Reducing rank of the adjacency matrix by graph modification

Meesum, Syed Mohammad; Misra, Pranabendu; Saurabh, Saket

doi:10.1016/j.tcs.2016.02.020

Cited by 4 publications

(2 citation statements)

References 12 publications

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“…Meesum, Misra, and Saurabh [49], and Meesum and Saurabh [50] considered parameterized algorithms for related problems about editing of the adjacencies of a graph (or directed graph) targeting a graph with adjacency matrix of small rank.…”

Section: Related Workmentioning

confidence: 99%

Parameterized low-rank binary matrix approximation

Fomin

Golovach

Panolan

2020

Data Min Knowl Disc

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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) .

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Section: Related Workmentioning

confidence: 99%

Parameterized low-rank binary matrix approximation

Fomin

Golovach

Panolan

2020

Data Min Knowl Disc

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“…Using the H-Bag Editing problem from Damaschke and Mogren [85], Meesum, Misra, and Saurabh [220] show that the r-Rank Reduction Editing problem is solvable in time (1) . In this problem, we are asked to edit the input graph G to a graph G by modifying at most k edges so that rank(A G ), the rank of the adjacency matrix of G is at most r. A similar result is shown in [221] for the directed case.…”

Section: Subexponential Time Algorithmsmentioning

confidence: 99%

A survey of parameterized algorithms and the complexity of edge modification

Crespelle¹,

Drange²,

Fomin³

et al. 2020

Preprint

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The survey provides an overview of the developing area of parameterized algorithms for graph modification problems. We concentrate on edge modification problems, where the task is to change a small number of adjacencies in a graph in order to satisfy some required property. IMPORTANT NOTICE:This survey is still in a tentative version. If you find some mistakes or missing results, please contact us as soon as possible so that we can correct before final publication.

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Rank Reduction of Oriented Graphs by Vertex and Edge Deletions

Meesum

Saurabh

2017

Algorithmica

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Reducing rank of the adjacency matrix by graph modification

Cited by 4 publications

References 12 publications

Parameterized low-rank binary matrix approximation

Parameterized low-rank binary matrix approximation

A survey of parameterized algorithms and the complexity of edge modification

Rank Reduction of Oriented Graphs by Vertex and Edge Deletions

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