Optimal Electrical Oblivious Routing on Expanders

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 |∂ S| ≥ Φ ⋅ vol(S) for every S ⊆ V, vol(S) ≤ 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 𝓁_∞ 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 𝓁₁- and 𝓁_∞-norms. We further observe that the oblivious routing is O(log² m)-competitive in the 𝓁₂-norm and, in fact, O(log m)-competitive if 𝓁₂-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 ∈ [2, ∞] and q given by 1/p + 1/q = 1. Assuming 𝓁₂-localization in O(log m), we obtain that in 𝓁_p and 𝓁_q, the electrical oblivious routing is O(Φ^{-(1-2/p)}log m) competitive. Using the currently known result for 𝓁₂-localization, this ratio deteriorates by at most a sublogarithmic factor for every p, q ≠ 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 𝓁₁ and 𝓁_∞ unconditionally tight up to constants, and the result in any 𝓁_p- and 𝓁_q-norm to be tight in case of 𝓁₂-localization in O(log m).