Efficiently enumerating hitting sets of hypergraphs arising in data profiling

Bläsius, Thomas; Friedrich, Tobias; Lischeid, Julius; Meeks, Kitty; Schirneck, Martin

doi:10.1016/j.jcss.2021.10.002

Cited by 9 publications

(11 citation statements)

References 43 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Notice that there are few natural parameterized problems higher up in the W-hierarchy; see Chen and Zhang (2006). However, the very idea of extension problems (not only applicable to formal language problems) seems to lead to such problems; we also refer to Bläsius et al (2019), Casel et al (2018.…”

Section: Extensions and Orderings Of Wordsmentioning

confidence: 99%

Synchronizing words and monoid factorization, yielding a new parameterized complexity class?

Fernau

Bruchertseifer²

2022

Math. Struct. Comp. Sci.

View full text Add to dashboard Cite

The concept of a synchronizing word is a very important notion in the theory of finite automata. We consider the associated decision problem to decide if a given DFA possesses a synchronizing word of length at most k, where k is the standard parameter. We show that this problem DFA-SW is equivalent to the problem Monoid Factorization introduced by Cai, Chen, Downey, and Fellows. Apart from the known $\textsf{W}[2]$ -hardness results, we show that these problems belong to $\textsf{A}[2]$ , $\textsf{W}[\textsf{P}],$ and $\textsf{WNL}$ . This indicates that DFA-SW is not complete for any of these classes, and hence, we suggest a new parameterized complexity class $\textsf{W}[\textsf{Sync}]$ as a proper home for these (and more) problems. We present quite a number of problems that belong to $\textsf{W}[\textsf{Sync}]$ or are hard or complete for this new class.

show abstract

Section: Extensions and Orderings Of Wordsmentioning

confidence: 99%

Synchronizing words and monoid factorization, yielding a new parameterized complexity class?

Fernau

Bruchertseifer²

2022

Math. Struct. Comp. Sci.

View full text Add to dashboard Cite

show abstract

“…7 More precisely, we also have to check that the possibly newly introduced vertices in V0(f ) or V1(f ) by the branching or by the reduction rules up to this point do maintain the invariant, but this is nothing else then re-checking the induction step of the correctness proof of this invariant, see the proof of Proposition 8. 8 Again, one has to partially follow the induction step of the proof of Proposition 8.…”

Section: ∀X ∈mentioning

confidence: 99%

“…In particular for Extension Dominating Set, this hope is not fulfilled. To the contrary, with this parameterization |U |, Extension Dominating Set is one of the few problems known to be complete for the parameterized complexity class W [3], as shown in [8].…”

Section: Introductionmentioning

confidence: 99%

Minimal Roman Dominating Functions: Extensions and Enumeration

Abu-Khzam¹,

Fernau²,

Mann³

2022

Preprint

View full text Add to dashboard Cite

Roman domination is one of the many variants of domination that keeps most of the complexity features of the classical domination problem. We prove that Roman domination behaves differently in two aspects: enumeration and extension. We develop non-trivial enumeration algorithms for minimal Roman domination functions with polynomial delay and polynomial space. Recall that the existence of a similar enumeration result for minimal dominating sets is open for decades. Our result is based on a polynomial-time algorithm for Extension Roman Domination: Given a graph G = (V, E) and a function f : V → {0, 1, 2}, is there a minimal Roman domination function f with f ≤ f ? Here, ≤ lifts 0 < 1 < 2 pointwise; minimality is understood in this order. Our enumeration algorithm is also analyzed from an input-sensitive viewpoint, leading to a run-time estimate of O(1.9332 n ) for graphs of order n; this is complemented by a lower bound example of Ω(1.7441 n ).

show abstract

“…Furthermore, the Connected Dominating Set Extension with the standard parameterization |U |, where U is the given subset of V , is even W[3]-hard on split graphs. To show this, we need the W[3]-completeness of Hitting Set Extension [4,5,6,8] with standard parameterization; in this problem, the input consists of a hypergraph (X, S), with S ⊆ 2 X , and a set U ⊆ X, and the question if there exists a minimal hitting set H (this means that H ∩ S = ∅ for all S ∈ S) that extends U , i.e., U ⊆ H. Theorem 4. Connected Dominating Set Extension (with standard parameterization) is W[3]-hard, even on split graphs.…”

Section: ⊓ ⊔mentioning

confidence: 99%

Enumerating Connected Dominating Sets

Abu-Khzam¹,

Fernau²,

Gras³

et al. 2022

Preprint

View full text Add to dashboard Cite

The question to enumerate all (inclusion-wise) minimal connected dominating sets in a graph of order n in time significantly less than 2 n is an open question that was asked in many places. We answer this question affirmatively, by providing an enumeration algorithm that runs in time O(1.9896 n ), using polynomial space only. The key to this result is the consideration of this enumeration problem on 2-degenerate graphs, which is proven to be possible in time O(1.9767 n ). Apart from solving this old open question, we also show new lower bound results. More precisely, we construct a family of graphs of order n with Ω(1.4890 n ) many minimal connected dominating sets, while previous examples achieved Ω(1.4422 n ). Our example happens to yield 4-degenerate graphs. Additionally, we give lower bounds for the previously not considered classes of 2-degenerate and of 3-degenerate graphs, which are Ω(1.3195 n ) and Ω(1.4723 n ), respectively. We also address essential questions concerning output-sensitive enumeration. Namely, we give reasons why our algorithm cannot be turned into an enumeration algorithm that guarantees polynomial delay without much efforts. More precisely, we prove that it is NP-complete to decide, given a graph G and a vertex set U , if there exists a minimal connected dominating set D with U ⊆ D, even if G is known to be 2-degenerate. Our reduction also shows that even any subexponential delay is not easy to achieve for enumerating minimal connected dominating sets. Another reduction shows that no FPT-algorithms can be expected for this extension problem concerning minimal connected dominating sets, parameterized by |U |. This also adds one more problem to the still rather few natural parameterized problems that are complete for the class W[3]. We also relate our enumeration problem to the famous open Hitting Set Transversal problem, which can be phrased in our context as the question to enumerate all minimal dominating sets of a graph with polynomial delay by showing that a polynomial-delay enumeration algorithm for minimal connected dominating sets implies an affirmative algorithmic solution to the Hitting Set Transversal problem.

show abstract

Efficiently enumerating hitting sets of hypergraphs arising in data profiling

Cited by 9 publications

References 43 publications

Synchronizing words and monoid factorization, yielding a new parameterized complexity class?

Synchronizing words and monoid factorization, yielding a new parameterized complexity class?

Minimal Roman Dominating Functions: Extensions and Enumeration

Enumerating Connected Dominating Sets

Contact Info

Product

Resources

About