London School of Economics and Political Science Structural and decomposition results for binet matrices, bidirected graphs and signed-graphic matroids Konstantinos Papalamprou Thesis submitted for the degree of Doctor of Philosophy in Operational Research London, 2009 UMI Number: U613425 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI U613425 Published by ProQuest LLC 2014. Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 TW ^es p W ° \ r n Librar\ ' >US^y of Political ancfEeftWVnic Soenrp IZ3IS4 ' Declaration I certify that the thesis I have presented for examination for the MPhil/PhD degree of the London School of Economics and Political Science is solely my own work other than where I have clearly indicated that it is the work of others (in which case the extent of any work carried out jointly by me and any other person is clearly identified in it). The copyright of this thesis rests with the author. Quotation from it is permitted, provided that full acknowledgement is made. This thesis may not be reproduced without the prior written consent of the author. I warrant that this authorization does not, to the best of my belief, infringe the rights of any third party. 3 Abstract In this thesis we deal with binet matrices and the class of signed-graphic matroids which is the class of matroids represented over R by binet matrices. The thesis is divided in three parts. In the first part, we provide the vast majority of the notions used throughout the thesis and some results regarding the class of binet matrices. In this part, we focus on the class of linear and integer programming problems in which the constraint matrix is binet and provide methods and algorithms which solve these problems efficiently. Results of the part regarding the optimization with binet matrices are joint work with G. Appa, B. Kotnyek and L. Pitsoulis and have been published in [5]. The main new result is that the existing combinatorial methods can not solve the { 0 , \ }-separation problem (special case of the well known separation problem) with integral binet matrices. The main new results of the whole thesis are provided in the next two parts. In the second part, we present a polynomial time algorithm to construct a bidirected graph for any totally unimodular matrix B by finding node-edge incidence matrices Q and S such that QB = S. Seymour’s famous decomposition theorem for regular matroids states that any totally unimodular matrix can be constructed through a series of composition operations called fc-sums starting from network matrices and their transposes and two compact representation matrices B\ and B 2 of a certain ten element matroid. Given that B\ and B 2 are binet matrices, we examine the /c-sums of network and binet matrices (k = 1,2,3). It is shown that the k- sum of a network and a binet matrix is a binet matrix, but binet matrices are not closed under this operation for k = 2,3. A new class of matrices is introduced, the so-called tour matrices, which generalises network and totally unimodular matrices. For any such matrix there exists a bidirected graph such that the columns represent a collection of closed tours in the graph. It is shown that tour matrices are closed under 1-, 2- and ©3-sum as well as under elementary operations on their rows and columns. Given the constructive proofs of the above results regarding the fc-sum operations and existing recognition algorithms for network and binet matrices, an algorithm is presented which constructs a bidirected graph for any totally unimodular matrix. I should note here that many results of this part are joint work with G. Appa, B. Kotnyek and L. Pitsoulis; these results can be found in a joint journal article [61]. In the third part of this thesis we deal with the frame matroid of a signed graph, or simply the signed- graphic matroid. Several new results are provided in this last part of the thesis. Specifically, given a signed graph, we provide methods to find representation matrices of the associated signed-graphic matroid over GF(2), GF(3) and R. Furthermore, two new matroid recognition algorithms are presented in this last part. The first one determines whether a binary matroid is signed-graphic or not and the second one determines whether a (general) matroid is binary signed-graphic or not. Finally, one of the most important new results of this thesis is the decomposition theory for the class of binary signed-graphic 4 Abstract 5 matroids which is provided in the last chapter. In order to achieve this result, we employed Tutte’s theory of bridges. The proposed decomposition differs from previous decomposition results on matroids that have appeared in the literature in the sense that it is not based on k-sums, but rather on the operation of deletion of a cocircuit. Specifically, it is shown that certain minors resulting from the deletion of a cocircuit of a binary matroid will be graphic matroids except for one that will be signed-graphic if and only if the matroid is signed-graphic. The decomposition theory for binary signed-graphic matroids is a joint work with G. Appa and L. Pitsoulis. Dedicated to my parents: Athanassios Papalamprou and Eleni Mparpagianni to my brother: Marios Papalamprou and to Vasiliki Samartzi Acknowledgements First and foremost, I would like to thank my supervisor Professor Gautam Appa for giving me the op­ portunity and motivation to write this thesis. His guidance and assistance were invaluable in completing this work. I would also like to thank Dr Leonidas Pitsoulis who assisted me during these years and whose personal attention, support and long hours of discussion were essential for my work. The mathematical insight, generosity and tremendous encouragement of these two people were overwhelming during the preparation of this thesis. I would also like to thank the two examiners of this thesis, Professor Graham Brightwell and Dr Kristina Vuskovic, for their valuable comments/suggestions and for bringing to my attention recent results regard­ ing the Well-Quasi-Ordering conjecture and the Minor-Recognition conjecture for matroids. The Operational Research Group at the LSE is an exciting and inspiring environment for someone to do research and, for that reason, I would like to thank all the members of the Operational Research Group. My special thanks go to my fellow PhD students for their friendship and help during these years. I was lucky enough to have very good friends during my years as a PhD student; in particular I would like to thank Mr Petros Machlis, Mr Vassilis Papatheodorou and Mr Michael Xanthakis for their support and understanding. I should also like to acknowledge the financial support of the Bodossaki Foundation, the Greek State Scholarships’ Foundation, and the London School of Economics and Political Science. I am forever indebted to my family, whose love, support, and patience are invaluable for me. There­ fore, thanks go to my father Athanassios, my mother Eleni and my brother Marios. Finally, I would like to thank Miss Vasiliki Samartzi for her love and support all these years. I feel that I can find her love and support behind every single word of this thesis. Konstantinos Papalamprou London, 2009 7 Contents I Introduction 13 1 Basic definitions and notation 14 1.1 Sets and fields ..................................................................................................................... 14 1.2 Matrices and polyhedra ..................................................................................................... 16 1.3 Graphs................................................................................................................................ 18 1.4 Digraphs and network matrices .......................................................................................... 24 1.5 Totally unimodular matrices ............................................................................................ 25 1.6 Relevant matroid theory ..................................................................................................... 28 1.6 .1 Definitions and important classes of matroids ....................................................... 28 1.6.2 Duality and m inors ................................................................................................. 30 1.6.3 Connectivity .......................................................................................................... 32 2 Bidirected graphs and binet matrices 34 2.1 Bidirected graphs ............................................................................................................... 34 2.1.1 Basic notions .......................................................................................................... 35 2.1.2 Some operations and the circuits of bidirected graphs ............................................ 37 2.2 Binet matrices and related classes of m atrices .................................................................. 39 2.2.1 Binet matrices ......................................................................................................