Smoothed Analysis of Dynamic Graph Algorithms
Uri Meir, Ami Paz
TL;DR
This work investigates how robust lower bounds for dynamic graph algorithms are under smoothed analysis, introducing a per-round noise model with parameter $p$ that interpolates between worst-case and average-case inputs. It shows a mixed landscape: some problems (notably short subgraph counting like ${s}$-${t}$ ${3}$-paths and ${s}$-${t}$ ${4}$-paths) retain near-worst-case hardness under smoothing, with update-time lower bounds of at least $ ilde{ ext{Ω}}(p n^{1- ext{ε}})$ conditioned on the OMv conjecture, while decision problems such as connectivity often become easy even with nontrivial noise. The paper also develops a hierarchical framework over smoothing models (oblivious flip, oblivious add/remove, and adaptive adversaries), proving that some separations occur (e.g., adaptive adversaries can induce higher runtimes) and that for counting small subgraphs the three models can coincide in complexity, whereas others separate. The technical core combines Poissonization, reductions from the OMv to OuMv problems, and constructions on $P_3$-partite graphs to derive near-tight conditional lower bounds, thereby clarifying when noise stabilizes or preserves hardness and guiding robust algorithm design under uncertainty.
Abstract
Recent years have seen significant progress in the study of dynamic graph algorithms, and most notably, the introduction of strong lower bound techniques for them (e.g., Henzinger, Krinninger, Nanongkai and Saranurak, STOC 2015; Larsen and Yu, FOCS 2023). As worst-case analysis (adversarial inputs) may lead to the necessity of high running times, a natural question arises: in which cases are high running times really necessary, and in which cases these inputs merely manifest unique pathological cases? Early attempts to tackle this question were made by Nikoletseas, Reif, Spirakis and Yung (ICALP 1995) and by Alberts and Henzinger (Algorithmica 1998), who considered models with very little adversarial control over the inputs, and showed fast algorithms exist for them. The question was then overlooked for decades, until Henzinger, Lincoln and Saha (SODA 2022) recently addressed uniformly random inputs, and presented algorithms and impossibility results for several subgraph counting problems. To tackle the above question more thoroughly, we employ smoothed analysis, a celebrated framework introduced by Spielman and Teng (J. ACM, 2004). An input is proposed by an adversary but then a noisy version of it is processed by the algorithm instead. Parameterized by the amount of adversarial control, this input model fully interpolates between worst-case inputs and a uniformly random input. Doing so, we extend impossibility results for some problems to the smoothed model with only a minor quantitative loss. That is, we show that partially-adversarial inputs suffice to impose high running times for certain problems. In contrast, we show that other problems become easy even with the slightest amount of noise. In addition, we study the interplay between the adversary and the noise, leading to three natural models of smoothed inputs, for which we show a hierarchy of increasing complexity.