Convex Discrete Optimization

We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial optimization problems and convex integer programming problems in variable dimension. We discuss some of the many applications of this theory including to quadratic programming, matroids, bin packing and cutting-stock problems, vector partitioning and clustering, multiway transportation problems, and privacy and confidential statistical data disclosure. Highlights of our work include a strongly polynomial time algorithm for convex and linear combinatorial optimization over any family presented by a membership oracle when the underlying polytope has few edge-directions; a new theory of so-termed n-fold integer programming, yielding polynomial time solution of important and natural classes of convex and linear integer programming problems in variable dimension; and a complete complexity classification of high dimensional transportation problems, with practical applications to fundamental problems in privacy and confidential statistical data disclosure.

[1]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[2]  Uriel G. Rothblum,et al.  A Polynomial Time Algorithm for Shaped Partition Problems , 1999, SIAM J. Optim..

[3]  L. Cox.  On properties of multi-dimensional statistical tables , 2003 .

[4]  Shmuel Onn,et al.  An Adaptive Algorithm for Vector Partitioning , 2003, J. Glob. Optim..

[5]  Raymond Hemmecke,et al.  On the positive sum property and the computation of Graver test sets , 2003, Math. Program..

[6]  Erich Steiner,et al.  A polynomial case of unconstrained zero-one quadratic optimization , 2001, Math. Program..

[7]  L. G. H. Cijan.  A polynomial algorithm in linear programming , 1979 .

[8]  Micha Sharir,et al.  On the Zone Theorem for Hyperplane Arrangements , 1991, SIAM J. Comput..

[9]  Shmuel Onn,et al.  Entry Uniqueness in Margined Tables , 2006, Privacy in Statistical Databases.

[10]  R. Gomory,et al.  A Linear Programming Approach to the Cutting-Stock Problem , 1961 .

[11]  Frits C. R. Spieksma,et al.  Approximation Algorithms for Multi-index Transportation Problems with Decomposable Costs , 1997, Discret. Appl. Math..

[12]  Jesús A. De Loera,et al.  All Linear and Integer Programs Are Slim 3-Way Transportation Programs , 2006, SIAM J. Optim..

[13]  Jesús A. De Loera,et al.  Markov bases of three-way tables are arbitrarily complicated , 2006, J. Symb. Comput..

[14]  Alan J. Hoffman,et al.  Integral Boundary Points of Convex Polyhedra , 2010, 50 Years of Integer Programming.

[15]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[16]  Jesús A. De Loera,et al.  The Complexity of Three-Way Statistical Tables , 2002, SIAM J. Comput..

[17]  P. Pardalos,et al.  Handbook of Combinatorial Optimization , 1998 .

[18]  Jack E. Graver,et al.  On the foundations of linear and integer linear programming I , 1975, Math. Program..

[19]  Jesús A. De Loera,et al.  N-fold integer programming , 2006, Discret. Optim..

[20]  László Lovász,et al.  Algorithmic theory of numbers, graphs and convexity , 1986, CBMS-NSF regional conference series in applied mathematics.

[21]  E. Harding.  The number of partitions of a set of N points in k dimensions induced by hyperplanes , 1967, Proceedings of the Edinburgh Mathematical Society.

[22]  Andreas S. Schulz,et al.  0/1-Integer Programming: Optimization and Augmentation are Equivalent , 1995, ESA.

[23]  M. Grötschel,et al.  Combinatorial optimization , 1996 .

[24]  T. Zaslavsky.  Facing Up to Arrangements: Face-Count Formulas for Partitions of Space by Hyperplanes , 1975 .

[25]  P. Doyle,et al.  Confidentiality, Disclosure and Data Access: Theory and Practical Applications for Statistical Agencies , 2001 .

[26]  B. Sturmfels.  Gröbner bases and convex polytopes , 1995 .

[27]  Shmuel Onn,et al.  Convex Matroid Optimization , 2002, SIAM J. Discret. Math..

[28]  Richard M. Karp,et al.  Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems , 1972, Combinatorial Optimization.

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

[30]  Michel Balinski,et al.  Signature classes of transportation polytopes , 1993, Math. Program..

[31]  Endre Boros,et al.  On clustering problems with connected optima in euclidean spaces , 1989, Discret. Math..

[32]  A. Takemura,et al.  Minimal Basis for a Connected Markov Chain over 3 × 3 ×K Contingency Tables with Fixed Two‐Dimensional Marginals , 2003 .

[33]  Bernd Sturmfels,et al.  Higher Lawrence configurations , 2003, J. Comb. Theory, Ser. A.

[34]  Jack Edmonds,et al.  Matroids and the greedy algorithm , 1971, Math. Program..

[35]  Andreas S. Schulz,et al.  The Complexity of Generic Primal Algorithms for Solving General Integer Programs , 2002, Math. Oper. Res..

[36]  V. A. Yemelicher,et al.  Polytopes, Graphs and Optimisation , 1984 .

[37]  Raimund Seidel,et al.  Constructing arrangements of lines and hyperplanes with applications , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[38]  Seth Sullivant,et al.  A finiteness theorem for Markov bases of hierarchical models , 2007, J. Comb. Theory, Ser. A.

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

[40]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

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

[42]  Uwe T. Zimmermann,et al.  A combinatorial interior point method for network flow problems , 1992, Math. Program..

[43]  Uriel G. Rothblum,et al.  Optimal partitions having disjoint convex and conic hulls , 1992, Math. Program..

[44]  Milan Vlach,et al.  Conditions for the existence of solutions of the three-dimensional planar transportation problem , 1986, Discret. Appl. Math..

[45]  B. Sturmfels,et al.  Cutting Corners , 1999, Food Fights.

[46]  Jesús A. De Loera,et al.  All Rational Polytopes Are Transportation Polytopes and All Polytopal Integer Sets Are Contingency Tables , 2004, IPCO.

[47]  Uriel G. Rothblum,et al.  Convex Combinatorial Optimization , 2003, Discret. Comput. Geom..

[48]  J. Humphreys.  Polytopes, Graphs and Optimisation , 2022 .

[49]  Refael Hassin,et al.  Maximizing Classes of Two-Parameter Objectives Over Matroids , 1989, Math. Oper. Res..

[50]  Vasek Chvátal,et al.  Edmonds polytopes and a hierarchy of combinatorial problems , 1973, Discret. Math..

[51]  Carl W. Lee,et al.  Transportation problems which can be solved by the use of hirsch-paths for the dual problems , 1987, Math. Program..

[52]  Leonard J. Schulman,et al.  The Vector Partition Problem for Convex Objective Functions , 2001, Math. Oper. Res..

[53]  András Frank,et al.  An application of simultaneous diophantine approximation in combinatorial optimization , 1987, Comb..

[54]  G. C. Shephard,et al.  Convex Polytopes , 1969, The Mathematical Gazette.

[55]  Éva Tardos,et al.  A Strongly Polynomial Algorithm to Solve Combinatorial Linear Programs , 1986, Oper. Res..

[56]  V. Klee,et al.  FACETS AND VERTICES OF TRANSPORTATION POLYTOPES , 1967 .

[57]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[58]  Rekha R. Thomas,et al.  The Hilbert zonotope and a polynomial time algorithm for universal Gröbner bases , 2002, Adv. Appl. Math..

[59]  Noga Alon,et al.  Separable Partitions , 1999, Discret. Appl. Math..