Computing the integer programming gap

We determine the maximal gap between the optimal values of an integer program and its linear programming relaxation, where the matrix and cost function are fixed but the right hand side is unspecified. Our formula involves irreducible decomposition of monomial ideals. The gap can be computed in polynomial time when the dimension is fixed.

[1]  S. Sullivant,et al.  Markov Bases of Binary Graph Models , 2003, math/0308280.

[2]  DepartmentGeorge Mason UniversityFairfax.  Standard Pairs and Group Relaxations in Integer Programming , 1998 .

[3]  Seth Sullivant,et al.  Gröbner Bases and Polyhedral Geometry of Reducible and Cyclic Models , 2002, J. Comb. Theory, Ser. A.

[4]  Bernd Sturmfels,et al.  Gröbner bases of lattices, corner polyhedra, and integer programming. , 1995 .

[5]  Jesús A. De Loera,et al.  Effective lattice point counting in rational convex polytopes , 2004, J. Symb. Comput..

[6]  B. Sturmfels,et al.  Grbner Deformations of Hypergeometric Differential Equations , 2000 .

[7]  Rekha R. Thomas,et al.  Computing gröbner fans of toric ideals , 2000, SIGS.

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

[9]  S E Fienberg,et al.  INAUGURAL ARTICLE by a Recently Elected Academy Member:Bounds for cell entries in contingency tables given marginal totals and decomposable graphs , 2000 .

[10]  A. Barvinok,et al.  An Algorithmic Theory of Lattice Points in Polyhedra , 1999 .

[11]  Rekha R. Thomas,et al.  The associated primes of initial ideals of lattice ideals , 1999 .

[12]  L. O'Carroll.  GRÖBNER BASES AND CONVEX POLYTOPES (University Lecture Series 8) , 1997 .

[13]  David A. Cox,et al.  Using Algebraic Geometry , 1998 .

[14]  Rekha R. Thomas.  The Structure of Group Relaxations , 2005 .

[15]  David K. Smith.  Theory of Linear and Integer Programming , 1987 .

[16]  Rekha R. Thomas,et al.  Standard pairs and group relaxations in integer programming , 1999 .

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

[18]  A. Barvinok,et al.  Short rational generating functions for lattice point problems , 2002, math/0211146.

[19]  Jean B. Lasserre,et al.  Duality and a Farkas lemma for integer programs , 2009 .

[20]  Jean B. Lasserre,et al.  Generating functions and duality for integer programs , 2004, Discret. Optim..

[21]  L. H. Cox,et al.  Controlled rounding for tables with subtotals , 1989 .