31

What is the computational complexity of solving a linear program with $m$ constraints in $n$ variables?

Rodrigo de Azevedo
  • 23,223
userAN82
  • 441
  • 1
    There are (weak)polynomial times algorithms that find the solutions by moving a point in the interior of the simplex. I don't know what is the minimum exponent possible. – user647486 Apr 26 '19 at 17:53
  • 3
    Though it’s worth mentioning that the most common linear programming algorithm—the simplex—method, os exponential in its worst case. – David M. Apr 28 '19 at 03:43
  • 1
    Thank you, but here I am speaking about the theoretical complexity of linear programming not algorithms. For example, it is known (to the best of my knowledge) that solving a quadratic program is equivalent to solving a system of linear equations, so the complexity of quadratic programming is about O(n^3). Is there any similar result for linear programming ? – userAN82 Apr 28 '19 at 17:40
  • 1
    The result due to Spielman and Teng has explained why simplex method is efficient in practice https://arxiv.org/pdf/cs/0111050.pdf Title : Smoothed Analysis of Algorithms: Why the Simplex Algorithm usually Takes Polynomial Time. – Zhao Song Apr 21 '20 at 23:18
  • 1
    This lecture notes contain a detailed history of LP algorithm in theory, https://www.cs.utexas.edu/~ecprice/courses/331/slides/04-20.pdf – Zhao Song Apr 29 '20 at 00:06

2 Answers2

14

The best possible (I believe) is by Michael Cohen, Yin Tat Lee, and Zhao Song: Solving linear program in the current matrix multiplication time. https://arxiv.org/abs/1810.07896 (STOC 2019) Hope this helps.

Zhao Song
  • 172
Jeff Linderoth
  • 526
  • 1
    The result by Cohen, Lee and Song has been reproduced via a different method, see this paper https://arxiv.org/pdf/1905.04447.pdf – Zhao Song Apr 21 '20 at 23:13
  • 1
    The 2+1/6 has been improved to 2+1/18 in a recent work by Shunhua Jiang, Zhao Song, Omri Weinstein, and Hengjie Zhang. The title is Faster Dynamic Matrix Inverse for Faster LPs, and the link is https://arxiv.org/pdf/2004.07470.pdf – Zhao Song May 19 '20 at 00:32
8

Brand's 2020 result derandomized the Cohen, Lee and Song result. Here is the link https://arxiv.org/pdf/1910.11957.pdf

Zhao Song
  • 172
  • Thank you very much for all these useful pointers; this is an impressive set of recent improvements that I was not aware of. I am wondering what model of computation is assumed in these papers? I cannot seem to find this explicitly stated in these papers, probably because there is some implicit standard in this branch of literature. I am guessing the assumption is a RAM model, but I'm not sure what the details are of this model: What set of basic arithmetic operations are supported and how long do they take? – Bart Apr 22 '21 at 18:06