169 research outputs found
Max-Leaves Spanning Tree is APX-hard for Cubic Graphs
We consider the problem of finding a spanning tree with maximum number of
leaves (MaxLeaf). A 2-approximation algorithm is known for this problem, and a
3/2-approximation algorithm when restricted to graphs where every vertex has
degree 3 (cubic graphs). MaxLeaf is known to be APX-hard in general, and
NP-hard for cubic graphs. We show that the problem is also APX-hard for cubic
graphs. The APX-hardness of the related problem Minimum Connected Dominating
Set for cubic graphs follows
Scale-free spanning trees: complexity, bounds and algorithms
We introduce and study the general problem of finding a most
"scale-free-like" spanning tree of a connected graph. It is motivated by a
particular problem in epidemiology, and may be useful in studies of various
dynamical processes in networks. We employ two possible objective functions for
this problem and introduce the corresponding algorithmic problems termed -SF
and -SF Spanning Tree problems. We prove that those problems are APX- and
NP-hard, respectively, even in the classes of cubic, bipartite and split
graphs. We study the relations between scale-free spanning tree problems and
the max-leaf spanning tree problem, which is the classical algorithmic problem
closest to ours. For split graphs, we explicitly describe the structure of
optimal spanning trees and graphs with extremal solutions. Finally, we propose
two Integer Linear Programming formulations and two fast heuristics for the
-SF Spanning Tree problem, and experimentally assess their performance using
simulated and real data
Approximability of Connected Factors
Finding a d-regular spanning subgraph (or d-factor) of a graph is easy by
Tutte's reduction to the matching problem. By the same reduction, it is easy to
find a minimal or maximal d-factor of a graph. However, if we require that the
d-factor is connected, these problems become NP-hard - finding a minimal
connected 2-factor is just the traveling salesman problem (TSP).
Given a complete graph with edge weights that satisfy the triangle
inequality, we consider the problem of finding a minimal connected -factor.
We give a 3-approximation for all and improve this to an
(r+1)-approximation for even d, where r is the approximation ratio of the TSP.
This yields a 2.5-approximation for even d. The same algorithm yields an
(r+1)-approximation for the directed version of the problem, where r is the
approximation ratio of the asymmetric TSP. We also show that none of these
minimization problems can be approximated better than the corresponding TSP.
Finally, for the decision problem of deciding whether a given graph contains
a connected d-factor, we extend known hardness results.Comment: To appear in the proceedings of WAOA 201
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
Computing graph gonality is hard
There are several notions of gonality for graphs. The divisorial gonality
dgon(G) of a graph G is the smallest degree of a divisor of positive rank in
the sense of Baker-Norine. The stable gonality sgon(G) of a graph G is the
minimum degree of a finite harmonic morphism from a refinement of G to a tree,
as defined by Cornelissen, Kato and Kool. We show that computing dgon(G) and
sgon(G) are NP-hard by a reduction from the maximum independent set problem and
the vertex cover problem, respectively. Both constructions show that computing
gonality is moreover APX-hard.Comment: The previous version only dealt with hardness of the divisorial
gonality. The current version also shows hardness of stable gonality and
discusses the relation between the two graph parameter
Approximating Maximum Weight Cycle Covers in Directed Graphs with Weights Zero and One
A cycle cover of a graph is a spanning subgraph, each node of which is part of exactly one simple cycle. A k-cycle cover is a cycle cover where each cycle has length at least k. Given a complete directed graph with edge weights zero and one, Max-k-DDC(0,1) is the problem of finding a k-cycle cover with maximum weight. We present a 2/3 approximation algorithm for Max-k-DDC(0,1) with running time O(n 5/2). This algorithm yields a 4/3 approximation algorithm for Max-k-DDC(1,2) as well. Instances of the latter problem are complete directed graphs with edge weights one and two. The goal is to find a k-cycle cover with minimum weight. We particularly obtain a 2/3 approximation algorithm for the asymmetric maximum traveling salesman problem with distances zero and one and a 4/3 approximation algorithm for the asymmetric minimum traveling salesman problem with distances one and two. As a lower bound, we prove that Max-k-DDC(0,1) for k ≥ 3 and Max-k-UCC(0,1) (finding maximum weight cycle covers in undirected graphs) for k ≥ 7 are \APX-complet
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