Recent progress on the combinatorial diameter of polytopes and simplicial complexes

The Hirsch Conjecture, posed in 1957, stated that the graph of a d-dimensional polytope or polyhedron with n facets cannot have diameter greater than n−d. The conjecture itself has been disproved, but what we know about the underlying question is quite scarce. Most notably, no polynomial upper bound is known for the diameters that were conjectured to be linear. In contrast, no polyhedron violating the conjecture by more than 25 % is known.This paper reviews several recent attempts and progress on the question. Some work is in the world of polyhedra or (more often) bounded polytopes, but some try to shed light on the question by generalizing it to simplicial complexes. In particular, we include here our recent and previously unpublished proof that the maximum diameter of arbitrary simplicial complexes is in nΘ(d), and we summarize the main ideas in the polymath 3 project, a web-based collective effort trying to prove an upper bound of type nd for the diameters of polyhedra and of more general objects (including, e.g., simplicial manifolds).

[1]  V. Klee.  Diameters of Polyhedral Graphs , 1964, Canadian Journal of Mathematics.

[2]  George B. Dantzig,et al.  Linear programming and extensions , 1965 .

[3]  V. Klee.  Paths on polyhedra. II. , 1964 .

[4]  V. Klee,et al.  Thed-step conjecture for polyhedra of dimensiond<6 , 1967 .

[5]  D. Larman.  Paths on Polytopes , 1970 .

[6]  V. Klee,et al.  HOW GOOD IS THE SIMPLEX ALGORITHM , 1970 .

[7]  P. Goodey,et al.  Some upper bounds for the diameters of convex polytopes , 1972 .

[8]  David W. Barnette.  An upper bound for the diameter of a polytope , 1974, Discret. Math..

[9]  G. Dantzig,et al.  Maximum Diameter of Abstract Polytopes , 1974 .

[10]  Michael J. Todd,et al.  The Monotonic Bounded Hirsch Conjecture is False for Dimension at Least 4 , 1980, Math. Oper. Res..

[11]  Amos Altshuler,et al.  The classification of simplicial 3-spheres with nine vertices into polytopes and nonpolytopes , 1980, Discret. Math..

[12]  J. Scott Provan,et al.  Decompositions of Simplicial Complexes Related to Diameters of Convex Polyhedra , 1980, Math. Oper. Res..

[13]  L. Khachiyan.  Polynomial algorithms in linear programming , 1980 .

[14]  Nesa L'abbe Wu,et al.  Linear programming and extensions , 1981 .

[15]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..

[16]  Victor Klee,et al.  The d-Step Conjecture and Its Relatives , 1987, Math. Oper. Res..

[17]  Denis Naddef,et al.  The hirsch conjecture is true for (0, 1)-polytopes , 1989, Math. Program..

[18]  G. Kalai,et al.  A quasi-polynomial bound for the diameter of graphs of polyhedra , 1992, math/9204233.

[19]  Martin E. Dyer,et al.  Random walks, totally unimodular matrices, and a randomised dual simplex algorithm , 1994, IPCO.

[20]  Gil Kalai.  Upper bounds for the diameter and height of graphs of convex polyhedra , 1992, Discret. Comput. Geom..

[21]  Peter Kleinschmidt,et al.  On the diameter of convex polytopes , 1992, Discret. Math..

[22]  Derek Allan Holton,et al.  The Petersen graph , 1993, Australian mathematical society lecture series.

[23]  G. Ziegler.  Lectures on Polytopes , 1994 .

[24]  S. Smale.  Mathematical problems for the next century , 1998 .

[25]  Jack J. Dongarra,et al.  Guest Editors Introduction to the top 10 algorithms , 2000, Comput. Sci. Eng..

[26]  John C. Nash,et al.  The (Dantzig) simplex method for linear programming , 2000, Comput. Sci. Eng..

[27]  Michael Joswig,et al.  Branched Coverings, Triangulations, and 3-Manifolds , 2001, math/0108202.

[28]  Santosh S. Vempala,et al.  A simple polynomial-time rescaling algorithm for solving linear programs , 2004, STOC '04.

[29]  Ulrich Betke.  Relaxation, New Combinatorial and Polynomial Algorithms for the Linear Feasibility Problem , 2004, Discret. Comput. Geom..

[30]  Roman Vershynin,et al.  Beyond Hirsch Conjecture: Walks on Random Polytopes and Smoothed Complexity of the Simplex Method , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[31]  Santosh S. Vempala,et al.  A simple polynomial-time rescaling algorithm for solving linear programs , 2008, Math. Program..

[32]  Tamás Terlaky,et al.  Polytopes and arrangements: Diameter and curvature , 2008, Oper. Res. Lett..

[33]  Tamás Terlaky,et al.  A Continuous d-Step Conjecture for Polytopes , 2009, Discret. Comput. Geom..

[34]  Alexander A. Razborov,et al.  Diameter of polyhedra: limits of abstraction , 2009, SCG '09.

[35]  Tamás Terlaky,et al.  Central Path Curvature and Iteration-Complexity for Redundant Klee—Minty Cubes , 2009 .

[36]  Francisco Santos,et al.  A counterexample to the Hirsch conjecture , 2010, ArXiv.

[37]  Francisco Santos,et al.  Companion to \An update on the Hirsch conjecture" , 2009, 0912.4235.

[38]  Edward D. Kim,et al.  An Update on the Hirsch Conjecture , 2009, 0907.1186.

[39]  J. D. Loera,et al.  Triangulations: Structures for Algorithms and Applications , 2010 .

[40]  A. Razborov,et al.  Diameter of Polyhedra: Limits of Abstraction , 2010 .

[41]  Lars Schewe,et al.  Nonrealizable Minimal Vertex Triangulations of Surfaces: Showing Nonrealizability Using Oriented Matroids and Satisfiability Solvers , 2008, Discret. Comput. Geom..

[42]  Uri Zwick,et al.  Subexponential lower bounds for randomized pivoting rules for the simplex algorithm , 2011, STOC '11.

[43]  David Bremner,et al.  Edge-Graph Diameter Bounds for Convex Polytopes with Few Facets , 2008, Exp. Math..

[44]  Edward D. Kim.  Polyhedral graph abstractions and an approach to the Linear Hirsch Conjecture , 2011, Electron. Notes Discret. Math..

[45]  Oliver Friedmann,et al.  A Subexponential Lower Bound for Zadeh's Pivoting Rule for Solving Linear Programs and Games , 2011, IPCO.

[46]  John Dunagan,et al.  Smoothed analysis of condition numbers and complexity implications for linear programming , 2011, Math. Program..

[47]  Tamon Stephen,et al.  Embedding a Pair of Graphs in a Surface, and the Width of 4-dimensional Prismatoids , 2012, Discret. Comput. Geom..

[48]  S. Klee,et al.  Obstructions to weak decomposability for simplicial polytopes , 2012, 1206.6143.

[49]  Nicolai Hähnle.  Constructing subset partition graphs with strong adjacency and end-point count properties , 2012, ArXiv.

[50]  Sergei Chubanov.  A strongly polynomial algorithm for linear systems having a binary solution , 2012, Math. Program..

[51]  Friedrich Eisenbrand,et al.  On Sub-determinants and the Diameter of Polyhedra , 2011, SoCG '12.

[52]  Jesús A. De Loera,et al.  Transportation Problems and Simplicial Polytopes That Are Not Weakly Vertex-Decomposable , 2012, Math. Oper. Res..

[53]  Etienne de Klerk,et al.  Special issue in honour of Professor Kees Roos’ 70th Birthday , 2013, Optim. Methods Softw..

[54]  David Bremner,et al.  More bounds on the diameters of convex polytopes , 2009, Optim. Methods Softw..

[55]  Bruno Benedetti,et al.  The Hirsch Conjecture Holds for Normal Flag Complexes , 2014, Math. Oper. Res..