Online matrix-vector multiplication problem: Difference between revisions

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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 |journal=Itcs '24 |series=Leibniz International Proceedings in Informatics (LIPIcs) |volume=287 |pages=62:1–62:25 |language=en |doi=10.4230/LIPIcs.ITCS.2024.62|doi-access=free |isbn=978-3-95977-309-6 }}</ref>

== 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 |series=STOC '15 |date=2015 |publisher=Association for Computing Machinery |pages=21–30 |doi=10.1145/2746539.2746609 |isbn=978-1-4503-3536-2|arxiv=1511.06773 }}</ref> Formally, they presented the following conjecture:

{{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. A faster algorithm for OMv is implied by a result of Williams and runs in time <math>O(n^3/\log^2 n)</math>.<ref>{{Cite journal |last=Williams |first=Ryan |date=2007-01-07 |title=Matrix-vector multiplication in sub-quadratic time: (some preprocessing required) |url=https://dl.acm.org/doi/10.5555/1283383.1283490 |journal=Proceedings of the ACM-SIAM Symposium on Discrete Algorithms |series=SODA '07 |___location=USA |pages=995–1001 |isbn=978-0-89871-624-5}}</ref> The fastest known algorithm for OMv runs in time <math>n^3/2^{\Omega{\sqrt{\log n}}}</math>, due to Larsen and Williams.<ref>{{Cite journal |last1=Larsen |first1=Kasper Green |last2=Williams |first2=Ryan |date=2017-01-16 |title=Faster online matrix-vector multiplication

|url=https://dl.acm.org/doi/10.5555/3039686.3039828 |journal=Proceedings of the ACM-SIAM Symposium on Discrete Algorithms |series=SODA '17 |___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 conferencebook

| last1 = Abboud | first1 = Amir

| last2 = Williams | first2 = Virginia Vassilevska | author2-linktitle = Virginia2014 VassilevskaIEEE Williams55th Annual Symposium on Foundations of Computer Science

| chapter = Popular Conjectures Imply Strong Lower Bounds for Dynamic Problems

| author2-link = Virginia Vassilevska Williams

| arxiv = 1402.0054

| contribution = Popular conjectures imply strong lower bounds for dynamic problems

| doi = 10.1109/FOCS.2014.53

| pages = 434–443

| publisher = IEEE Computer Society

| title = 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18–21, 2014

| year = 2014| isbn = 978-1-4799-6517-5

}}</ref> many of the lower bounds that follow from OMv are stronger or new.

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 journalbook |last1=Henzinger |first1=Monika |last2=Lincoln |first2=Andrea |last3=Saha|first3=Barna |title=Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) |chapter=The Complexity of Average-Case Dynamic Subgraph Counting |chapter-url=https://doi.org/10.1137/1.9781611977073.23 |journal=ACM-SIAM Symposium on Discrete Algorithms (SODA) |series=SODA '22 |date=2022 |pages=459–498 |doi=10.1137/1.9781611977073.23 |isbn=978-1-61197-707-3 }}</ref>

==== Lower bounds from OMv ====

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{{Authority control}}

[[Category:ProblemsUnsolved problems in computer science]]

[[Category:Computational problems]]