EMIS 8373 Complexity of Linear Programming

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.

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.

Growth Rates of Complexity Functions

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

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|.