Diagonal matrix scaling is NP-hard

Abstract

A symmetric matrix A is said to be scalable if there exists a positive diagonal matrix X such that the row and column sums of XAX are all ones. We show that testing the scalability of arbitrary matrices is NP-hard. Equivalently, it is NP-hard to check for a given symmetric matrix A whether the logarithmic barrier function 1 2 x TAx − ∑ ln x i has a stationary point in the positive orthant x > 0.