94,093 research outputs found
Algorithms for Cut Problems on Trees
We study the {\sc multicut on trees} and the {\sc generalized multiway Cut on
trees} problems. For the {\sc multicut on trees} problem, we present a
parameterized algorithm that runs in time , where is the positive root of the polynomial
. This improves the current-best algorithm of Chen et al. that runs
in time . For the {\sc generalized multiway cut on trees}
problem, we show that this problem is solvable in polynomial time if the number
of terminal sets is fixed; this answers an open question posed in a recent
paper by Liu and Zhang. By reducing the {\sc generalized multiway cut on trees}
problem to the {\sc multicut on trees} problem, our results give a
parameterized algorithm that solves the {\sc generalized multiway cut on trees}
problem in time , where time
Parameterized Complexity of Critical Node Cuts
We consider the following natural graph cut problem called Critical Node Cut
(CNC): Given a graph on vertices, and two positive integers and
, determine whether has a set of vertices whose removal leaves
with at most connected pairs of vertices. We analyze this problem in the
framework of parameterized complexity. That is, we are interested in whether or
not this problem is solvable in time (i.e., whether
or not it is fixed-parameter tractable), for various natural parameters
. We consider four such parameters:
- The size of the required cut.
- The upper bound on the number of remaining connected pairs.
- The lower bound on the number of connected pairs to be removed.
- The treewidth of .
We determine whether or not CNC is fixed-parameter tractable for each of
these parameters. We determine this also for all possible aggregations of these
four parameters, apart from . Moreover, we also determine whether or not
CNC admits a polynomial kernel for all these parameterizations. That is,
whether or not there is an algorithm that reduces each instance of CNC in
polynomial time to an equivalent instance of size , where
is the given parameter
Counting and enumerating optimum cut sets for hypergraph -partitioning problems for fixed
We consider the problem of enumerating optimal solutions for two hypergraph
-partitioning problems -- namely, Hypergraph--Cut and
Minmax-Hypergraph--Partition. The input in hypergraph -partitioning
problems is a hypergraph with positive hyperedge costs along with a
fixed positive integer . The goal is to find a partition of into
non-empty parts -- known as a -partition -- so as
to minimize an objective of interest.
1. If the objective of interest is the maximum cut value of the parts, then
the problem is known as Minmax-Hypergraph--Partition. A subset of hyperedges
is a minmax--cut-set if it is the subset of hyperedges crossing an optimum
-partition for Minmax-Hypergraph--Partition.
2. If the objective of interest is the total cost of hyperedges crossing the
-partition, then the problem is known as Hypergraph--Cut. A subset of
hyperedges is a min--cut-set if it is the subset of hyperedges crossing an
optimum -partition for Hypergraph--Cut.
We give the first polynomial bound on the number of minmax--cut-sets and a
polynomial-time algorithm to enumerate all of them in hypergraphs for every
fixed . Our technique is strong enough to also enable an -time
deterministic algorithm to enumerate all min--cut-sets in hypergraphs, thus
improving on the previously known -time deterministic algorithm,
where is the number of vertices and is the size of the hypergraph. The
correctness analysis of our enumeration approach relies on a structural result
that is a strong and unifying generalization of known structural results for
Hypergraph--Cut and Minmax-Hypergraph--Partition. We believe that our
structural result is likely to be of independent interest in the theory of
hypergraphs (and graphs).Comment: Accepted to ICALP'22. arXiv admin note: text overlap with
arXiv:2110.1481
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