Report Number: CS-TR-72-284
Institution: Stanford University, Department of Computer Science
Title: Edmonds polyhedra and a hierarchy of combinatorial problems.
Author: Chvatal, Vaclav
Date: May 1972
Abstract: Let S be a set of linear inequalities that determine a bounded polyhedron P. The closure of S is the smallest set of inequalities that contains S and is closed under two operations: (i) taking linear combinations of inequalities, (ii) replacing an inequality $\sum\ a_j x_j \leq\ a_0$, where $a_1, a_2, ... , a_n$ are integers, by the inequality $\sum\ a_j x_j \leq\ a$ with $a \geq\ [a_0]$. Obviously, if integers $x_1, x_2, ... , x_n$ satisfy all the inequalities in S then they satisfy also all the inequalities in the closure of S. Conversely, let $\sum\ c_j x_j \leq\ c_0$ hold for all choices of integers $x_1, x_2, ... , x_n$, that satisfy all the inequalities in S. Then we prove that $\sum\ c_j x_j \leq\ c_0$ belongs to the closure of S. To each integer linear programming problem, we assign a nonnegative integer, called its rank. (The rank is the minimum number of iterations of the operation (ii) that are required in order to eliminate the integrality constraint.) We prove that there is no upper bound on the rank of problems arising from the search for largest independent sets in graphs.
http://i.stanford.edu/pub/cstr/reports/cs/tr/72/284/CS-TR-72-284.pdf