379 research outputs found
On Minimum Average Stretch Spanning Trees in Polygonal 2-trees
A spanning tree of an unweighted graph is a minimum average stretch spanning
tree if it minimizes the ratio of sum of the distances in the tree between the
end vertices of the graph edges and the number of graph edges. We consider the
problem of computing a minimum average stretch spanning tree in polygonal
2-trees, a super class of 2-connected outerplanar graphs. For a polygonal
2-tree on vertices, we present an algorithm to compute a minimum average
stretch spanning tree in time. This algorithm also finds a
minimum fundamental cycle basis in polygonal 2-trees.Comment: 17 pages, 12 figure
On unrooted and root-uncertain variants of several well-known phylogenetic network problems
The hybridization number problem requires us to embed a set of binary rooted
phylogenetic trees into a binary rooted phylogenetic network such that the
number of nodes with indegree two is minimized. However, from a biological
point of view accurately inferring the root location in a phylogenetic tree is
notoriously difficult and poor root placement can artificially inflate the
hybridization number. To this end we study a number of relaxed variants of this
problem. We start by showing that the fundamental problem of determining
whether an \emph{unrooted} phylogenetic network displays (i.e. embeds) an
\emph{unrooted} phylogenetic tree, is NP-hard. On the positive side we show
that this problem is FPT in reticulation number. In the rooted case the
corresponding FPT result is trivial, but here we require more subtle
argumentation. Next we show that the hybridization number problem for unrooted
networks (when given two unrooted trees) is equivalent to the problem of
computing the Tree Bisection and Reconnect (TBR) distance of the two unrooted
trees. In the third part of the paper we consider the "root uncertain" variant
of hybridization number. Here we are free to choose the root location in each
of a set of unrooted input trees such that the hybridization number of the
resulting rooted trees is minimized. On the negative side we show that this
problem is APX-hard. On the positive side, we show that the problem is FPT in
the hybridization number, via kernelization, for any number of input trees.Comment: 28 pages, 8 Figure
A Note on the Approximability of Deepest-Descent Circuit Steps
Linear programs (LPs) can be solved by polynomially many moves along the
circuit direction improving the objective the most, so-called deepest-descent
steps (dd-steps). Computing these steps is NP-hard (De Loera et al., arXiv,
2019), a consequence of the hardness of deciding the existence of an optimal
circuit-neighbor (OCNP) on LPs with non-unique optima.
We prove OCNP is easy under the promise of unique optima, but already
-approximating dd-steps remains hard even for totally
unimodular -dimensional 0/1-LPs with a unique optimum. We provide a matching
-approximation
Hardness and Approximation of Submodular Minimum Linear Ordering Problems
The minimum linear ordering problem (MLOP) generalizes well-known
combinatorial optimization problems such as minimum linear arrangement and
minimum sum set cover. MLOP seeks to minimize an aggregated cost due
to an ordering of the items (say ), i.e., , where is the set of items
mapped by to indices . Despite an extensive literature on MLOP
variants and approximations for these, it was unclear whether the graphic
matroid MLOP was NP-hard. We settle this question through non-trivial
reductions from mininimum latency vertex cover and minimum sum vertex cover
problems. We further propose a new combinatorial algorithm for approximating
monotone submodular MLOP, using the theory of principal partitions. This is in
contrast to the rounding algorithm by Iwata, Tetali, and Tripathi [ITT2012],
using Lov\'asz extension of submodular functions. We show a
-approximation for monotone submodular MLOP where
satisfies . Our theory provides new approximation bounds for special cases of the
problem, in particular a -approximation for the
matroid MLOP, where is the rank function of a matroid. We further show
that minimum latency vertex cover (MLVC) is -approximable, by
which we also lower bound the integrality gap of its natural LP relaxation,
which might be of independent interest
A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms
Parameterization and approximation are two popular ways of coping with
NP-hard problems. More recently, the two have also been combined to derive many
interesting results. We survey developments in the area both from the
algorithmic and hardness perspectives, with emphasis on new techniques and
potential future research directions
A New Bound on the Length of Minimum Cycle Bases
For any weighted graph we construct a cycle basis of length O(W*log(n)*log(log(n))), where W denotes the sum of the weights of the edges. This improves the upper bound that was obtained only recently by Elkin et al. (2005) by a logarithmic factor. From below, our result has to be compared with Ω(W*log(n)), being the length of the minimum cycle bases (MCB) of a class of graphs with large girth
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