Fair allocation of indivisible goods with minimum inequality or minimum envy

We provide a new mathematical framework for the classic problem of fair allocation of indivisible goods, showing that it can be formulated as the problem of finding an optimal column rearrangement of multi- ple matrices. Based on this formulation, we design two novel algorithms called MinCov and MinCovTar- get to find optimal allocations under the newly introduced notion of minimum social inequality, and the popular notion of minimum envy. Numerical illustrations show an excellent performance of the newly developed algorithms also with respect to other allocation criteria, in particular for the maximum Nash welfare.