe that the oblivious routing is $O(\log^2 m)$-competitive in the $\ell_2$-norm and, in fact, $O(\log m)$-competitive if $\ell_2$-localization is $O(\log m)$ which is widely believed. Using these three upper bounds, we can smoothly interpolate to obtain upper bounds for every $p \in [2, \infty]$ and $q$ given by $1/p + 1/q = 1$. Assuming $\ell_2$-localization in $O(\log m)$, we obtain that in $\ell_p$ and $\ell_q$, the electrical oblivious routing is $O(Φ^{-(1-2/p)}\log m)$ competitive. Using the currently known result for $\ell_2$-localization, this ratio deteriorates by at most a sublogarithmic factor for every $p, q \neq 2$. We complement our upper bounds with lower bounds that show that the electrical routing for any such $p$ and $q$ is $Ω(Φ^{-(1-2/p)}\log m)$-competitive. This renders our results in $\ell_1$ and $\ell_{\infty}$ unconditionally tight up to constants, and the result in any $\ell_p$- and $\ell_q$-norm to be tight in case of $\ell_2$-localization in $O(\log m)$.

Figures (1)

• Figure 1: An illustration of the result given in Corollary \ref{['thm:mainUpperBoundViaLocalization']} of the different competitive ratios achieved with respect to each $\ell_p$-norm, where $n$ and $\Phi$ are fixed and $\Phi \ll 1/\log m$. The red curve shows the optimal ratios, achieved if localization is in $O(\log m)$, which is also obtained up to constant factors by our lower bound in Theorem \ref{['thm:mainLowerBound']}. The lilac curve shows the current trade-off where we use the known result that localization is in $O(\log^2 n)$. For $\Phi \gg 1/\log m$ the upper bound of Theorem \ref{['thm:mainLp']} achieves values between the two curves.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3: implied by Riesz-Thorin
• Theorem 1.4
• Corollary 1.5: (Informal) Electrical Oblivious $\ell_{\infty}$- and $\ell_1$-Routing on Weighted Expanders
• Remark 1.6
• Corollary 1.7: (Informal) Localization of Electrical Flow
• Theorem 3.1: $\ell_{\infty}$ Competitive bound of electrical flows
• proof
• Lemma 4.1: Competitive ratio of $\ell_p$-norms