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EMIS 8373 Complexity of Linear Programming

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 2 d -1 pivots to find the optimal BFS.

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EMIS 8373 Complexity of Linear Programming

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  1. EMIS 8373Complexity 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|.

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