Separations between Oblivious and Adaptive Adversaries for Natural Dynamic Graph Problems

TL;DR

This paper proves exponential separation in update times between oblivious and adaptive adversaries for natural dynamic graph problems, based on popular complexity hypotheses, highlighting fundamental differences in algorithmic capabilities.

Contribution

It establishes the first exponential separation between oblivious and adaptive adversaries for natural dynamic graph problems under standard hypotheses.

Findings

Adaptive algorithms require near-linear update time under certain hypotheses.

Oblivious algorithms achieve polylogarithmic update time for the same problems.

A new separation between incremental and decremental algorithms for triangle detection is demonstrated.

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 $\Delta/n^{o(1)}$ amortized update time when the initial maximum degree is $\Delta \le \sqrt{n}$. 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 $\tilde{O}(n^{\omega})$ 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.