Line-Sum Symmetry via the DomEig Algorithm
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Discrete Applied Mathematics
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An $n \times n$ matrix with nonnegative entries is said to be $balanced$ if for each $i=1,\dots,n$ the sum of the entries of its $i$th row is equal to the sum of the entries of its $i$th column. An $n \times n$ matrix $A$ with nonnegative entries is said to be $balancable$ $via$ $diagonal$ $similarity$ $scaling$ if there exists a diagonal matrix $X$ with positive diagonal entries such that $XAX^{-1}$ is balanced. We give upper and lower bounds on the entries of $X$ and prove the necessary sensitivity analysis in the required accuracy of the minimization of an associated convex programming problem. These results are used to prove the polynomial-time solvability of computing $X$ to any prescribed accuracy.