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On the Graph of the Pedigree Polytope [article]

Abdullah Makkeh and Mozhgan Pourmoradnasseri and Dirk Oliver Theis
2016 arXiv   pre-print
Based on this characterization, we study the graphs of Pedigree polytopes asymptotically, for large numbers of cities.  ...  Pedigree polytopes are extensions of the classical Symmetric Traveling Salesman Problem polytopes whose graphs (1-skeletons) contain the TSP polytope graphs as spanning subgraphs.  ...  Acknowledgments The authors would like to thank Kaveh Khoshkhah for pointing us to the idea of analyzing the pair pS, T q of random variables.  ... 
arXiv:1611.08431v1 fatcat:qag7etyx3jchldpozkpvyv2ewq

The Graph of the Pedigree Polytope is Asymptotically Almost Complete (Extended Abstract) [article]

Abdullah Makkeh and Mozhgan Pourmoradnasseri and Dirk Oliver Theis
2016 arXiv   pre-print
We show that in the graph of the pedigree polytope, the quotient minimum degree over number of vertices tends to 1 as the number of cities tends to infinity.  ...  Pedigree polytopes are extensions of the classical Symmetric Traveling Salesman Problem polytopes (Arthanari 2000) whose graphs contain the TSP polytope graphs as spanning subgraphs (Arthanari 2013).  ...  Moreover, the graphs of the TSP polytopes are spanning subgraphs of the graphs of the Pedigree polytopes [3] .  ... 
arXiv:1611.08419v2 fatcat:sfwymizqynaszo4er6fkoob2fe

On Pedigree Polytopes and Hamiltonian Cycles

Tim S. Arthanari
2003 Electronic Notes in Discrete Mathematics  
Interestingly, the pedigree polytope seems to differ from the standard tour polytope, Q n with respect to the complexity of testing whether two given vertices of the polytope are nonadjacent.  ...  Pedigrees are in one-to-one correspondence with the Hamiltonian cycles on K n .  ...  Acknowledgments The author thanks Prof. Ananth Srinivasan, Head, Department of ISOM, for the support to attend the Symposium and for the encouragement during the preparation of the paper.  ... 
doi:10.1016/s1571-0653(04)00515-3 fatcat:awrrr6wuaref3is4esuwc2v36e

On pedigree polytopes and Hamiltonian cycles

T.S. Arthanari
2006 Discrete Mathematics  
Interestingly, the pedigree polytope seems to differ from the standard tour polytope, Q n with respect to the complexity of testing whether two given vertices of the polytope are nonadjacent.  ...  Pedigrees are in one-to-one correspondence with the Hamiltonian cycles on K n .  ...  Acknowledgments The author thanks Prof. Ananth Srinivasan, Head, Department of ISOM, for the support to attend the Symposium and for the encouragement during the preparation of the paper.  ... 
doi:10.1016/j.disc.2005.11.030 fatcat:kguxuwjmrfbchjwmrmmpis7aze

Two Algorithmic Results for the Traveling Salesman Problem

Alexander I. Barvinok
1996 Mathematics of Operations Research  
Polyhedral combinatorics deals with the study of polytopes with the corner points (vertices) corresponding to objects of interest like the tours (Korte and Vygen 2012).  ...  We focus on one compact formulation for the symmetric TSP, known as the "multistage insertion" formulation [T. S.  ...  Maria Grazia Scutellà for access to a copy of a paper on the experiments with Hypergraph Simplex method.  ... 
doi:10.1287/moor.21.1.65 fatcat:revgcqn765bqheijf3i6o7tl7q

Contents

2006 Discrete Mathematics  
Arthanari On pedigree polytopes and Hamiltonian cycles 1474 V. Swaminathan and P. Jeyanthi Super edge-magic strength of fire crackers, banana trees and unicyclic graphs 1624 A. Baartmans and S.  ...  Vijayakumar On sorting by 3-bounded transpositions 1569 S. Mishra and S.B. Rao Minimum monopoly in regular and tree graphs 1586 A.R. Rao The number of reachable pairs in a digraph 1595 M.  ... 
doi:10.1016/s0012-365x(06)00448-1 fatcat:omtgfgm4vrf5lkty3znachfwp4

Hamiltonian decomposition and verifying vertex adjacency in 1-skeleton of the traveling salesperson polytope by variable neighborhood search [article]

Andrei Nikolaev, Anna Kozlova
2020 arXiv   pre-print
A sufficient condition for vertex adjacency in the 1-skeleton of the traveling salesperson polytope can be formulated as the Hamiltonian decomposition problem in a 4-regular multigraph.  ...  We consider a Hamiltonian decomposition problem of partitioning a regular graph into edge-disjoint Hamiltonian cycles.  ...  We are very grateful to the anonymous reviewers for their comments and suggestions which helped to improve the presentation of the results in this paper.  ... 
arXiv:2001.04683v2 fatcat:4vyd7dh3evgohegnty7c5kazsa

G. M. Ziegler Lectures on polytopes (Graduate Texts in Mathematics, Vol. 152, Springer-Verlag, Berlin-Heidelberg-New York-London-Paris-Tokyo-Hong Kong 1995), ix + 370 pp., softcover: 3 540 94365 X, £21, hardcover: 3 540 94329 3, £47

P. McMullen
1996 Proceedings of the Edinburgh Mathematical Society  
n facets and his proof of the fact that the graph of a simple dpolytope (one with just d facets through each vertex) determines its combinatorial type.  ...  Graphs of polytopes (formed by their vertices and edges) are dealt with in Lecture 3; here we see recent important theorems such as Kalai's pseudopolynomial bound on the edgediameter of a rf-polytope with  ...  ordering of the facets of a polytope.  ... 
doi:10.1017/s0013091500022914 fatcat:cdfypucoujfglar7k4c7ovjzyu

Page 559 of The Journal of the Operational Research Society Vol. 50, Issue 5 [page]

1999 The Journal of the Operational Research Society  
This group is regarded by the scientific community as one of the centres of investigastion into polynomially solvable cases of NP-hard combinatorial problems. So the book has an excellent pedigree.  ...  in some ways, because, while Padberg and Rial concentrated on investigating the QAP polytope, Cela’s book is devoted in large measure to polynomially solvable cases of the QAP.  ... 

On Polyhedral Approximations of Polytopes for Learning Bayesian Networks

Milan Studený, David C. Haws
2013 Journal of Algebraic Statistics  
Asa consequence, we confirm a conjecture from [21] that the above-mentioned implicit polyhedralapproximation of the standard imset polytope is an LP relaxation of that polytope.  ...  The topic is the comparison of outer polyhedral approximationsof the corresponding polytopes. We show how to transform the inequalities suggested byJaakkola et al. [9] into the framework of imsets.  ...  Acknowledgements Our thanks are devoted to reviewers, whose comments helped us to improve the readability of the paper. Milan Studený was supported by GAČR grant n. 201/08/0539.  ... 
doi:10.18409/jas.v4i1.19 fatcat:o7752omvvvgmvkxtziiu6g6q5e

An iterative ILP approach for constructing a Hamiltonian decomposition of a regular multigraph [article]

Andrey Kostenko, Andrei Nikolaev
2021 arXiv   pre-print
Our motivation for this problem comes from the field of polyhedral combinatorics, as a sufficient condition for vertex nonadjacency in the 1-skeleton of the traveling salesperson polytope can be formulated  ...  A Hamiltonian decomposition of a regular graph is a partition of its edge set into Hamiltonian cycles.  ...  We are very grateful to the anonymous reviewers for their comments and suggestions which helped to improve the presentation of the results in this paper.  ... 
arXiv:2102.12242v2 fatcat:5dx2lga2jvg2rkdsr24izwjwhe

How matroids occur in the context of learning Bayesian network structure

Milan Studený
2015 Conference on Uncertainty in Artificial Intelligence  
It is shown that any connected matroid having a non-trivial cluster of BN variables as its ground set induces a facet-defining inequality for the polytope(s) used in the ILP approach to globally optimal  ...  The result applies to well-known k-cluster inequalities, which play a crucial role in the ILP approach.  ...  Acknowledgements The research on this topic has been supported by the grant GA ČR n. 13-20012S. I am indebted to my colleague Fero Matúš for giving me some guidance in matroid theory.  ... 
dblp:conf/uai/Studeny15 fatcat:fmqgjbniavb2nd4wi7kuhs6cv4

Finding a second Hamiltonian decomposition of a 4-regular multigraph by integer linear programming [article]

Andrei V. Nikolaev, Egor V. Klimov
2022 arXiv   pre-print
A Hamiltonian decomposition of a regular graph is a partition of its edge set into Hamiltonian cycles.  ...  This problem arises in polyhedral combinatorics as a sufficient condition for non-adjacency in the 1-skeleton of the travelling salesperson polytope.  ...  Acknowledgement(s) We acknowledge the contribution of Andrey N. Kostenko who participated in the development of the previous version of the algorithm presented at the conference "MOTOR 2021" [35] .  ... 
arXiv:2201.03846v1 fatcat:3denxy42rrerxp77imksn5ukn4

Decomposition Bounds for Marginal MAP [article]

Wei Ping, Qiang Liu, Alexander Ihler
2015 arXiv   pre-print
We demonstrate our approach on marginal MAP queries defined on real-world problems from the UAI approximate inference challenge, showing that our framework is faster and more reliable than previous methods  ...  It is significantly more difficult than pure marginalization and MAP tasks, for which a large class of efficient and convergent variational algorithms, such as dual decomposition, exist.  ...  Alexander Ihler is also funded in part by the United States Air Force under Contract No. FA8750-14-C-0011 under the DARPA PPAML program.  ... 
arXiv:1511.02619v1 fatcat:7xjbc2oupra6jnqpttd2hjrunq

Graphical Models, Exponential Families, and Variational Inference

Martin J. Wainwright, Michael I. Jordan
2007 Foundations and Trends® in Machine Learning  
basic statistical quantities such as likelihoods and score functions have often been expressed in terms of recursions operating on these graphs; examples include phylogenies, pedigrees, hidden Markov  ...  of the problems of computing likelihoods, marginal probabilities and most probable configurations.  ...  The intellectual contributions and support of Alan Willsky and Tommi Jaakkola were particularly significant in the development of the ideas presented here.  ... 
doi:10.1561/2200000001 fatcat:3f33bwasgvg5ndjfqezocaxxfa
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