3,270 research outputs found
Structured Connectivity Augmentation
We initiate the algorithmic study of the following "structured augmentation" question: is it possible to increase the connectivity of a given graph G by superposing it with another given graph H? More precisely, graph F is the superposition of G and H with respect to injective mapping phi:V(H)->V(G) if every edge uv of F is either an edge of G, or phi^{-1}(u)phi^{-1}(v) is an edge of H. Thus F contains both G and H as subgraphs, and the edge set of F is the union of the edge sets of G and phi(H). We consider the following optimization problem. Given graphs G, H, and a weight function omega assigning non-negative weights to pairs of vertices of V(G), the task is to find phi of minimum weight omega(phi)=sum_{xyin E(H)}omega(phi(x)phi(y)) such that the edge connectivity of the superposition F of G and H with respect to phi is higher than the edge connectivity of G. Our main result is the following ``dichotomy\u27\u27 complexity classification. We say that a class of graphs C has bounded vertex-cover number, if there is a constant t depending on C only such that the vertex-cover number of every graph from C does not exceed t. We show that for every class of graphs C with bounded vertex-cover number, the problems of superposing into a connected graph F and to 2-edge connected graph F, are solvable in polynomial time when Hin C. On the other hand, for any hereditary class C with unbounded vertex-cover number, both problems are NP-hard when Hin C. For the unweighted variants of structured augmentation problems, i.e. the problems where the task is to identify whether there is a superposition of graphs of required connectivity, we provide necessary and sufficient combinatorial conditions on the existence of such superpositions. These conditions imply polynomial time algorithms solving the unweighted variants of the problems
Approximating subset -connectivity problems
A subset of terminals is -connected to a root in a
directed/undirected graph if has internally-disjoint -paths for
every ; is -connected in if is -connected to every
. We consider the {\sf Subset -Connectivity Augmentation} problem:
given a graph with edge/node-costs, node subset , and
a subgraph of such that is -connected in , find a
minimum-cost augmenting edge-set such that is
-connected in . The problem admits trivial ratio .
We consider the case and prove that for directed/undirected graphs and
edge/node-costs, a -approximation for {\sf Rooted Subset -Connectivity
Augmentation} implies the following ratios for {\sf Subset -Connectivity
Augmentation}: (i) ; (ii) , where
b=1 for undirected graphs and b=2 for directed graphs, and is the th
harmonic number. The best known values of on undirected graphs are
for edge-costs and for
node-costs; for directed graphs for both versions. Our results imply
that unless , {\sf Subset -Connectivity Augmentation} admits
the same ratios as the best known ones for the rooted version. This improves
the ratios in \cite{N-focs,L}
Spider covers for prize-collecting network activation problem
In the network activation problem, each edge in a graph is associated with an
activation function, that decides whether the edge is activated from
node-weights assigned to its end-nodes. The feasible solutions of the problem
are the node-weights such that the activated edges form graphs of required
connectivity, and the objective is to find a feasible solution minimizing its
total weight. In this paper, we consider a prize-collecting version of the
network activation problem, and present first non- trivial approximation
algorithms. Our algorithms are based on a new LP relaxation of the problem.
They round optimal solutions for the relaxation by repeatedly computing
node-weights activating subgraphs called spiders, which are known to be useful
for approximating the network activation problem
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
Approximating Source Location and Star Survivable Network Problems
In Source Location (SL) problems the goal is to select a mini-mum cost source
set such that the connectivity (or flow) from
to any node is at least the demand of . In many SL problems
if , namely, the demand of nodes selected to is
completely satisfied. In a node-connectivity variant suggested recently by
Fukunaga, every node gets a "bonus" if it is selected to
. Fukunaga showed that for undirected graphs one can achieve ratio for his variant, where is the maximum demand. We
improve this by achieving ratio \min\{p^*\lnk,k\}\cdot O(\ln (k/q^*)) for a
more general version with node capacities, where is
the maximum bonus and is the minimum capacity. In
particular, for the most natural case considered by Fukunaga, we
improve the ratio from to . We also get ratio
for the edge-connectivity version, for which no ratio that depends on only
was known before. To derive these results, we consider a particular case of the
Survivable Network (SN) problem when all edges of positive cost form a star. We
give ratio for this variant, improving over the best
ratio known for the general case of Chuzhoy and Khanna
Non-Uniform Robust Network Design in Planar Graphs
Robust optimization is concerned with constructing solutions that remain
feasible also when a limited number of resources is removed from the solution.
Most studies of robust combinatorial optimization to date made the assumption
that every resource is equally vulnerable, and that the set of scenarios is
implicitly given by a single budget constraint. This paper studies a robustness
model of a different kind. We focus on \textbf{bulk-robustness}, a model
recently introduced~\cite{bulk} for addressing the need to model non-uniform
failure patterns in systems.
We significantly extend the techniques used in~\cite{bulk} to design
approximation algorithm for bulk-robust network design problems in planar
graphs. Our techniques use an augmentation framework, combined with linear
programming (LP) rounding that depends on a planar embedding of the input
graph. A connection to cut covering problems and the dominating set problem in
circle graphs is established. Our methods use few of the specifics of
bulk-robust optimization, hence it is conceivable that they can be adapted to
solve other robust network design problems.Comment: 17 pages, 2 figure
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
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