Who Solved the Hirsch Conjecture?
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On the Circuit Diameter Conjecture
On the Circuit Diameter Conjecture Steffen Borgwardt1, Tamon Stephen2, and Timothy Yusun2 1 University of Colorado Denver [email protected] 2 Simon Fraser University {tamon,tyusun}@sfu.ca Abstract. From the point of view of optimization, a critical issue is relating the combina- torial diameter of a polyhedron to its number of facets f and dimension d. In the seminal paper of Klee and Walkup [KW67], the Hirsch conjecture of an upper bound of f − d was shown to be equivalent to several seemingly simpler statements, and was disproved for unbounded polyhedra through the construction of a particular 4-dimensional polyhedron U4 with 8 facets. The Hirsch bound for bounded polyhedra was only recently disproved by Santos [San12]. We consider analogous properties for a variant of the combinatorial diameter called the circuit diameter. In this variant, the walks are built from the circuit directions of the poly- hedron, which are the minimal non-trivial solutions to the system defining the polyhedron. We are able to prove that circuit variants of the so-called non-revisiting conjecture and d-step conjecture both imply the circuit analogue of the Hirsch conjecture. For the equiva- lences in [KW67], the wedge construction was a fundamental proof technique. We exhibit why it is not available in the circuit setting, and what are the implications of losing it as a tool. Further, we show the circuit analogue of the non-revisiting conjecture implies a linear bound on the circuit diameter of all unbounded polyhedra – in contrast to what is known for the combinatorial diameter. Finally, we give two proofs of a circuit version of the 4-step conjecture. -
Diameter of Simplicial Complexes, a Computational Approach
University of Cantabria Master’s thesis. Masters degree in Mathematics and Computing Diameter of simplicial complexes, a computational approach Author: Advisor: Francisco Criado Gallart Francisco Santos Leal 2015-2016 This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. Desvarío laborioso y empobrecedor el de componer vastos libros; el de explayar en quinientas páginas una idea cuya perfecta exposición oral cabe en pocos minutos. Mejor procedimiento es simular que esos libros ya existen y ofrecer un resumen, un comentario. Borges, Ficciones Acknowledgements First, I would like to thank my advisor, professor Francisco Santos. He has been supportive years before I arrived in Santander, and even more when I arrived. Also, I want to thank professor Michael Joswig and his team in the discrete ge- ometry group of the Technical University of Berlin, for all the help and advice that they provided during my (short) visit to their department. Finally, I want to thank my friends at Cantabria, and my “disciples”: Álvaro, Ana, Beatriz, Cristina, David, Esther, Jesús, Jon, Manu, Néstor, Sara and Susana. i Diameter of simplicial complexes, a computational approach Abstract The computational complexity of the simplex method, widely used for linear program- ming, depends on the combinatorial diameter of the edge graph of a polyhedron with n facets and dimension d. Despite its popularity, little is known about the (combinatorial) diameter of polytopes, and even for simpler types of complexes. For the case of polytopes, it was conjectured by Hirsch (1957) that the diameter is smaller than (n − d), but this was disproved by Francisco Santos (2010). -
Victor L. Klee 1925–2007 Peter Gritzmann and Bernd Sturmfels
Victor L. Klee 1925–2007 Peter Gritzmann and Bernd Sturmfels ictor L. Klee passed away on August student, respectively) edited the volume Applied 17, 2007, in Lakewood, Ohio. Born in Geometry and Discrete Mathematics, which was San Francisco in 1925, he received his published by the American Mathematical Society. Ph.D. in mathematics from the Univer- For this obituary, we invited a group of former sity of Virginia in 1949. In 1953 he colleagues and mentees to contribute short pieces Vmoved to the University of Washington in Seattle, on Klee’s mathematical life. This resulted in ten where he was a faculty member for 54 years. individual spotlights, followed by some person- Klee specialized in convex sets, functional anal- al remarks by the editors. The emphasis lies on ysis, analysis of algorithms, optimization, and Klee’s work in the more recent decades of his combinatorics, writing more than 240 research rich scientific life, and hence they focus on finite- papers. He received many honors, including a dimensional convexity, discrete mathematics, and Guggenheim Fellowship; the Ford Award (1972), optimization. His bibliography, however, makes it the Allendoerfer Award (1980 and 1999), and clear that by the late 1960s he already had more the Award for Distinguished Service (1977) from than a career’s worth of papers in continuous and the Mathematical Association of America; and the infinite dimensional convexity. Humboldt Research Award (1980); as well as hon- orary doctorates from Pomona College (1965) and the Universities of Liège (1984) and Trier (1995). Louis J. Billera For collaborations with the first listed editor he received the Max Planck Research Award (1992). -
Report on 8Th EUROPT Workshop on Advances in Continuous
8th EUROPT Workshop on Advances in Continuous Optimization Aveiro, Portugal, July 9-10, 2010 Final Report The 8th EUROPT Workshop on Advances in Continuous Optimization was hosted by the Univer- sity of Aveiro, Portugal. Workshop started with a Welcome Reception of the participants on July 8, fol- lowed by two working days, July 9 and July 10. This Workshop continued the line of the past EUROPT workshops. The first one was held in 2000 in Budapest, followed by the workshops in Rotterdam in 2001, Istanbul in 2003, Rhodes in 2004, Reykjavik in 2006, Prague in 2007, and Remagen in 2009. All these international workshops were organized by the EURO Working Group on Continuous Optimization (EUROPT; http://www.iam.metu.edu.tr/EUROPT) as a satellite meeting of the EURO Conferences. This time, the 8th EUROPT Workshop, was a satellite meeting before the EURO XXIV Conference in Lisbon, (http://www.euro2010lisbon.org/). During the 8th EUROPT Workshop, the organizational duties were distributed between a Scientific Committee, an Organizing Committee, and a Local Committee. • The Scientific Committee was composed by Marco Lopez (Chair), Universidad de Alicante; Domin- gos Cardoso, Universidade de Aveiro; Emilio Carrizosa, Universidad de Sevilla; Joaquim Judice, Uni- versidade de Coimbra; Diethard Klatte, Universitat Zurich; Olga Kostyukova, Belarusian Academy of Sciences; Marco Locatelli, Universita’ di Torino; Florian Potra, University of Maryland Baltimore County; Franz Rendl, Universitat Klagenfurt; Claudia Sagastizabal, CEPEL (Research Center for Electric Energy); Oliver Stein, Karlsruhe Institute of Technology; Georg Still, University of Twente. • The Organizing Committee was composed by Domingos Cardoso (Chair), Universidade de Aveiro; Tatiana Tchemisova (co-Chair), Universidade de Aveiro; Miguel Anjos, University of Waterloo; Edite Fernandes, Universidade do Minho; Vicente Novo, UNED (Spain); Juan Parra, Universidad Miguel Hernandez´ de Elche; Gerhard-Wilhelm Weber, Middle East Technical University. -
Arxiv:Math/0403494V1
ONE-POINT SUSPENSIONS AND WREATH PRODUCTS OF POLYTOPES AND SPHERES MICHAEL JOSWIG AND FRANK H. LUTZ Abstract. It is known that the suspension of a simplicial complex can be realized with only one additional point. Suitable iterations of this construction generate highly symmetric simplicial complexes with various interesting combinatorial and topological properties. In particular, infinitely many non-PL spheres as well as contractible simplicial complexes with a vertex-transitive group of automorphisms can be obtained in this way. 1. Introduction McMullen [34] constructed projectively unique convex polytopes as the joint convex hulls of polytopes in mutually skew affine subspaces which are attached to the vertices of yet another polytope. It is immediate that if the polytopes attached are pairwise isomorphic one can obtain polytopes with a large group of automorphisms. In fact, if the polytopes attached are simplices, then the resulting polytope can be obtained by successive wedging (or rather its dual operation). This dual wedge, first introduced and exploited by Adler and Dantzig [1] in 1974 for the study of the Hirsch conjecture of linear programming, is essentially the same as the one-point suspension in combinatorial topology. It is striking that this simple construction makes several appearances in the literature, while it seems that never before it had been the focus of research for its own sake. The purpose of this paper is to collect what is known (for the polytopal as well as the combinatorial constructions) and to fill in several gaps, most notably by introducing wreath products of simplicial complexes. In particular, we give a detailed analysis of wreath products in order to provide explicit descriptions of highly symmetric polytopes which previously had been implicit in McMullen’s construction. -
Program of the Sessions San Diego, California, January 9–12, 2013
Program of the Sessions San Diego, California, January 9–12, 2013 AMS Short Course on Random Matrices, Part Monday, January 7 I MAA Short Course on Conceptual Climate Models, Part I 9:00 AM –3:45PM Room 4, Upper Level, San Diego Convention Center 8:30 AM –5:30PM Room 5B, Upper Level, San Diego Convention Center Organizer: Van Vu,YaleUniversity Organizers: Esther Widiasih,University of Arizona 8:00AM Registration outside Room 5A, SDCC Mary Lou Zeeman,Bowdoin upper level. College 9:00AM Random Matrices: The Universality James Walsh, Oberlin (5) phenomenon for Wigner ensemble. College Preliminary report. 7:30AM Registration outside Room 5A, SDCC Terence Tao, University of California Los upper level. Angles 8:30AM Zero-dimensional energy balance models. 10:45AM Universality of random matrices and (1) Hans Kaper, Georgetown University (6) Dyson Brownian Motion. Preliminary 10:30AM Hands-on Session: Dynamics of energy report. (2) balance models, I. Laszlo Erdos, LMU, Munich Anna Barry*, Institute for Math and Its Applications, and Samantha 2:30PM Free probability and Random matrices. Oestreicher*, University of Minnesota (7) Preliminary report. Alice Guionnet, Massachusetts Institute 2:00PM One-dimensional energy balance models. of Technology (3) Hans Kaper, Georgetown University 4:00PM Hands-on Session: Dynamics of energy NSF-EHR Grant Proposal Writing Workshop (4) balance models, II. Anna Barry*, Institute for Math and Its Applications, and Samantha 3:00 PM –6:00PM Marina Ballroom Oestreicher*, University of Minnesota F, 3rd Floor, Marriott The time limit for each AMS contributed paper in the sessions meeting will be found in Volume 34, Issue 1 of Abstracts is ten minutes. -
ADDENDUM the Following Remarks Were Added in Proof (November 1966). Page 67. an Easy Modification of Exercise 4.8.25 Establishes
ADDENDUM The following remarks were added in proof (November 1966). Page 67. An easy modification of exercise 4.8.25 establishes the follow ing result of Wagner [I]: Every simplicial k-"complex" with at most 2 k ~ vertices has a representation in R + 1 such that all the "simplices" are geometric (rectilinear) simplices. Page 93. J. H. Conway (private communication) has established the validity of the conjecture mentioned in the second footnote. Page 126. For d = 2, the theorem of Derry [2] given in exercise 7.3.4 was found earlier by Bilinski [I]. Page 183. M. A. Perles (private communication) recently obtained an affirmative solution to Klee's problem mentioned at the end of section 10.1. Page 204. Regarding the question whether a(~) = 3 implies b(~) ~ 4, it should be noted that if one starts from a topological cell complex ~ with a(~) = 3 it is possible that ~ is not a complex (in our sense) at all (see exercise 11.1.7). On the other hand, G. Wegner pointed out (in a private communication to the author) that the 2-complex ~ discussed in the proof of theorem 11.1 .7 indeed satisfies b(~) = 4. Page 216. Halin's [1] result (theorem 11.3.3) has recently been genera lized by H. A. lung to all complete d-partite graphs. (Halin's result deals with the graph of the d-octahedron, i.e. the d-partite graph in which each class of nodes contains precisely two nodes.) The existence of the numbers n(k) follows from a recent result of Mader [1] ; Mader's result shows that n(k) ~ k.2(~) . -
Majors in Mathematics and Chem Istry
Vita of Victor Klee Personal: Born in San Francisco, 1925 Education: B.A. (with high honors), Pomona College, 1945 (majors in Mathematics and Chem istry) Ph.D. (Mathematics), University of Virginia, 1949 Honorary Degrees: D.Sc., Universitat Trier, 1995 D.Sc., Universite de Liege, 1984 D.Sc., Pomona College, 1965 Awards: American Academy of Arts and Sciences " . Fellow, 1997 American Association for the Advancement of Science Fellow, 1976 Mathematical Association of America: Annual Award for Distinguished Service to Mathematics, 1977 C.B. Allendoerfer Award, 1980, 1999 L.R. Ford Award, 1972 Max-Planck-Gesellschaft: Max-Planck-Forschungspreis, 1992 Alexander von Humboldt Stiftung: Preistrager, 1980-81 Pomona College: David Prescott Barrows Award for Distinguished Achievement, 1988 Reed College: Vollum Award for Distinguished Accomplishment in Science and Technology, 1982 University of Virginia: President's and Visitor's Research Prize, 1952 1 "'lNf\""". 4 sZUA=eJUPLJ 2 Full-Time Employment: University of Washington: Professor of Mathematics, 1957-98, then Professor Emeritus; \ Associate Professor, 1954-57; Assistant Professor, 1953-54; Adjunct Professor of Computer Science, 1974-presentj Professor or Adjunct Professor of Applied Mathematics, 1976-84; University of Western Australia: Visiting Professor, 1979. University of Victoria: Visiting Professor, 1975. T.J. Watson Research Center, IBM: Full-Time Consultant, 1972. University of Colorado: Visiting Professor, 1971. University of California, Los Angeles: Visiting Associate Professor, 1955-56. University of Virginia: Assistant Professor, 1949-53; Instructor, 1947-48. Fellowships: Fulbright Research Scholar, University of Trier, 1992 Senior Fellow, Institute for Mathematics and its Applications, Minneapolis, 1987. Mathematical Sciences Research Institute, Berkeley, 1985-86. Guggenheim Fellow, University of Erlangen-Nurnberg, 1980-8l. -
Ten Problems in Geometry
Ten Problems in Geometry Moritz W. Schmitt G¨unter M. Ziegler∗ Institut f¨urMathematik Institut f¨urMathematik Freie Universit¨atBerlin Freie Universit¨atBerlin Arnimallee 2 Arnimallee 2 14195 Berlin, Germany 14195 Berlin, Germany [email protected] [email protected] May 1, 2011 Abstract We describe ten problems that wait to be solved. Introduction Geometry is a field of knowledge, but it is at the same time an active field of research | our understanding of space, about shapes, about geometric structures develops in a lively dialog, where problems arise, new questions are asked every day. Some of the problems are settled nearly immediately, some of them need years of careful study by many authors, still others remain as challenges for decades. Contents 1 Unfolding polytopes 2 2 Almost disjoint triangles 4 3 Representing polytopes with small coordinates 6 4 Polyhedra that tile space 8 5 Fatness 10 6 The Hirsch conjecture 12 7 Unimodality for cyclic polytopes 14 8 Decompositions of the cube 16 9 A problem concerning the ball and the cube 19 10 The 3d conjecture 21 ∗Partially supported by DFG, Research Training Group \Methods for Discrete Structures" 1 1 Unfolding polytopes In 1525 Albrecht D¨urer'sfamous geometry masterpiece \Underweysung der Messung mit dem Zirckel und Richtscheyt"1 was published in Nuremberg. Its fourth part contains many drawings of nets of 3-dimensional polytopes. Implicitly it contains the following conjecture: Every 3-dimensional convex polytope can be cut open along a spanning tree of its graph and then unfolded into the plane without creating overlaps. -
Interview with Gil Kalai Toufik Mansour
numerative ombinatorics A pplications Enumerative Combinatorics and Applications ECA 2:2 (2022) Interview #S3I7 ecajournal.haifa.ac.il Interview with Gil Kalai Toufik Mansour Gil Kalai received his Ph.D. at the Einstein Institute of the Hebrew University in 1983, under the supervi- sion of Micha A. Perles. After a postdoctoral position at the Massachusetts Institute of Technology (MIT), he joined the Hebrew University in 1985, (Professor Emeritus since 2018) and he holds the Henry and Manya Noskwith Chair. Since 2018, Kalai is a Pro- fessor of Computer Science at the Efi Arazi School of Computer Science in IDC, Herzliya. Since 2004, he has Photo by Muli Safra also been an Adjunct Professor at the Departments of Mathematics and Computer Science, Yale University. He has held visiting positions at MIT, Cornell, the Institute of Advanced Studies in Prince- ton, the Royal Institute of Technology in Stockholm, and in the research centers of IBM and Microsoft. Professor Gil Kalai has made significant contributions to the fields of discrete math- ematics and theoretical computer science. Two breakthrough contributions include his work in understanding randomized simplex algorithms and disproving Borsuk's famous conjecture of 1933. In recognition of his contributions, professor Kalai has received several awards in- cluding the P´olya Prize in 1992, the Erd}osPrize of the Israel Mathematical Society in 1993, the Fulkerson Prize in 1994, and the Rothschild Prize in mathematics in 2012. He has given lectures and talks at many conferences, including a plenary talk at the International Congress of Mathematicians in 2018. From 1995 to 2001, he was the Editor-in-Chief of the Israel Journal of Mathematics. -
Nonsimplicities and the Perturbed Wedge
NONSIMPLICITIES AND THE PERTURBED WEDGE FRED B. HOLT Abstract. In producing a counterexample to the Hirsch Conjecture, Francisco Santos has described a new construction, the perturbed wedge. Santos starts in dimension 5 with a counterexample to the nonrevisiting conjecture. If this counterexample were a simple polytope, we could repeatedly apply wedging in order to produce a counterexample to the Hirsch conjecture. However, the 5-dimensional polytope is not simple, so we need a different way to proceed - thus the perturbed wedge. In the perturbed wedge, we take the wedge over one facet of a non- simple polytope and then perturb one or more of the facets of the wedge to move the resulting polytope closer to simplicity. In this paper, we examine some of the technical underpinnings of the perturbed wedge, using Santos and Weibel's counterexample to the Hirsch conjecture as the guiding example. Introduction Coincidence in high dimensions is a delicate issue. Using nonsimple poly- topes and introducing the perturbed wedge construction, Santos [9] has produced a counterexample to the Hirsch conjecture. As a complement to Santos' exposition, we take the specific counterexample in dimension 20 provided by Matschke, Santos, and Weibel [10, 8], and study some of the mechanics behind its construction. We show first that the 5-dimensional polytope with which Santos begins his construction is a counterexample to the nonrevisiting conjecture [7, 6]. If Santos and Weibel's counterexamples were simple, then repeated wedging arXiv:1501.02517v2 [math.CO] 1 Mar 2015 would produce a corresponding counterexample to the Hirsch conjecture [7]. Since the counterexample to the nonrevisiting conjecture is not simple, we need an alternate method to produce the corresponding counterexample to the Hirsch conjecture, and the perturbed wedge provides this method. -
Preface Issue 4-2012
Jahresber Dtsch Math-Ver (2012) 114:187–188 DOI 10.1365/s13291-012-0054-y PREFACE Preface Issue 4-2012 Hans-Christoph Grunau © Deutsche Mathematiker-Vereinigung and Springer-Verlag Berlin Heidelberg 2012 “Foundations of Discrete Optimization: in transition from linear to non-linear models and methods.” This is the title of the survey article by Jesús A. De Loera, Raymond Hemmecke, and Matthias Köppe in which the authors summarise a number of re- cent developments in discrete optimisation. These range from improving, extending, and discussing classical algorithms from linear optimisation to nonlinear transporta- tion problems. A common feature of most real life optimisation problems is the huge number of variables and hence, the need for efficient, i.e. fast, algorithms. For a long time there has been an intimate relation between optimisation and convex geometry and in this respect it would be particularly interesting to find good bounds for the so called diameter of polytopes. The related prominent Hirsch conjecture was discussed in Issue 2-2010 of the Jahresbericht by Edward Kim and Francisco Santos. Although this particular conjecture was disproved by the latter author while their survey article was in press, the struggle to find good diameter bounds remains. Jesús De Loera, Ray- mond Hemmecke, and Matthias Köppe explain further how a more refined modelling changes linear problems into nonlinear ones and how algebraic, geometric, and topo- logical techniques enter discrete optimisation in order to tackle the new challenges. For more detailed information and proofs the authors refer to their corresponding recent monograph. Anybody, irrespective of the individual fields of interest, who has ever visited the Oberwolfach institute will know the name of Horst Tietz.