Online Graph Balancing and the Power of Two Choices

Abstract

In the classic online graph balancing problem, edges arrive sequentially and must be oriented immediately upon arrival, to minimize the maximum in-degree. For adversarial arrivals, the natural greedy algorithm is $O(\log n)$-competitive, and this bound is the best possible for any algorithm, even with randomization. We study this problem in the i.i.d. model where a base graph $G$ is known in advance and each arrival is an independent uniformly random edge of $G$. This model generalizes the standard power-of-two choices setting, corresponding to $G = K_n$, where the greedy algorithm achieves an $O(\log\!\log n)$ guarantee. We ask whether a similar bound is possible for arbitrary base graphs. While the greedy algorithm is optimal for adversarial arrivals and also for i.i.d. arrivals from regular base graphs (such as $G = K_n$), we show that it can perform poorly in general: there exist mildly irregular graphs $G$ for which greedy is $\widetildeΩ(\log n)$-competitive under i.i.d. arrivals. In sharp contrast, our main result is an $O(\log\!\log n)$-competitive online algorithm for every base graph $G$; this is optimal up to constant factors, since an $Ω(\log\!\log n)$ lower bound already holds even for the complete graph $G = K_n$. The key new idea is a notion of log-skewness for graphs, which captures the irregular substructures in $G$ that force the offline optimum to be large. Moreover, we show that any base graph can be decomposed into ``skew-biregular'' pieces at only $O(\log\!\log n)$ scales of log-skewness, and use this to design a decomposition-based variant of greedy that is $O(\log\!\log n)$-competitive.

Figures (3)

• Figure 1: Imbalanced biregular obstruction motivating log-skewness. A large low-degree side $A$ and a small high-degree side $B$ create, after sampling, a dense core of load $\Omega(s)$.
• Figure 2: Left: a rooted pattern, with edge colors indicating decomposition classes. Right: a sampled graph containing a tree of this pattern. Thick colored edges are greedy, and thin dotted edges are threshold. Black circles are left vertices and gray circles are right vertices. Each left vertex incident to a class-$i$ greedy edge has at least $\alpha_i$ class-$i$ threshold edges.
• Figure 3: Lower-bound construction for Greedy. The graph has layers $V_1,\ldots,V_b$ with $|V_i|=t^i\sqrt n$, where $t=(\log n)^3$, and between consecutive layers the edges form $t^i$ disjoint copies of $K_{\sqrt n,\,\sqrt n t}$. The figure shows the batch-wise propagation of load by Greedy: batch $B_1$ pushes all of $V_{b-1}$ to load at least $1$, batch $B_2$ pushes all of $V_{b-2}$ to load at least $2$, and so on.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1: Graph Balancing
• Theorem 2.2
• Claim 2.3
• Lemma 2.4
• proof
• Claim 2.5
• proof
• Example 3.1: Imbalanced biregular subgraph