Dynamic parameterized problems and algorithms

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 paper we provide fixed-parameter algorithms and kernelizations for fundamental NPhard problems with dynamic inputs. We consider a variety of parameterized graph and hitting set problems which are known to have f(k)n^1+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) n 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 n^o(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.