17 research outputs found
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
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 rooted -connectivity problems in quasi-bipartite digraphs
We consider the directed Rooted Subset -Edge-Connectivity problem: given a
set of terminals in a digraph with edge costs and
an integer , find a min-cost subgraph of that contains edge disjoint
-paths for all . The case when every edge of positive cost has
head in admits a polynomial time algorithm due to Frank, and the case when
all positive cost edges are incident to is equivalent to the -Multicover
problem. Recently, [Chan et al. APPROX20] obtained ratio for
quasi-bipartite instances, when every edge in has an end in . We give
a simple proof for the same ratio for a more general problem of covering an
arbitrary -intersecting supermodular set function by a minimum cost edge
set, and for the case when only every positive cost edge has an end in
Approximating survivable networks with β-metric costs
AbstractThe Survivable Network Design (SND) problem seeks a minimum-cost subgraph that satisfies prescribed node-connectivity requirements. We consider SND on both directed and undirected complete graphs with β-metric costs when c(xz)⩽β[c(xy)+c(yz)] for all x,y,z∈V, which varies from uniform costs (β=1/2) to metric costs (β=1).For the k-Connected Subgraph (k-CS) problem our ratios are: 1+2βk(1−β)−12k−1 for undirected graphs, and 1+4β3k(1−3β2)−12k−1 for directed graphs and 12⩽β<13. For undirected graphs this improves the ratios β1−β of Böckenhauer et al. (2008) [3] and 2+βkn of Kortsarz and Nutov (2003) [11] for all k⩾4 and 12+3k−22(4k2−7k+2)⩽β⩽k2(k+1)2−2. We also show that SND admits the ratios 2β1−β for undirected graphs, and 4β31−3β2 for directed graphs with 1/2⩽β<1/3. For two important particular cases of SND, so-called Subset k-CS and Rooted SND, our ratios are 2β31−3β2 for directed graphs and β1−β for subset k-CS on undirected graphs