On Line Sum Optimization

TL;DR

This paper demonstrates polynomial-time solutions for the column sum optimization problem under certain conditions and conjectures similar tractability for the more general line sum optimization problem.

Contribution

It provides polynomial algorithms for specific cases of the column sum optimization problem and conjectures polynomial solvability for the broader line sum problem.

Findings

Column sum optimization can be solved in polynomial time when all functions are identical.

Polynomial solutions exist when row sums are bounded by a constant.

Conjecture that line sum optimization may also be polynomial-time solvable.

Abstract

We show that the {\em column sum optimization problem}, of finding a $(0,1)$-matrix with prescribed row sums which minimizes the sum of evaluations of given functions at its column sums, can be solved in polynomial time, either when all functions are the same or when all row sums are bounded by any constant. We conjecture that the more general {\em line sum optimization problem}, of finding a matrix minimizing the sum of given functions evaluated at its row sums and column sums, can also be solved in polynomial time.