arXiv subjects

George B. Mertzios

Publications and source records attributed to George B. Mertzios.

58 records · Page 4Linked to original sources

arXiv2012

On the Fixation Probability of Superstars

The Moran process models the spread of genetic mutations through a population. A mutant with relative fitness $r$ is introduced into a population and the system evolves, either reaching fixation (in which every individual is a mutant) or extinction (in which none is). In a widely cited paper (Nature, 2005), Lieberman, Hauert and Nowak generalize the model to populations on the vertices of graphs. They describe a class of graphs (called "superstars"), with a parameter $k$. Superstars are designed to have an increasing fixation probability as $k$ increases. They state that the probability of fixation tends to $1-r^{-k}$ as graphs get larger but we show that this claim is untrue as stated. Specifically, for $k=5$, we show that the true fixation probability (in the limit, as graphs get larger) is at most $1-1/j(r)$ where $j(r)=Θ(r^4)$, contrary to the claimed result. We do believe that the qualitative claim of Lieberman et al.\ --- that the fixation probability of superstars tends to 1 as $k$ increases --- is correct, and that it can probably be proved along the lines of their sketch. We were able to run larger computer simulations than the ones presented in their paper. However, simulations on graphs of around 40,000 vertices do not support their claim. Perhaps these graphs are too small to exhibit the limiting behaviour.

cs.CE↗

arXiv2010

A Simple Polynomial Algorithm for the Longest Path Problem on Cocomparability Graphs

Given a graph $G$, the longest path problem asks to compute a simple path of $G$ with the largest number of vertices. This problem is the most natural optimization version of the well known and well studied Hamiltonian path problem, and thus it is NP-hard on general graphs. However, in contrast to the Hamiltonian path problem, there are only few restricted graph families such as trees and some small graph classes where polynomial algorithms for the longest path problem have been found. Recently it has been shown that this problem can be solved in polynomial time on interval graphs by applying dynamic programming to a characterizing ordering of the vertices of the given graph \cite{longest-int-algo}, thus answering an open question. In the present paper, we provide the first polynomial algorithm for the longest path problem on a much greater class, namely on cocomparability graphs. Our algorithm uses a similar - but essentially simpler - dynamic programming approach, which is applied to a Lexicographic Depth First Search (LDFS) characterizing ordering of the vertices of a cocomparability graph. Therefore, our results provide evidence that this general dynamic programming approach can be used in a more general setting, leading to efficient algorithms for the longest path problem on greater classes of graphs. LDFS has recently been introduced in \cite{Corneil-LDFS08}. Since then, a similar phenomenon of extending an existing interval graph algorithm to cocomparability graphs by using an LDFS preprocessing step has also been observed for the minimum path cover problem \cite{Corneil-MPC}. Therefore, more interestingly, our results also provide evidence that cocomparability graphs present an interval graph structure when they are considered using an LDFS ordering of their vertices, which may lead to other new and more efficient combinatorial algorithms.

cs.DM↗

arXiv2010

The Recognition of Tolerance and Bounded Tolerance Graphs

Tolerance graphs model interval relations in such a way that intervals can tolerate a certain degree of overlap without being in conflict. This subclass of perfect graphs has been extensively studied, due to both its interesting structure and its numerous applications. Several efficient algorithms for optimization problems that are NP-hard on general graphs have been designed for tolerance graphs. In spite of this, the recognition of tolerance graphs - namely, the problem of deciding whether a given graph is a tolerance graph - as well as the recognition of their main subclass of bounded tolerance graphs, have been the most fundamental open problems on this class of graphs (cf. the book on tolerance graphs \cite{GolTol04}) since their introduction in 1982 \cite{GoMo82}. In this article we prove that both recognition problems are NP-complete, even in the case where the input graph is a trapezoid graph. The presented results are surprising because, on the one hand, most subclasses of perfect graphs admit polynomial recognition algorithms and, on the other hand, bounded tolerance graphs were believed to be efficiently recognizable as they are a natural special case of trapezoid graphs (which can be recognized in polynomial time) and share a very similar structure with them. For our reduction we extend the notion of an \emph{acyclic orientation} of permutation and trapezoid graphs. Our main tool is a new algorithm that uses \emph{vertex splitting} to transform a given trapezoid graph into a permutation graph, while preserving this new acyclic orientation property. This method of vertex splitting is of independent interest; very recently, it has been proved a powerful tool also in the design of efficient recognition algorithms for other classes of graphs \cite{MC-Trapezoid}.

cs.CC↗

arXiv2008

A polynomial algorithm for the k-cluster problem on interval graphs

This paper deals with the problem of finding, for a given graph and a given natural number k, a subgraph of k nodes with a maximum number of edges. This problem is known as the k-cluster problem and it is NP-hard on general graphs as well as on chordal graphs. In this paper, it is shown that the k-cluster problem is solvable in polynomial time on interval graphs. In particular, we present two polynomial time algorithms for the class of proper interval graphs and the class of general interval graphs, respectively. Both algorithms are based on a matrix representation for interval graphs. In contrast to representations used in most of the previous work, this matrix representation does not make use of the maximal cliques in the investigated graph.

cs.DS↗