32,726 research outputs found
On Single-Objective Sub-Graph-Based Mutation for Solving the Bi-Objective Minimum Spanning Tree Problem
We contribute to the efficient approximation of the Pareto-set for the
classical -hard multi-objective minimum spanning tree problem
(moMST) adopting evolutionary computation. More precisely, by building upon
preliminary work, we analyse the neighborhood structure of Pareto-optimal
spanning trees and design several highly biased sub-graph-based mutation
operators founded on the gained insights. In a nutshell, these operators
replace (un)connected sub-trees of candidate solutions with locally optimal
sub-trees. The latter (biased) step is realized by applying Kruskal's
single-objective MST algorithm to a weighted sum scalarization of a sub-graph.
We prove runtime complexity results for the introduced operators and
investigate the desirable Pareto-beneficial property. This property states that
mutants cannot be dominated by their parent. Moreover, we perform an extensive
experimental benchmark study to showcase the operator's practical suitability.
Our results confirm that the sub-graph based operators beat baseline algorithms
from the literature even with severely restricted computational budget in terms
of function evaluations on four different classes of complete graphs with
different shapes of the Pareto-front
A simple linear time algorithm for the locally connected spanning tree problem on maximal planar chordal graphs
A locally connected spanning tree (LCST) T of a graph G is a spanning tree of G such that, for each node, its neighborhood in T induces a connected sub- graph in G. The problem of determining whether a graph contains an LCST or not has been proved to be NP-complete, even if the graph is planar or chordal. The main result of this paper is a simple linear time algorithm that, given a maximal planar chordal graph, determines in linear time whether it contains an LCST or not, and produces one if it exists. We give an anal- ogous result for the case when the input graph is a maximal outerplanar graph
A Local Algorithm for Constructing Spanners in Minor-Free Graphs
Constructing a spanning tree of a graph is one of the most basic tasks in
graph theory. We consider this problem in the setting of local algorithms: one
wants to quickly determine whether a given edge is in a specific spanning
tree, without computing the whole spanning tree, but rather by inspecting the
local neighborhood of . The challenge is to maintain consistency. That is,
to answer queries about different edges according to the same spanning tree.
Since it is known that this problem cannot be solved without essentially
viewing all the graph, we consider the relaxed version of finding a spanning
subgraph with edges (where is the number of vertices and
is a given sparsity parameter). It is known that this relaxed
problem requires inspecting edges in general graphs, which
motivates the study of natural restricted families of graphs. One such family
is the family of graphs with an excluded minor. For this family there is an
algorithm that achieves constant success probability, and inspects
edges (for each edge it is queried
on), where is the maximum degree in the graph and is the size of the
excluded minor. The distances between pairs of vertices in the spanning
subgraph are at most a factor of larger than in
.
In this work, we show that for an input graph that is -minor free for any
of size , this task can be performed by inspecting only edges. The distances between pairs of vertices in the spanning
subgraph are at most a factor of larger
than in . Furthermore, the error probability of the new algorithm is
significantly improved to . This algorithm can also be easily
adapted to yield an efficient algorithm for the distributed setting
- …