0

Linear programming is the very common problem of computing $$\min_{Ax\leq b}c^\top x,$$ where $A\in\mathbb{R}^{n\times m}$, $b\in\mathbb{R}^n$, and $c\in\mathbb{R}^m$. This is an optimization problem, but it can be turned into a decision problem with an additional parameter $s$, by asking: Does a vector $x$ exist such as $Ax\leq b$ and $c^\top x \leq s$?

In various places, it is stated that linear programming can be "solved in polynomial time"; what this actually means is that it admits an FPTAS, i.e., the optimum value can be determined up to arbitrary accuracy in polynomial time w.r.t. $n$, $m$, and the accuracy.

Does this mean that it is unknown whether the decision formulation of linear programming is in $\mathsf{P}$? (I know that the exact solution can be computed using the simplex algorithm, but since it has exponential runtime in the worst case, it can not be used to prove that linear programming is in $\mathsf{P}$)

Linear programming is pretty much the easiest kind of (continuous) optimization problem that I can think of, which is why I am curious at the possibility of its decision formulation not being in $\mathsf{P}$...

Firavox
  • 178
  • 6

2 Answers2

1

From Wikipedia:

The linear programming problem was first shown to be solvable in polynomial time by Leonid Khachiyan in 1979, but a larger theoretical and practical breakthrough in the field came in 1984 when Narendra Karmarkar introduced a new interior-point method for solving linear-programming problems.

The complexity class P contains the set of decision problems for which we have at least one deterministic polynomial time algorithm. If we have a decision version problem in P, we also have a deterministic polynomial time algorithm for the (original) optimization problem (with possibly some additional complexity than the decision version).

FPTAS and other related terms are usually associated with NP-hard problems where we do expect to have deterministic polynomial time algorithms, and thus we compromise with respect to solution quality to gain an advantage in time complexity.

Also see this discussion on LP vs ILP.

codeR
  • 1,983
  • 7
  • 17
1

DISCLAIMER: The question was resolved in the comments of codeR's answer, but since I am not allowed to point to that in a small note after the question, here is the answer to my own question. Thank you in particular to Neal Young for the last piece of the puzzle!

The decision-version of the linear programming problem is indeed in $\mathsf{P}$, and the key idea is to solve the linear program using the ellipsoid method (or interior-point methods) up to a small accuracy parameter $\varepsilon>0$. If $\varepsilon$ is small enough, one can guarantee that the (a priori approximative) solution is, in fact, exact. The details are discussed in lecture notes by Ben-Tal & Nemirovski (it is in Section 8.4.2, specifically in the discussion right after Eq. (8.4.6)).

Firavox
  • 178
  • 6