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### Convergence to the Tracy-Widom distribution for longest paths in a directed random graph [article]

Takis Konstantopoulos, Katja Trinajstić
2013 arXiv   pre-print
Let L_n,m denote the maximum length of all paths contained in an n × m rectangle.  ...  We consider a directed graph on the 2-dimensional integer lattice, placing a directed edge from vertex (i_1,i_2) to (j_1,j_2), whenever i_1 < j_1, i_2 < j_2, with probability p, independently for each  ...  Consider the directed random graph on Z × Z and let L n,m be the maximum length of all paths between two vertices in [0, n] × [1, m] .  ...

### The longest path in a random graph

Miklós Ajtai, János Komlós, Endre Szemerédi
1981
A random graph with (1+e)n/2-edges contains a path of length cn. A random directed graph with (1+e)n edges contains a directed path of length cn. This settles a conjecture of Erdôs.  ...  Since the longest path in a random graph with (1e)-^-edges (resp. the longest directed path in a random directed graph with (1-e)n edges) is of length O(logn) with a fl probability near 1, while in a random  ...  Introduction A. We are going to investigate the length of the longest path (directed path) in a random graph.  ...

### Convergence to the Tracy-Widom distribution for longest paths in a directed random graph

Alea, Lat
2013 Am. J. Probab. Math. Stat   unpublished
Let L n,m denote the maximum length of all paths contained in an n × m rectangle.  ...  We consider a directed graph on the 2-dimensional integer lattice, placing a directed edge from vertex (i 1 , i 2) to (j 1 , j 2), whenever i 1 ≤ j 1 , i 2 ≤ j 2 , with probability p, independently for  ...  Consider the directed random graph on Z × Z and let L n,m be the maximum length of all paths between two vertices in [0, n] × [1, m].  ...

### Random graph generation for scheduling simulations

Daniel Cordeiro, Grégory Mounié, Swann Perarnau, Denis Trystram, Jean-Marc Vincent, Frédéric Wagner
2010 Proceedings of the 3rd International ICST Conference on Simulation Tools and Techniques
In order to provide a comparison basis we propose GGen -a unified and standard implementation of classical task graph generation methods used in the scheduling domain.  ...  Moreover, different implementations of the same randomized generation method may produce slightly different graphs. These problems can harm the experimental comparison of scheduling algorithms.  ...  ACKNOWLEDGMENTS This work was supported by the GdR RO (Operational Research) of the CNRS.  ...

### Continuum Cascade Model of Directed Random Graphs: Traveling Wave Analysis [article]

Yoshiaki Itoh, P. L. Krapivsky
2012 arXiv   pre-print
We study a class of directed random graphs.  ...  We analyze the length of the longest directed path starting from the origin. In the large x limit, we employ traveling wave techniques to extract the asymptotic behavior of this quantity.  ...  Continuum Cascade Model of Directed Random Graphs: Traveling Wave Analysis 11  ...

### Longest-path attacks on complex networks [article]

Cunlai Pu, Wei Cui
2014 arXiv   pre-print
We propose two algorithms, the random augmenting approach (RPA) and the Hamilton-path based approach (HPA), for finding the approximately longest simple path in a network.  ...  We investigate the longest-path attacks on complex networks. Specifically, we remove approximately the longest simple path from a network iteratively until there are no paths left in the network.  ...  Finding the longest paths in general graphs is a well-known NP-hard problem in the literature  .  ...

### The Longest Path in the Price Model [article]

Tim Evans
2019 Figshare
Mathematically, one distinctive property of a Directed Acyclic Graph is that the longest path between nodes is both well defined and often more meaningful than the shortest path.  ...  Edges are therefore directed and there are no cycles in the network so a citation network is an example of a Directed Acyclic Graph.  ...  1-p) Random Attachment p Preferential Attachment The Longest Path in Price Model • The longest path is well-defined in a DAG • Approximates geodesic in Minkowski random DAGs [Brightwell & Gregory  ...

### Limit Distribution for the Existence of Hamiltonian Cycles in Random Bipartite Graphs

A.M. Frieze
1985 European journal of combinatorics (Print)
A random bipartite graph D with 2n vertices is generated by allowing each of the n 2 possible edges to occur with probability p = (log n + log log n + e n )/ n.  ...  We show that lim P(D contains a Hamiltonian cycle) = {  ...  longest path (2.lb) In the next lemma we have a bipartite graph BG E EVEN.  ...

### Longest Path in the Price Model [article]

Tim S. Evans, Lucille Calmon, Vaiva Vasiliauskaite
2020 arXiv   pre-print
In such networks, the longest path is well defined and in some cases is known to be a better approximation to geodesics than the shortest path.  ...  This is a lower bound on the length of the longest path to any given vertex and we show numerically that the longest path also scales with the logarithm of the size of the network but with a larger coefficient  ...  Author contributions statement T.S.E. derived the analytical solution. L.C. and V.V. conducted the data analysis. T.S.E. and V.V. wrote the manuscript.  ...

### Game-theoretic timing analysis

Sanjit A. Seshia, Alexander Rakhlin
2008 2008 IEEE/ACM International Conference on Computer-Aided Design
We model the estimation problem as a game between our algorithm (player) and the environment of the program (adversary), where the player seeks to find the longest path through the program while the adversary  ...  On the theoretical side, we prove that our algorithm can converge to find the longest path with high probability.  ...  Jha, and the anonymous reviewers for their inputs. The first author was supported in part by NSF CAREER grant CNS-0644436, and the second author by DARPA grant FA8750-05-2-0249.  ...

### First Passage Percolation on Random Geometric Graphs and an Application to Shortest-Path Trees

C. Hirsch, D. Neuhäuser, C. Gloaguen, V. Schmidt
2015 Advances in Applied Probability
length of the longest branch in the shortest-path tree extracted from a typical segment system if the intensity of network stations converges to 0.  ...  In particular, we establish a strong linear growth property for shortest-path lengths on random geometric graphs which are generated by point processes.  ...  the edge set G (1) formally, G * is the Palm version of G Shortest-path tree consider the graph G * obtained from G by placing the origin at random on the edge set G (1) formally, G * is the Palm  ...

### Finding long simple paths in a weighted digraph using pseudo-topological orderings [article]

Miguel Raggi
2018 arXiv   pre-print
Given a weighted digraph D, finding the longest simple path is well known to be NP-hard. Furthermore, even giving an approximation algorithm is known to be NP-hard.  ...  acyclic graphs, via a process we call "opening edges".  ...  Acknowledgements We would like to thank the organizers of the Oracle MDC coding challenge for providing a very interesting problem to work on (and for the prize of a drone and camera, of course).  ...

### The longest path in the Price model

Tim S. Evans, Lucille Calmon, Vaiva Vasiliauskaite
2020 Scientific Reports
In such networks, the longest path is well defined and in some cases is known to be a better approximation to geodesics than the shortest path.  ...  This is a lower bound on the length of the longest path to any given vertex and we show numerically that the longest path also scales with the logarithm of the size of the network but with a larger coefficient  ...  In directed graphs with cycles, the longest path is infinite, if multiple visits to a node are allowed.  ...

### Going Far From Degeneracy [article]

Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Meirav Zehavi
2019 arXiv   pre-print
In other words, deciding whether a 2-connected n-vertex G contains a cycle of length at least d+log n can be done in polynomial time. Similar algorithmic results hold for long paths in graphs.  ...  We observe that deciding whether a graph has a path of length at least d+1 is NP-complete.  ...  Acknowledgement We thank Nikolay Karpov for communicating to us the question of finding a path above the degeneracy bound and Proposition 3.  ...

### On the existence of Hamiltonian cycles in a class of random graphs

T.I. Fenner, A.M. Frieze
1983 Discrete Mathematics
We consider the probability of the existence of a Hamiltonian cycle in the graph obtained by ignoring arc orientation.  ...  A digraph with n vertices and fixed outdegree m is generated randomly so that each such digraph is equally likely to be chosen.  ...  A graph G has property LC if a longest cycle of G has the same number of vertices as a longest path of G. Suppose then that Di E I3 and GDi has a longest path P with k vertices.  ...
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