14 research outputs found
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}
A logarithmic approximation algorithm for the activation edge multicover problem
In the Activation Edge-Multicover problem we are given a multigraph
with activation costs for every edge , and
degree requirements . The goal is to find an edge subset of minimum activation cost ,such that every has at least neighbors in the graph
. Let be the maximum requirement and let
be
the maximum quotient between the two costs of an edge. For the
problem admits approximation ratio . For it generalizes the
Set Cover problem (when ), and admits a tight approximation
ratio . This implies approximation ratio for general
and , and no better approximation ratio was known. We obtain the
first logarithmic approximation ratio , that
bridges between the two known ratios -- for and for . This implies approximation ratio for the Activation
-Connected Subgraph problem, where is the best known approximation
ratio for the ordinary min-cost version of the problem
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
Approximating Minimum-Cost k-Node Connected Subgraphs via Independence-Free Graphs
We present a 6-approximation algorithm for the minimum-cost -node
connected spanning subgraph problem, assuming that the number of nodes is at
least . We apply a combinatorial preprocessing, based on the
Frank-Tardos algorithm for -outconnectivity, to transform any input into an
instance such that the iterative rounding method gives a 2-approximation
guarantee. This is the first constant-factor approximation algorithm even in
the asymptotic setting of the problem, that is, the restriction to instances
where the number of nodes is lower bounded by a function of .Comment: 20 pages, 1 figure, 28 reference
Approximating k-Connected m-Dominating Sets
A subset of nodes in a graph is a -connected -dominating set
(-cds) if the subgraph induced by is -connected and every
has at least neighbors in . In the -Connected
-Dominating Set (-CDS) problem the goal is to find a minimum weight
-cds in a node-weighted graph. For we obtain the following
approximation ratios. For general graphs our ratio improves the
previous best ratio and matches the best known ratio for unit
weights. For unit disc graphs we improve the ratio to
-- this is the
first sublinear ratio for the problem, and the first polylogarithmic ratio
when ; furthermore, we obtain ratio
for uniform
weights. These results are obtained by showing the same ratios for the Subset
-Connectivity problem when the set of terminals is an -dominating set
with
Approximating minimum cost connectivity problems
We survey approximation algorithms of connectivity problems.
The survey presented describing various techniques. In the talk the following techniques and results are presented.
1)Outconnectivity: Its well known that there exists a polynomial time algorithm to solve the problems of finding an edge k-outconnected from r subgraph [EDMONDS] and a vertex k-outconnectivity subgraph from r [Frank-Tardos] .
We show how to use this to obtain a ratio 2 approximation for the min cost edge k-connectivity
problem.
2)The critical cycle theorem of Mader: We state a fundamental theorem of Mader and use it to provide a 1+(k-1)/n ratio approximation for the min cost vertex k-connected subgraph, in the metric case.
We also show results for the min power vertex k-connected problem using this lemma.
We show that the min power is equivalent to the min-cost case with respect to approximation.
3)Laminarity and uncrossing: We use the well known laminarity of a BFS solution and show a simple new proof due to Ravi et al for Jain\u27s 2 approximation for Steiner network