← 研踪 PaperTrailarXiv:2611.05932 · v1
Faster Dynamic Maximal Independent Set against Adaptive Adversaries
首次提交 2026-11-04 12:19
版本更新 2026-11-04 12:19
雷达观测 2026-11-05 10:08
摘要
We study the fully dynamic maximal independent set (MIS) problem: maintaining an MIS in an undirected graph on $n$ vertices with at most $m$ edges under edge insertions and deletions. Against an oblivious adversary, which fixes the edge update sequence in advance, the randomized algorithms of Behnezhad et al. (2019) and Chechik and Zhang (2019) achieve polylogarithmic update time. Against an adaptive adversary, which may choose each update after observing the previously reported MISs, the best previously known update bound was $O(m^{2/3})$, given by the deterministic algorithm of Gupta and Khan (2018, 2021). Assuming the combinatorial Boolean matrix multiplication hypothesis, Bernstein et al. (2026) proved that any combinatorial algorithm against an adaptive adversary requires at least $m^{1/2-o(1)}$ update time even under edge insertions alone, leaving a polynomial gap between upper and lower bounds. We present a randomized fully dynamic MIS algorithm with $\widetilde O(m^{5/8})=\widetilde O(m^{2/3-1/24})$ update time against adaptive adversaries. This gives the first polynomial improvement over the $m^{2/3}$ bound for adaptive fully dynamic MIS.
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v1 · 2026-11-04 12:19 · 最新观测版本
Faster Dynamic Maximal Independent Set against Adaptive Adversaries
We study the fully dynamic maximal independent set (MIS) problem: maintaining an MIS in an undirected graph on $n$ vertices with at most $m$ edges under edge insertions and deletions. Against an oblivious adversary, which fixes the edge update sequence in advance, the randomized algorithms of Behnezhad et al. (2019) and Chechik and Zhang (2019) achieve polylogarithmic update time. Against an adaptive adversary, which may choose each update after observing the previously reported MISs, the best previously known update bound was $O(m^{2/3})$, given by the deterministic algorithm of Gupta and Khan (2018, 2021). Assuming the combinatorial Boolean matrix multiplication hypothesis, Bernstein et al. (2026) proved that any combinatorial algorithm against an adaptive adversary requires at least $m^{1/2-o(1)}$ update time even under edge insertions alone, leaving a polynomial gap between upper and lower bounds. We present a randomized fully dynamic MIS algorithm with $\widetilde O(m^{5/8})=\widetilde O(m^{2/3-1/24})$ update time against adaptive adversaries. This gives the first polynomial improvement over the $m^{2/3}$ bound for adaptive fully dynamic MIS.
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