10 research outputs found
Fixed-parameter algorithms for minimum-cost edge-connectivity augmentation
We consider connectivity-augmentation problems in a setting where each potential new edge has a non-negative cost associated with it, and the task is to achieve a certain connectivity target with at most p new edges of minimum total cost. The main result is that the minimum cost augmentation of edge-connectivity from k − 1 to k with at most p new edges is fixed-parameter tractable parameterized by p and admits a polynomial kernel. We also prove the fixed-parameter tractability of increasing edge connectivity from 0 to 2 and increasing node connectivity from 1 to 2
On the fixed-parameter tractability of the maximum connectivity improvement problem
In the Maximum Connectivity Improvement (MCI) problem, we are given a
directed graph and an integer and we are asked to find new
edges to be added to in order to maximize the number of connected pairs of
vertices in the resulting graph. The MCI problem has been studied from the
approximation point of view. In this paper, we approach it from the
parameterized complexity perspective in the case of directed acyclic graphs. We
show several hardness and algorithmic results with respect to different natural
parameters. Our main result is that the problem is -hard for parameter
and it is FPT for parameters and , the matching number of
. We further characterize the MCI problem with respect to other
complementary parameters.Comment: 15 pages, 1 figur
Path-Contractions, Edge Deletions and Connectivity Preservation
We study several problems related to graph modification problems under
connectivity constraints from the perspective of parameterized complexity: {\sc
(Weighted) Biconnectivity Deletion}, where we are tasked with deleting~
edges while preserving biconnectivity in an undirected graph, {\sc
Vertex-deletion Preserving Strong Connectivity}, where we want to maintain
strong connectivity of a digraph while deleting exactly~ vertices, and {\sc
Path-contraction Preserving Strong Connectivity}, in which the operation of
path contraction on arcs is used instead. The parameterized tractability of
this last problem was posed by Bang-Jensen and Yeo [DAM 2008] as an open
question and we answer it here in the negative: both variants of preserving
strong connectivity are -hard. Preserving biconnectivity, on the
other hand, turns out to be fixed parameter tractable and we provide a
-algorithm that solves {\sc Weighted Biconnectivity
Deletion}. Further, we show that the unweighted case even admits a randomized
polynomial kernel. All our results provide further interesting data points for
the systematic study of connectivity-preservation constraints in the
parameterized setting
Breaching the 2-Approximation Barrier for Connectivity Augmentation: a Reduction to Steiner Tree
The basic goal of survivable network design is to build a cheap network that
maintains the connectivity between given sets of nodes despite the failure of a
few edges/nodes. The Connectivity Augmentation Problem (CAP) is arguably one of
the most basic problems in this area: given a (-edge)-connected graph
and a set of extra edges (links), select a minimum cardinality subset of
links such that adding to increases its edge connectivity to .
Intuitively, one wants to make an existing network more reliable by augmenting
it with extra edges. The best known approximation factor for this NP-hard
problem is , and this can be achieved with multiple approaches (the first
such result is in [Frederickson and J\'aj\'a'81]).
It is known [Dinitz et al.'76] that CAP can be reduced to the case ,
a.k.a. the Tree Augmentation Problem (TAP), for odd , and to the case ,
a.k.a. the Cactus Augmentation Problem (CacAP), for even . Several better
than approximation algorithms are known for TAP, culminating with a recent
approximation [Grandoni et al.'18]. However, for CacAP the best known
approximation is .
In this paper we breach the approximation barrier for CacAP, hence for
CAP, by presenting a polynomial-time
approximation. Previous approaches exploit properties of TAP that do not seem
to generalize to CacAP. We instead use a reduction to the Steiner tree problem
which was previously used in parameterized algorithms [Basavaraju et al.'14].
This reduction is not approximation preserving, and using the current best
approximation factor for Steiner tree [Byrka et al.'13] as a black-box would
not be good enough to improve on . To achieve the latter goal, we ``open the
box'' and exploit the specific properties of the instances of Steiner tree
arising from CacAP.Comment: Corrected a typo in the abstract (in metadata
Fixed-parameter algorithms for minimum cost edge-connectivity augmentation
We consider connectivity-augmentation problems in a setting where each potential new edge has a nonnegative cost associated with it, and the task is to achieve a certain connectivity target with at most p new edges of minimum total cost. The main result is that the minimum cost augmentation of edge-connectivity from k − 1 to k with at most p new edges is fixed-parameter tractable parameterized by p and admits a polynomial kernel. We also prove the fixed-parameter tractability of increasing edge-connectivity from 0 to 2, and increasing node-connectivity from 1 to 2
A survey of parameterized algorithms and the complexity of edge modification
The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum