29 research outputs found
Local Maximum Stable Sets Greedoids Stemmed from Very Well-Covered Graphs
A maximum stable set in a graph G is a stable set of maximum cardinality. S
is called a local maximum stable set of G if S is a maximum stable set of the
subgraph induced by the closed neighborhood of S. A greedoid (V,F) is called a
local maximum stable set greedoid if there exists a graph G=(V,E) such that its
family of local maximum stable sets coinsides with (V,F). It has been shown
that the family local maximum stable sets of a forest T forms a greedoid on its
vertex set. In this paper we demonstrate that if G is a very well-covered
graph, then its family of local maximum stable sets is a greedoid if and only
if G has a unique perfect matching.Comment: 12 pages, 12 figure
Computing Unique Maximum Matchings in O(m) time for Konig-Egervary Graphs and Unicyclic Graphs
Let alpha(G) denote the maximum size of an independent set of vertices and
mu(G) be the cardinality of a maximum matching in a graph G. A matching
saturating all the vertices is perfect. If alpha(G) + mu(G) equals the number
of vertices of G, then it is called a Konig-Egervary graph. A graph is
unicyclic if it has a unique cycle.
In 2010, Bartha conjectured that a unique perfect matching, if it exists, can
be found in O(m) time, where m is the number of edges.
In this paper we validate this conjecture for Konig-Egervary graphs and
unicylic graphs. We propose a variation of Karp-Sipser leaf-removal algorithm
(Karp and Spiser, 1981), which ends with an empty graph if and only if the
original graph is a Konig-Egervary graph with a unique perfect matching
obtained as an output as well.
We also show that a unicyclic non-bipartite graph G may have at most one
perfect matching, and this is the case where G is a Konig-Egervary graph.Comment: 10 pages, 5 figure
Greedy and dynamic programming by calculation
Dissertação mestrado integrado em Informatics EngineeringThe mathematical study of the greedy algorithm provides a blueprint for the study of Dynamic Programming (DP),
whose body of knowledge is largely unorganized, remaining obscure to a large part of the software engineering
community. This study aims to structure this body of knowledge, narrowing the gap between a purely examplebased
approach to DP and its scientific foundations. To that effect, matroid theory is leveraged through a pointfree
relation algebra, which is applied to greedy and DP problems. A catalogue of such problems is compiled, and a
broad characterization of DP algorithms is given. Alongside, the theory underlying the thinning relational operator
is explored.O estudo matemático do algoritmo ganancioso («greedy») serve como guia para o estudo da programação
dinâmica, cujo corpo de conhecimento permanece desorganizado e obscuro a uma grande parte da comunidade
de engenharia de software. Este estudo visa estruturar esse corpo de conhecimento, fazendo a ponte entre a
abordagem popular baseada em exemplos e os métodos mais teóricos da literatura científica. Para esse efeito,
a teoria dos matroides é explorada pelo uso de uma álgebra de relações pointfree, e aplicada a problemas
«greedy» e de programação dinâmica. Um catálogo de tais problemas é compilado, e é feita uma caraterização
geral de algoritmos de programação dinâmica. Em paralelo, é explorada a teoria do combinador relacional de
«thinning».This work is financed by National Funds through the Portuguese funding agency, FCT - Fundação para a
Ciência e a Tecnologia, within project UIDB/50014/202
A system-theoretic approach to multi-agent models
A system-theoretic model for cooperative settings is presented that unifies and ex-
tends the models of classical cooperative games and coalition formation processes and
their generalizations. The model is based on the notions of system, state and transi-
tion graph. The latter describes changes of a system over time in terms of actions
governed by individuals or groups of individuals. Contrary to classic models, the pre-
sented model is not restricted to acyclic settings and allows the transition graph to have
cycles.
Time-dependent solutions to allocation problems are proposed and discussed. In par-
ticular, Weber’s theory of randomized values is generalized as well as the notion of
semi-values. Convergence assertions are made in some cases, and the concept of the
Cesàro value of an allocation mechanism is introduced in order to achieve convergence
for a wide range of allocation mechanisms. Quantum allocation mechanisms are de-
fined, which are induced by quantum random walks on the transition graph and it is
shown that they satisfy certain fairness criteria. A concept for Weber sets and two dif-
ferent concepts of cores are proposed in the acyclic case, and it is shown under some
mild assumptions that both cores are subsets of the Weber set.
Moreover, the model of non-cooperative games in extensive form is generalized such
that the presented model achieves a mutual framework for cooperative and non-co-
operative games. A coherency to welfare economics is made and to each allocation
mechanism a social welfare function is proposed