110 research outputs found
Exploiting -Closure in Kernelization Algorithms for Graph Problems
A graph is c-closed if every pair of vertices with at least c common
neighbors is adjacent. The c-closure of a graph G is the smallest number such
that G is c-closed. Fox et al. [ICALP '18] defined c-closure and investigated
it in the context of clique enumeration. We show that c-closure can be applied
in kernelization algorithms for several classic graph problems. We show that
Dominating Set admits a kernel of size k^O(c), that Induced Matching admits a
kernel with O(c^7*k^8) vertices, and that Irredundant Set admits a kernel with
O(c^(5/2)*k^3) vertices. Our kernelization exploits the fact that c-closed
graphs have polynomially-bounded Ramsey numbers, as we show
Approximation Algorithms for Mixed, Windy, and Capacitated Arc Routing Problems
We show that any alpha(n)-approximation algorithm for the n-vertex metric asymmetric Traveling Salesperson problem yields O(alpha(C))-approximation algorithms for various mixed, windy, and capacitated arc routing problems. Herein, C is the number of weakly-connected components in the subgraph induced by the positive-demand arcs, a number that can be expected to be small in applications. In conjunction with known results, we derive constant-factor approximations if C is in O(log n) and O(log(C)/log(log(C)))-approximations in general
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
A Multivariate Complexity Analysis of the Generalized Noah's Ark Problem
In the Generalized Noah's Ark Problem, one is given a phylogenetic tree on a
set of species and a set of conservation projects for each species. Each
project comes with a cost and raises the survival probability of the
corresponding species. The aim is to select for each species a conservation
project such that the total cost of the selected projects does not exceed some
given threshold and the expected phylogenetic diversity is as large as
possible. We study Generalized Noah's Ark Problem and some of its special cases
with respect to several parameters related to the input structure such as the
number of different costs, the number of different survival probabilities, or
the number of species,
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