11 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}
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 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
Streaming Algorithms for Connectivity Augmentation
We study the -connectivity augmentation problem (-CAP) in the
single-pass streaming model. Given a -edge connected graph
that is stored in memory, and a stream of weighted edges with weights in
, the goal is to choose a minimum weight subset such that is -edge connected. We give a
-approximation algorithm for this problem which requires to store
words. Moreover, we show our result is tight: Any
algorithm with better than -approximation for the problem requires
bits of space even when . This establishes a gap between the
optimal approximation factor one can obtain in the streaming vs the offline
setting for -CAP.
We further consider a natural generalization to the fully streaming model
where both and arrive in the stream in an arbitrary order. We show that
this problem has a space lower bound that matches the best possible size of a
spanner of the same approximation ratio. Following this, we give improved
results for spanners on weighted graphs: We show a streaming algorithm that
finds a -approximate weighted spanner of size at most
for integer , whereas the best prior
streaming algorithm for spanner on weighted graphs had size depending on . Using our spanner result, we provide an optimal -approximation for
-CAP in the fully streaming model with words of space.
Finally we apply our results to network design problems such as Steiner tree
augmentation problem (STAP), -edge connected spanning subgraph (-ECSS),
and the general Survivable Network Design problem (SNDP). In particular, we
show a single-pass -approximation for SNDP using
words of space, where is the maximum connectivity requirement