6,098 research outputs found
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
Greedy Randomized Adaptive Search and Variable Neighbourhood Search for the minimum labelling spanning tree problem
This paper studies heuristics for the minimum labelling spanning tree (MLST) problem. The purpose is to find a spanning tree using edges that are as similar as possible. Given an undirected labelled connected graph, the minimum labelling spanning tree problem seeks a spanning tree whose edges have the smallest number of distinct labels. This problem has been shown to be NP-hard. A Greedy Randomized Adaptive Search Procedure (GRASP) and a Variable Neighbourhood Search (VNS) are proposed in this paper. They are compared with other algorithms recommended in the literature: the Modified Genetic Algorithm and the Pilot Method. Nonparametric statistical tests show that the heuristics based on GRASP and VNS outperform the other algorithms tested. Furthermore, a comparison with the results provided by an exact approach shows that we may quickly obtain optimal or near-optimal solutions with the proposed heuristics
Pair-Linking for Collective Entity Disambiguation: Two Could Be Better Than All
Collective entity disambiguation aims to jointly resolve multiple mentions by
linking them to their associated entities in a knowledge base. Previous works
are primarily based on the underlying assumption that entities within the same
document are highly related. However, the extend to which these mentioned
entities are actually connected in reality is rarely studied and therefore
raises interesting research questions. For the first time, we show that the
semantic relationships between the mentioned entities are in fact less dense
than expected. This could be attributed to several reasons such as noise, data
sparsity and knowledge base incompleteness. As a remedy, we introduce MINTREE,
a new tree-based objective for the entity disambiguation problem. The key
intuition behind MINTREE is the concept of coherence relaxation which utilizes
the weight of a minimum spanning tree to measure the coherence between
entities. Based on this new objective, we design a novel entity disambiguation
algorithms which we call Pair-Linking. Instead of considering all the given
mentions, Pair-Linking iteratively selects a pair with the highest confidence
at each step for decision making. Via extensive experiments, we show that our
approach is not only more accurate but also surprisingly faster than many
state-of-the-art collective linking algorithms
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