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.

翻译：基于流行的细粒度复杂性假设，我们在自然动态图问题中首次建立了针对无意识敌手的动态算法与针对自适应敌手的动态算法之间的更新时间分离性。具体而言，在组合BMM假设下，我们证明了针对增量式最大独立集问题的、对抗自适应敌手的任何组合算法均需要$n^{1-o(1)}$的摊还更新时间。此外，假设3SUM或APSP假设成立，对于初始最大度$\Delta \le \sqrt{n}$的情况，针对递减式最大团问题的任何算法都需要$\Delta/n^{o(1)}$的摊还更新时间。这些下界与现有的对抗自适应敌手的算法相匹配。相比之下，这两个问题都存在对抗无意识敌手的算法，能够实现$\operatorname{polylog}(n)$的摊还更新时间[Behnezhad, Derakhshan, Hajiaghayi, Stein, Sudan; FOCS '19] [Chechik, Zhang; FOCS '19]。因此，我们得到的分离是指数级的。先前已知的动态算法分离性要么是针对人为构造的问题并依赖于强密码学假设[Beimel, Kaplan, Mansour, Nissim, Saranurak, Stemmer; STOC '22]，要么适用于那些输入并非显式给出而是通过预言机调用访问的问题[Bateni, Esfandiari, Fichtenberger, Henzinger, Jayaram, Mirrokni, Wiese; SODA '23]。作为副产品，我们还为三角形检测问题提供了增量式算法与递减式算法之间的分离性：我们展示了一个总更新时间为$\tilde{O}(n^{\omega})$的递减式算法，而在OMv假设下，任何增量式算法都需要$n^{3-o(1)}$的总更新时间。据我们所知，这是此类分离性的首次证明。