scholarly journals The Parameterised Complexity of Computing the Maximum Modularity of a Graph

Algorithmica ◽  
2019 ◽  
Vol 82 (8) ◽  
pp. 2174-2199 ◽  
Author(s):  
Kitty Meeks ◽  
Fiona Skerman
Keyword(s):  
Vertex Cover ◽  
Constant Factor ◽  
The Other ◽  
Input Graph ◽  
Feedback Vertex Set ◽  
Fpt Algorithm ◽  
Vertex Set ◽  
Fixed Constant ◽  
Minimum Feedback Vertex Set ◽  
Parameterised Complexity

Abstract The maximum modularity of a graph is a parameter widely used to describe the level of clustering or community structure in a network. Determining the maximum modularity of a graph is known to be $$\textsf {NP}$$ NP -complete in general, and in practice a range of heuristics are used to construct partitions of the vertex-set which give lower bounds on the maximum modularity but without any guarantee on how close these bounds are to the true maximum. In this paper we investigate the parameterised complexity of determining the maximum modularity with respect to various standard structural parameterisations of the input graph G. We show that the problem belongs to $$\textsf {FPT}$$ FPT when parameterised by the size of a minimum vertex cover for G, and is solvable in polynomial time whenever the treewidth or max leaf number of G is bounded by some fixed constant; we also obtain an FPT algorithm, parameterised by treewidth, to compute any constant-factor approximation to the maximum modularity. On the other hand we show that the problem is W[1]-hard (and hence unlikely to admit an FPT algorithm) when parameterised simultaneously by pathwidth and the size of a minimum feedback vertex set.

scholarly journals On Structural Parameterizations of the Edge Disjoint Paths Problem

Algorithmica ◽  
2021 ◽  
Author(s):  
Robert Ganian ◽  
Sebastian Ordyniak ◽  
M. S. Ramanujan
Keyword(s):  
Polynomial Time ◽  
Disjoint Paths ◽  
Structural Parameter ◽  
Input Graph ◽  
Feedback Vertex Set ◽  
Fpt Algorithms ◽  
Edge Disjoint ◽  
Fpt Algorithm ◽  
Vertex Set ◽  
Terminal Pair

AbstractIn this paper we revisit the classical edge disjoint paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or FPT) algorithms. As our first result, we answer an open question stated in Fleszar et al. (Proceedings of the ESA, 2016), by showing that the problem can be solved in polynomial time if the input graph has a feedback vertex set of size one. We also show that EDP parameterized by the treewidth and the maximum degree of the input graph is fixed-parameter tractable. Having developed two novel algorithms for EDP using structural restrictions on the input graph, we then turn our attention towards the augmented graph, i.e., the graph obtained from the input graph after adding one edge between every terminal pair. In constrast to the input graph, where EDP is known to remain -hard even for treewidth two, a result by Zhou et al. (Algorithmica 26(1):3--30, 2000) shows that EDP can be solved in non-uniform polynomial time if the augmented graph has constant treewidth; we note that the possible improvement of this result to an FPT-algorithm has remained open since then. We show that this is highly unlikely by establishing the [1]-hardness of the problem parameterized by the treewidth (and even feedback vertex set) of the augmented graph. Finally, we develop an FPT-algorithm for EDP by exploiting a novel structural parameter of the augmented graph.

An O(2 O(k) n 3) FPT Algorithm for the Undirected Feedback Vertex Set Problem

2005 ◽  
pp. 859-869 ◽  
Author(s):  
Frank Dehne ◽  
Michael Fellows ◽  
Michael A. Langston ◽  
Frances Rosamond ◽  
Kim Stevens
Keyword(s):  
Feedback Vertex Set ◽  
Fpt Algorithm ◽  
Vertex Set ◽  
Feedback Vertex Set Problem

scholarly journals Constant Factor Approximation for Tracking Paths and Fault Tolerant Feedback Vertex Set

2021 ◽  
pp. 23-38
Author(s):  
Václav Blažej ◽  
Pratibha Choudhary ◽  
Dušan Knop ◽  
Jan Matyáš Křišt’an ◽  
Ondřej Suchý ◽  
...  
Keyword(s):  
Approximation Algorithm ◽  
Fault Tolerant ◽  
Minimum Weight ◽  
Weighted Graph ◽  
Constant Factor ◽  
Weighted Graphs ◽  
Feedback Vertex Set ◽  
Fixed Integer ◽  
Vertex Set

AbstractConsider a vertex-weighted graph G with a source s and a target t. Tracking Paths requires finding a minimum weight set of vertices (trackers) such that the sequence of trackers in each path from s to t is unique. In this work, we derive a factor 66-approximation algorithm for Tracking Paths in weighted graphs and a factor 4-approximation algorithm if the input is unweighted. This is the first constant factor approximation for this problem. While doing so, we also study approximation of the closely related r-Fault Tolerant Feedback Vertex Set problem. There, for a fixed integer r and a given vertex-weighted graph G, the task is to find a minimum weight set of vertices intersecting every cycle of G in at least $$r+1$$ r + 1 vertices. We give a factor $$\mathcal {O}(r^2)$$ O ( r 2 ) approximation algorithm for r-Fault Tolerant Feedback Vertex Set if r is a constant.

scholarly journals Bounds for minimum feedback vertex sets in distance graphs and circulant graphs

2008 ◽  
Vol Vol. 10 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Hamamache Kheddouci ◽  
Olivier Togni
Keyword(s):  
Distance Graph ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Feedback Vertex Set ◽  
Distance Graphs ◽  
Vertex Set ◽  
Feedback Vertex Set Problem ◽  
International Audience ◽  
Minimum Feedback Vertex Set ◽  
Vertex Sets

Graphs and Algorithms International audience For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with |i − j| ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedback vertex set problem for some families of distance graphs and circulant graphs depending on the value of D.

scholarly journals Bootstrap Percolation in High Dimensions

2010 ◽  
Vol 19 (5-6) ◽  
pp. 643-692 ◽  
Author(s):  
JÓZSEF BALOGH ◽  
BÉLA BOLLOBÁS ◽  
ROBERT MORRIS
Keyword(s):  
Positive Root ◽  
Constant Factor ◽  
The Other ◽  
Bootstrap Percolation ◽  
Main Question ◽  
Critical Probability ◽  
High Dimensions ◽  
Critical Window ◽  
Fixed Constant ◽  
Image Position

In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices A ⊂ V(G) is chosen independently at random, with density p, and new vertices are subsequently infected if they have at least r infected neighbours. The set A is said to percolate if eventually all vertices are infected. Our aim is to understand this process on the grid, [n]d, for arbitrary functions n = n(t), d = d(t) and r = r(t), as t → ∞. The main question is to determine the critical probability pc([n]d, r) at which percolation becomes likely, and to give bounds on the size of the critical window. In this paper we study this problem when r = 2, for all functions n and d satisfying d ≫ log n.The bootstrap process has been extensively studied on [n]d when d is a fixed constant and 2 ⩽ r ⩽ d, and in these cases pc([n]d, r) has recently been determined up to a factor of 1 + o(1) as n → ∞. At the other end of the scale, Balogh and Bollobás determined pc([2]d, 2) up to a constant factor, and Balogh, Bollobás and Morris determined pc([n]d, d) asymptotically if d ≥ (log log n)2+ϵ, and gave much sharper bounds for the hypercube.Here we prove the following result. Let λ be the smallest positive root of the equation so λ ≈ 1.166. Then if d is sufficiently large, and moreover as d → ∞, for every function n = n(d) with d ≫ log n.

An O(2O(k)n3) FPT Algorithm for the Undirected Feedback Vertex Set Problem

2007 ◽  
Vol 41 (3) ◽  
pp. 479-492 ◽  
Author(s):  
Frank Dehne ◽  
Michael Fellows ◽  
Michael Langston ◽  
Frances Rosamond ◽  
Kim Stevens
Keyword(s):  
Feedback Vertex Set ◽  
Fpt Algorithm ◽  
Vertex Set ◽  
Feedback Vertex Set Problem
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