14,442 research outputs found
Solving hard cut problems via flow-augmentation
We present a new technique for designing FPT algorithms for graph cut
problems in undirected graphs, which we call flow augmentation. Our technique
is applicable to problems that can be phrased as a search for an (edge)
-cut of cardinality at most in an undirected graph with
designated terminals and .
More precisely, we consider problems where an (unknown) solution is a set of size at most such that (1) in , and are in
distinct connected components, (2) every edge of connects two distinct
connected components of , and (3) if we define the set as these edges for which there exists an -path with
, then separates from . We prove that
in this scenario one can in randomized time add a
number of edges to the graph so that with probability no
added edge connects two components of and becomes a minimum cut
between and .
We apply our method to obtain a randomized FPT algorithm for a notorious
"hard nut" graph cut problem we call Coupled Min-Cut. This problem emerges out
of the study of FPT algorithms for Min CSP problems, and was unamenable to
other techniques for parameterized algorithms in graph cut problems, such as
Randomized Contractions, Treewidth Reduction or Shadow Removal.
To demonstrate the power of the approach, we consider more generally Min
SAT(), parameterized by the solution cost. We show that every problem
Min SAT() is either (1) FPT, (2) W[1]-hard, or (3) able to express the
soft constraint , and thereby also the min-cut problem in directed
graphs. All the W[1]-hard cases were known or immediate, and the main new
result is an FPT algorithm for a generalization of Coupled Min-Cut
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}
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
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