Separations between Oblivious and Adaptive Adversaries for Natural Dynamic Graph Problems

TL;DR

This work establishes the first natural, explicit-input update-time separations between dynamic algorithms that are robust to adaptive adversaries and those that are non-robust to oblivious adversaries. Under standard fine-grained hypotheses, it proves near-optimal lower bounds for incremental MIS and decremental maximal clique against adaptive adversaries, while matching polylog-time algorithms exist against oblivious adversaries, yielding exponential gaps. A key byproduct is a first incremental-vs-decremental separation for triangle detection, illustrating fundamental differences between update directions. The results hinge on reductions from triangle-detection problems via well-known hypotheses (BMM, 3SUM, APSP, OMv), and are complemented by near-optimal robust and non-robust algorithms illustrating tightness. The paper closes with open questions on broader natural separations and potential improvements to robust dynamic MIS.

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 $Δ\le \sqrt{n}$. 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 $\tilde{O}(n^ω)$ 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.

Theorems & Definitions (35)

• theorem 1.1
• theorem 1.2
• theorem 1.3
• theorem 1.4: Incremental triangle detection is hard
• theorem 1.5: Decremental triangle detection is easy
• corollary 1.6
• proof
• definition 3.1: All-Edges Triangle Detection
• theorem 3.3: Patrascu10, WilliamsX20, ChanWX22
• lemma 3.4