Convex Combinatorial Optimization

Abstract We introduce the convex combinatorial optimization problem, a far-reaching generalization of the standard linear combinatorial optimization problem. We show that it is strongly polynomial time solvable over any edge-guaranteed family, and discuss several applications.

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  R. Buck.  Partition of Space , 1943 .

[3]  C. Zheng,et al.  ; 0 ; , 1951 .

[4]  Jack Edmonds,et al.  Maximum matching and a polyhedron with 0,1-vertices , 1965 .

[5]  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.

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

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

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

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

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

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

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

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

[14]  A. Barrett.  Network Flows and Monotropic Optimization. , 1984 .

[15]  Amiya K. Chakravarty,et al.  Consecutive Optimizers for a Partitioning Problem with Applications to Optimal Inventory Groupings for Joint Replenishment , 1985, Oper. Res..

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

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

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

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

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

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

[22]  David Avis,et al.  A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra , 1991, SCG '91.

[23]  U. Rothblum,et al.  The Pareto set of the partition bargaining problem , 1991 .

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

[25]  Peter Gritzmann,et al.  Minkowski Addition of Polytopes: Computational Complexity and Applications to Gröbner Basis , 1993, SIAM J. Discret. Math..

[26]  David Avis,et al.  A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra , 1992, Discret. Comput. Geom..

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

[28]  Shmuel Onn,et al.  Geometry, Complexity, and Combinatorics of Permutation Polytopes , 1993, J. Comb. Theory A.

[29]  Panos M. Pardalos,et al.  The Quadratic Assignment Problem: A Survey and Recent Developments , 1993, Quadratic Assignment and Related Problems.

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

[31]  Robert E. Tarjan,et al.  Improved Algorithms for Bipartite Network Flow , 1994, SIAM J. Comput..

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

[33]  Michel Deza,et al.  Lattice-free polytopes and their diameter , 1995, Discret. Comput. Geom..

[34]  Uriel G. Rothblum,et al.  Directional-Quasi-Convexity, Asymmetric Schur-Convexity and Optimality of Consecutive Partitions , 1996, Math. Oper. Res..

[35]  Louis J. Billera,et al.  All 0–1 polytopes are traveling salesman polytopes , 1996, Comb..

[36]  Gil Kalai,et al.  Linear programming, the simplex algorithm and simple polytopes , 1997, Math. Program..

[37]  Rekha R. Thomas,et al.  Variation of cost functions in integer programming , 1997, Math. Program..

[38]  M. R. Rao,et al.  Combinatorial Optimization , 1992, NATO ASI Series.

[39]  Yossi Azar,et al.  Ancient and New Algorithms for Load Balancing in the lp Norm , 1998, SODA '98.

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

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

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

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

[44]  David J. Rader,et al.  Maximizing the Product of Two Linear Functions In 0-1 Variables , 2002 .

[45]  S. Aviran,et al.  Momentopes, the Complexity of Vector Partitioning, and Davenport—Schinzel Sequences , 2002, Discret. Comput. Geom..

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

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

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

[49]  Uriel G. Rothblum,et al.  Edge-Directions of Standard Polyhedra with Applications to Network Flows , 2005, J. Glob. Optim..