Linear Programming with Reals (Linear Programming)

Description

In this case, we allow all of the variables to be any real number.

Related Problems

Generalizations: General Linear Programming

Subproblem: Integer Linear Programming

Related: 0-1 Linear Programming

Parameters

$n$: number of variables

$m$: number of constraints

$L$: length of input, in bits

Table of Algorithms

Name	Year	Time	Space	Approximation Factor	Model	Reference
Fourier–Motzkin elimination	1940	$O((m/{4})$^{({2}^n)})	$O((m/{4})$^{({2}^n)})	Exact	Deterministic
Khachiyan Ellipsoid algorithm	1979	$O(n^{6} * L^{2} \log L \log\log L)$	$O(nmL)$	Exact	Deterministic	Time
Karmarkar's algorithm	1984	$O(n^{3.5} L^{2} logL loglogL)$	$O(nmL)$	Exact	Deterministic	Time
Simplex Algorithm	1947	$O({2}^n*poly(n, m))$	$O(nm)$	Exact	Deterministic
Terlaky's Criss-cross algorithm	1985	$O({2}^n*poly(n, m))$	$O(nm)$	Exact	Deterministic
Affine scaling	1967	? (originally $O(n^{3.5} L)$ but seems unclear)	$O(nm+m^{2})$?	Exact	Deterministic
Cohen; Lee and Song	2018	$O(n^{max(omega, {2.5}-alpha/{2}, {13}/{6})}*polylog(n, m, L))$, where omega is the exponent on matrix multiplication, alpha is the dual exponent of matrix multiplication; currently $O(n^{2.37285956})$ || $O(nm+n^{2})$? || Exact || Deterministic || Time
Lee and Sidford	2015	$O((nnz(A) + n^{2}) n^{0.5})$	$O(nm+n^{2})$??	Exact	Deterministic	Time
Vaidya	1987	$O(((m+n)$n^{2}+(m+n)^{1.5}*n)L^{2} logL loglogL)	$O((nm+n^{2})$L)?	Exact	Deterministic	Time
Vaidya	1989	$O((m+n)$^{1.5}*n*L^{2} logL loglogL)	$O((nm+n^{2})$L)?	Exact	Deterministic	Time
Jiang, Song, Weinstein and Zhang	2020	$O(n^(max(omega, {2.5}-alpha/{2}, {37}/{18}))*polylog(n, m, L))$, where omega is the exponent on matrix multiplication, alpha is the dual exponent of matrix multiplication; currently $O(n^{2.37285956})$ || || Exact || Deterministic || Time

References/Citation

https://arxiv.org/abs/1810.07896