Dynamic Parameterized Problems and Algorithms
Author(s)
Alman, Josh
Mnich, Matthias
Williams, Virginia Vassilevska
Date Issued
2020
Journal
ACM Transactions on Algorithms
Publisher
Association for Computing Machinery (ACM)
Version
Final published version
Abstract
© 2020 ACM. Fixed-parameter algorithms and kernelization are two powerful methods to solve NP-hard problems. Yet so far those algorithms have been largely restricted to static inputs. In this article, we provide fixed-parameter algorithms and kernelizations for fundamental NP-hard problems with dynamic inputs. We consider a variety of parameterized graph and hitting set problems that are known to have f(k)n1+o(1) time algorithms on inputs of size n, and we consider the question of whether there is a data structure that supports small updates (such as edge/vertex/set/element insertions and deletions) with an update time of g(k)no(1); such an update time would be essentially optimal. Update and query times independent of n are particularly desirable. Among many other results, we show that FEEDBACK VERTEX SET and k-PATH admit dynamic algorithms with f(k)log O(1) update and query times for some function f depending on the solution size k only. We complement our positive results by several conditional and unconditional lower bounds. For example, we show that unlike their undirected counterparts, DIRECTED FEEDBACK VERTEX SET and DIRECTED k-PATH do not admit dynamic algorithms with no(1) update and query times even for constant solution sizes k ≤ 3, assuming popular hardness hypotheses. We also show that unconditionally, in the cell probe model, DIRECTED FEEDBACK VERTEX SET cannot be solved with update time that is purely a function of k.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
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Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1145/3395037
