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{{unsolved|computer science|Is there an algorithm for solving the OMv problem in time <math>O(n^{3-\varepsilon})</math>, for some constant <math>\varepsilon>0</math>?}}
In [[computational complexity theory]], the '''online matrix-vector multiplication problem''' (OMv) asks an online algorithm to return, at each round, the product of an <math>n\times n</math> matrix and a newly-arrived <math>n</math>-dimensional vector. OMv is conjectured to require roughly cubic time. This conjectured hardness implies lower bounds on the time needed to solve various [[Dynamic problem (algorithms)|dynamic problems]] and is of particular interest in [[Fine-grained reduction|fine-grained complexity]].<ref name=":1" /><ref name=":0" /><ref>{{Cite journal |last1=Henzinger |first1=Monika |last2=Saha |first2=Barna |last3=Seybold |first3=Martin P. |last4=Ye |first4=Christopher |date=2024 |title=On the Complexity of Algorithms with Predictions for Dynamic Graph Problems
== Definition ==
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== Conjectured hardness ==
In 2015, Henzinger, Krinninger, Nanongkai, and Saranurak conjectured that OMv cannot be solved in "truly subcubic" time.<ref name=":0">{{Cite book |last1=Henzinger |first1=Monika |last2=Krinninger |first2=Sebastian |last3=Nanongkai |first3=Danupon |last4=Saranurak |first4=Thatchaphol |chapter=Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture |title=Proceedings of the forty-seventh annual ACM symposium on Theory of Computing |chapter-url=https://doi.org/10.1145/2746539.2746609
{{Blockquote
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=== Algorithms for solving OMv ===
OMv can be solved in <math>O(n^3)</math> time by a naive algorithm that, in each of the <math>n</math> rounds, multiplies the matrix <math>M</math> and the new vector <math>v_i</math> in <math>O(n^2)</math> time.
|url=https://dl.acm.org/doi/10.5555/3039686.3039828 |journal=Proceedings of the ACM-SIAM Symposium on Discrete Algorithms |___location=USA |pages=2182–2189 |arxiv=1605.01695 |isbn= 978-1-61197-478-2}}</ref>
=== Implications of conjectured hardness ===
The OMv conjecture implies lower bounds on the time needed to solve a large class of dynamic graph problems, including [[reachability]] and [[Connectivity (graph theory)|connectivity]], [[Shortest path problem|shortest path]], and subgraph detection. For many of these problems, the implied lower bounds have matching upper bounds.<ref name=":0" /> While some of these lower bounds also followed from previous conjectures (e.g., [[3SUM]]),<ref>{{cite
| last1 = Abboud | first1 = Amir
| last2 = Williams | first2 = Virginia Vassilevska |
| chapter = Popular Conjectures Imply Strong Lower Bounds for Dynamic Problems
| author2-link = Virginia Vassilevska Williams
| arxiv = 1402.0054
| doi = 10.1109/FOCS.2014.53
| pages = 434–443
| publisher = IEEE Computer Society
| year = 2014| isbn = 978-1-4799-6517-5
Later work showed that the OMv conjecture implies lower bounds on the time needed for subgraph counting in [[Average-case complexity|average-case]] graphs.<ref name=":1">{{Cite
==== Lower bounds from OMv ====
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* [[Pagh's problem]] on <math>k</math> subsets from a size-<math>n</math> universe requires linear time.<ref name=":0" />
* Determining s-t [[reachability]] for a (worst-case) dynamic graph on a graph with <math>n</math> nodes and <math>m\leq n^2</math> edges requires <math>\widetilde{\Omega}(m)</math> time.<ref name=":0" />
* Counting 4-cycles in average-case, dynamic graphs with <math>n</math> nodes requires <math>\widetilde{\Omega}(n^2)</math> time.<ref name=":1" />
* Counting length-5 paths in average-case, dynamic graphs with <math>n</math> nodes requires <math>\widetilde{\Omega}(n^3)</math> time.<ref name=":1" />
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