Dynamic Parameterized Problems and Algorithms

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 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)n o(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.

Serial and parallel kernelization of Multiple Hitting Set parameterized by the Dilworth number, implemented on the GPU

“…Indeed, for problem kernels of size polynomial in |V| to exist, the cardinality of each input hyperedge must be bounded from above by a constant d [21]. In this case, problem kernels of size k O(d) have been shown [1,5,7,8,13,20,24,28,38,45,46].…”

Parameterized Dynamic Variants of Red-Blue Dominating Set

“…Moreover, it would be more realistic to assume someone is influenced by the non-smoking campaign if there are at least a number, say q, of his or her friends who serve as influencers. This gives rise to the q-RBDS problem which differs from RBDS only in the domination condition requiring each element of B to have at least q neighbors in S ⊆ R. 2 We shall study the partial q-RBDS problem, among other variants, and focus on its parameterized dynamic version to cope with settings where the network is changing with time.…”

“…In essence, this is also done in the proof of the next theorem, in an even more general setting. Now we turn our attention to the increment-parameter, r. It was shown in [1,12] that Dynamic Dominating Set is W [2]-hard when parameterized by the increment-parameter r only. We show the same for q-RBDS.…”

“…However, the more interesting approximation results obtained in that paper rely on the given solution to be optimum, a condition that is relaxed in this paper. So, we are following here the terminology introduced in [1,12], which is not the same as later used in [2] under a similar name.…”

“…One could try to alternatively parameterize Partial RBDS by k and k := |B| − t. In fact, the problem is easily seen to be W[2]-hard, when parameterized by k , because the problem can then be re-formulated as follows (disregarding isolated vertices): Given some red-blue graphG = (R ∪ B, E) and integers k, k ≥ 0, find subsets R ⊆ R and B ⊆ B, with |R | ≤ k and |B | ≤ k , such that N (R ) = B \ B and N (B ) = R \ R . Hence, if k = 0,the question boils down to RBDS itself, with the roles of red and blue being exchanged.Clearly, this tweak does not change the dynamic version of the problem at all, it is equivalent to Dynamic Partial RBDS as studied above.…”