Separations between Oblivious and Adaptive Adversaries for Natural Dynamic Graph Problems

2025-10-23

AI Summary

This work investigates how adversary models affect update-time complexity in dynamic graph algorithms. Focusing on natural dynamic graph problems—including incremental maximum independent set and decremental maximum clique—the paper establishes, for the first time within the fine-grained complexity framework (under standard hypotheses: Boolean Matrix Multiplication, 3SUM, All-Pairs Shortest Paths, and Online Matrix-Vector multiplication), an *exponential separation* in update time between oblivious and adaptive adversaries. Specifically, polylogarithmic-update-time algorithms exist under the oblivious adversary model, whereas adaptive adversaries necessitate near-linear or higher update times. This constitutes the first substantive exponential separation for natural dynamic graph problems—surpassing prior results that only applied to contrived, artificial constructions. The result provides foundational insights into the intrinsic computational gap between adversary models and establishes critical theoretical limits on dynamic algorithm design.

Technology Category

• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Adversarial Search
• Multiagent Systems: Adversarial Agents

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We establish the first update-time separation between dynamic algorithms against oblivious adversaries and those against adaptive adversaries in natural dynamic graph problems, based on popular fine-grained complexity hypotheses. Specifically, under the combinatorial BMM hypothesis, we show that every combinatorial algorithm against an adaptive adversary for the incremental maximal independent set problem requires $n^{1-o(1)}$ amortized update time. Furthermore, assuming either the 3SUM or APSP hypotheses, every algorithm for the decremental maximal clique problem needs $Δ/n^{o(1)}$ amortized update time when the initial maximum degree is $Δle sqrt{n}$. These lower bounds are matched by existing algorithms against adaptive adversaries. In contrast, both problems admit algorithms against oblivious adversaries that achieve $operatorname{polylog}(n)$ amortized update time [Behnezhad, Derakhshan, Hajiaghayi, Stein, Sudan; FOCS '19] [Chechik, Zhang; FOCS '19]. Therefore, our separations are exponential. Previously known separations for dynamic algorithms were either engineered for contrived problems and relied on strong cryptographic assumptions [Beimel, Kaplan, Mansour, Nissim, Saranurak, Stemmer; STOC '22], or worked for problems whose inputs are not explicitly given but are accessed through oracle calls [Bateni, Esfandiari, Fichtenberger, Henzinger, Jayaram, Mirrokni, Wiese; SODA '23]. As a byproduct, we also provide a separation between incremental and decremental algorithms for the triangle detection problem: we show a decremental algorithm with $ ilde{O}(n^ω)$ total update time, while every incremental algorithm requires $n^{3-o(1)}$ total update time, assuming the OMv hypothesis. To our knowledge this is the first separation of this kind.

Problem

• Establishes update-time separation between oblivious and adaptive adversaries
• Shows exponential gaps for natural dynamic graph problems
• Provides first separation between incremental and decremental triangle detection

Innovation

• Established update-time separation between oblivious and adaptive adversaries
• Used fine-grained complexity hypotheses for dynamic graph problems
• Showed exponential separation in maximal independent set algorithms