1. EMIS 8373 Complexity of Linear Programming

2. Complexity of LP
(Klee and Minty 1972) For every d > 1 there is an LP with 2d equations, 3d variables, and integer coefficients bounded by 4, such that the Simplex Method may take 2d-1 pivots to find the optimal BFS.
The Simplex Method has exponential worst-case complexity.

3. Complexity of LP (Khachian 1979)
The Ellipsoid Algorithm has worst-case complexity O(n6log(nU)) where n is the number variables and U is the absolute value of the largest number in the matrix A or vector b.
LP is polynomial.

4. Growth Rates of Complexity Functions

5. Easy vs. Hard Problems Easy (i.e., polynomial) Problems:
Uncapacitated Lot-Sizing (ULS)
Spanning Tree
Minimum Cost Network Flow
Linear Programming
Integer Programming with TU Constraint Matrix
Hard Problems:
TSP
Uncapacitated Facility Location (UFL)
Knapsack
Integer Programming with General Constraint Matrix

6. A Note About Representing Networks and Graphs
In practice we say that a graph G=(V,E) can be encoded by a string whose length is O(|E|).
Computers usually reserve a fixed number of bits (a word) to store any integer.
Storing a MCNF problem in adjacency list requires 4 |E| words.
Since we are interested growth rates, we say that the space required to store a network is bounded by a linear function of the number arcs (i.e. O(|E|)).
The size of a graph or network is generally taken to be |E|.