Cook, Kannan and Schrijver's example revisited

Authors:
Yanjun Li;Jean-Philippe P. Richard
Affiliations:
Krannert School of Management, Purdue University, West Lafayette, IN 47907, USA;School of Industrial Engineering, Purdue University, West Lafayette, IN 47907, USA
Venue:
Discrete Optimization
Year:
2008

Citing 4
Cited 4

Chva´tal closures for mixed integer programming problems

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Disjunctive programming: properties of the convex hull of feasible points

Discrete Applied Mathematics
Split closure and intersection cuts

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Inequalities from Two Rows of a Simplex Tableau

IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization

Finite Disjunctive Programming Characterizations for General Mixed-Integer Linear Programs

Operations Research
Split Rank of Triangle and Quadrilateral Inequalities

Mathematics of Operations Research
Computing with multi-row Gomory cuts

Operations Research Letters
On some generalizations of the split closure

IPCO'13 Proceedings of the 16th international conference on Integer Programming and Combinatorial Optimization

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Abstract

In 1990, Cook, Kannan and Schrijver [W. Cook, R. Kannan, A. Schrijver, Chvatal closures for mixed integer programming problems, Mathematical Programming 47 (1990) 155-174] proved that the split closure (the 1st 1-branch split closure) of a polyhedron is again a polyhedron. They also gave an example of a mixed-integer polytope in R^2^+^1 whose 1-branch split rank is infinite. We generalize this example to a family of high-dimensional polytopes and present a closed-form description of the kth 1-branch split closure of these polytopes for any k=1. Despite the fact that the m-branch split rank of the (m+1)-dimensional polytope in this family is 1, we show that the 2-branch split rank of the (m+1)-dimensional polytope is infinite when m=3. We conjecture that the t-branch split rank of the (m+1)-dimensional polytope of the family is infinite for any 1@?t@?m-1 and m=2.