Dynamic Parameterized Problems and Algorithms

TL;DR: This article provides fixed-parameter algorithms and kernelizations for fundamental NP-hard problems with dynamic inputs and shows 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.

Abstract: 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 DIRECTEDk-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.