Tight bounds for parameterized complexity of Cluster Editing with a small number of clusters

Fomin, Fedor V.; Kratsch, Stefan; Pilipczuk, Marcin; Pilipczuk, Michał; Villanger, Yngve

doi:10.1016/j.jcss.2014.04.015

Cited by 49 publications

(31 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Respectively, Low GF(2)-Rank Approximation asks whether it is possible to edit at most k entries of the input matrix to obtain a matrix of rank at most r. In P-Matrix Approximation, we ask whether we can edit at most k elements to obtain a P-matrix. A lot of work in graph algorithms has been done on graph editing problems, in particular parameterized subexponential time algorithms were developed for a number of problems, including various cluster editing problems [21,24].…”

Section: Related Workmentioning

confidence: 99%

Parameterized low-rank binary matrix approximation

Fomin

Golovach

Panolan

2020

Data Min Knowl Disc

Self Cite

View full text Add to dashboard Cite

We provide a number of algorithmic results for the following family of problems: For a given binary m × n matrix A and integer k, decide whether there is a "simple" binary matrix B which differs from A in at most k entries. For an integer r, the "simplicity" of B is characterized as follows.• Binary r-Means: Matrix B has at most r different columns. This problem is known to be NP-complete already for r = 2. We show that the problem is solvable in time 2 O(k log k) · (nm) O(1) and thus is fixed-parameter tractable parameterized by k. We prove that the problem admits a polynomial kernel when parameterized by r and k but it has no polynomial kernel when parameterized by k only unless NP ⊆ coNP /poly. We also complement these result by showing that when being parameterized by r and k, the problem admits an algorithm of running time 2 O(r· √ k log (k+r)) (nm) O(1) , which is subexponential in k for r ∈ O(k 1/2−ε ) for any ε > 0.• Low GF(2)-Rank Approximation: Matrix B is of GF(2)-rank at most r. This problem is known to be NP-complete already for r = 1. It also known to be W[1]-hard when parameterized by k. Interestingly, when parameterized by r and k, the problem is not only fixed-parameter tractable, but it is solvable in time 2 O(r 3/2 · √ k log k) (nm) O(1) , which is subexponential in k.• Low Boolean-Rank Approximation: Matrix B is of Boolean rank at most r. The problem is known to be NP-complete for k = 0 as well as for r = 1. We show that it is solvable in subexponential in k time 2 O(r2 r · √ k log k) (nm) O(1) .

show abstract

Section: Related Workmentioning

confidence: 99%

Parameterized low-rank binary matrix approximation

Fomin

Golovach

Panolan

2020

Data Min Knowl Disc

Self Cite

View full text Add to dashboard Cite

show abstract

“…However, such type of algorithms are rarely observed in other contexts except for very few examples [51,52] .…”

Section: Subexponential Algorithms and Polynomialtime Approximation Smentioning

confidence: 99%

Social choice meets graph drawing: How to get subexponential time algorithms for ranking and drawing problems

Fernau

Fomin

Lokshtanov

et al. 2014

Tinshhua Sci. Technol.

Self Cite

View full text Add to dashboard Cite

We analyze a common feature of p-Kemeny AGGregation (p-KAGG) and p-One-Sided Crossing Minimization (p-OSCM) to provide new insights and findings of interest to both the graph drawing community and the social choice community. We obtain parameterized subexponential-time algorithms for p-KAGG-a problem in social choice theory-and for p-OSCM-a problem in graph drawing. These algorithms run inwhere k is the parameter, and significantly improve the previous best algorithms with running times O .1.403 k / and O .1.4656 k /, respectively. We also study natural "above-guarantee" versions of these problems and show them to be fixed parameter tractable. In fact, we show that the above-guarantee versions of these problems are equivalent to a weighted variant of p-directed feedback arc set. Our results for the above-guarantee version of p-KAGG reveal an interesting contrast. We show that when the number of "votes" in the input to p-KAGG is odd the above guarantee version can still be solved in time O .2 O. p k log k/ /, while if it is even then the problem cannot have a subexponential time algorithm unless the exponential time hypothesis fails (equivalently, unless FPT D MOE1).

show abstract

“…Ghosh et al [22] showed that Split Completion is solvable in the same running time. Although Komusiewicz and Uhlmann [27] showed that we cannot expect Cluster Editing to be solvable in subexponential parameterized time, as shown by Fomin et al [17], when the number of clusters in the target graph is sublinear in the number of allowed edits, this is possible nonetheless.…”

Section: Introductionmentioning

confidence: 97%