Karmarkar's algorithm is an algorithm introduced by Narendra Karmarkar in 1984 for solving linear programming problems. It was the first reasonably efficient algorithm that solves these problems in polynomial time. The ellipsoid method is also polynomial time but proved to be inefficient in practice.

Denoting by ${\displaystyle n}$ the number of variables, m the number of inequality constraints, and ${\displaystyle L}$ the number of bits of input to the algorithm, Karmarkar's algorithm requires ${\displaystyle O(m^{1.5}n^{2}L)}$ operations on ${\displaystyle O(L)}$-digit numbers, as compared to ${\displaystyle O(n^{3}(n+m)L)}$ such operations for the ellipsoid algorithm. In "square" problems, when m is in O(n), Karmarkar's algorithm requires ${\displaystyle O(n^{3.5}L)}$ operations on ${\displaystyle O(L)}$-digit numbers, as compared to ${\displaystyle O(n^{4}L)}$ such operations for the ellipsoid algorithm. The runtime of Karmarkar's algorithm is thus $${\displaystyle O(n^{3.5}L^{2}\cdot \log L\cdot \log \log L),}$$ using FFT-based multiplication (see Big O notation).

Karmarkar's algorithm falls within the class of interior-point methods: the current guess for the solution does not follow the boundary of the feasible set as in the simplex method, but moves through the interior of the feasible region, improving the approximation of the optimal solution by a definite fraction with every iteration and converging to an optimal solution with rational data.

Consider a linear programming problem in matrix form:

Karmarkar's algorithm determines the next feasible direction toward optimality and scales back by a factor 0 < γ ≤ 1. It is described in a number of sources. Karmarkar also has extended the method to solve problems with integer constraints and non-convex problems.

Since the actual algorithm is rather complicated, researchers looked for a more intuitive version of it, and in 1985 developed affine scaling, a version of Karmarkar's algorithm that uses affine transformations where Karmarkar used projective ones, only to realize four years later that they had rediscovered an algorithm published by Soviet mathematician I. I. Dikin in 1967. The affine-scaling method can be described succinctly as follows. While applicable to small scale problems, it is not a polynomial time algorithm.

Consider the linear program $${\displaystyle {\begin{array}{lrclr}{\text{maximize}}&x_{1}+x_{2}\\{\text{subject to}}&2px_{1}+x_{2}&\leq &p^{2}+1,&p=0.0,0.1,0.2,\ldots ,0.9,1.0.\end{array}}}$$ That is, there are 2 variables ${\displaystyle x_{1},x_{2}}$ and 11 constraints associated with varying values of ${\displaystyle p}$. This figure shows each iteration of the algorithm as red circle points. The constraints are shown as blue lines.

At the time he invented the algorithm, Karmarkar was employed by IBM as a postdoctoral fellow in the IBM San Jose Research Laboratory in California. On August 11, 1983 he gave a seminar at Stanford University explaining the algorithm, with his affiliation still listed as IBM. By the fall of 1983 Karmarkar started to work at AT&T and submitted his paper to the 1984 ACM Symposium on Theory of Computing (STOC, held April 30 - May 2, 1984) stating AT&T Bell Laboratories as his affiliation. After applying the algorithm to optimizing AT&T's telephone network, they realized that his invention could be of practical importance. In April 1985, AT&T promptly applied for a patent on his algorithm.