Deterministic Dynamic Maximal Matching in Sublinear Update Time
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912558383890432 |
|---|---|
| author | Bernstein, Aaron Bhattacharya, Sayan Kiss, Peter Saranurak, Thatchaphol |
| author_facet | Bernstein, Aaron Bhattacharya, Sayan Kiss, Peter Saranurak, Thatchaphol |
| contents | We give a fully dynamic deterministic algorithm for maintaining a maximal matching of an $n$-vertex graph in $\tilde{O}(n^{8/9})$ amortized update time. This breaks the long-standing $Ω(n)$-update-time barrier on dense graphs, achievable by trivially scanning all incident vertices of the updated edge, and affirmatively answers a major open question repeatedly asked in the literature [BGS15, BCHN18, Sol22]. We also present a faster randomized algorithm against an adaptive adversary with $\tilde{O}(n^{3/4})$ amortized update time.
Our approach employs the edge degree constrained subgraph (EDCS), a central object for optimizing approximation ratio, in a completely novel way; we instead use it for maintaining a matching that matches all high degree vertices in sublinear update time so that it remains to handle low degree vertices rather straightforwardly. To optimize this approach, we employ tools never used in the dynamic matching literature prior to our work, including sublinear-time algorithms for matching high degree vertices, random walks on directed expanders, and the monotone Even-Shiloach tree for dynamic shortest paths. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20780 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Deterministic Dynamic Maximal Matching in Sublinear Update Time Bernstein, Aaron Bhattacharya, Sayan Kiss, Peter Saranurak, Thatchaphol Data Structures and Algorithms We give a fully dynamic deterministic algorithm for maintaining a maximal matching of an $n$-vertex graph in $\tilde{O}(n^{8/9})$ amortized update time. This breaks the long-standing $Ω(n)$-update-time barrier on dense graphs, achievable by trivially scanning all incident vertices of the updated edge, and affirmatively answers a major open question repeatedly asked in the literature [BGS15, BCHN18, Sol22]. We also present a faster randomized algorithm against an adaptive adversary with $\tilde{O}(n^{3/4})$ amortized update time. Our approach employs the edge degree constrained subgraph (EDCS), a central object for optimizing approximation ratio, in a completely novel way; we instead use it for maintaining a matching that matches all high degree vertices in sublinear update time so that it remains to handle low degree vertices rather straightforwardly. To optimize this approach, we employ tools never used in the dynamic matching literature prior to our work, including sublinear-time algorithms for matching high degree vertices, random walks on directed expanders, and the monotone Even-Shiloach tree for dynamic shortest paths. |
| title | Deterministic Dynamic Maximal Matching in Sublinear Update Time |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2504.20780 |