A Strongly Polynomial Label-Correcting Algorithm for Linear Systems with Two Variables per Inequality

We present a strongly polynomial label-correcting algorithm for solving the feasibility of linear systems with two variables per inequality (2VPI). The algorithm is based on the Newton-Dinkelbach method for fractional combinatorial optimization. We extend and strengthen previous work of Madani (2002) that showed a weakly polynomial bound for a variant of the Newton-Dinkelbach method for solving deterministic Markov decision processes (DMDPs), a special class of 2VPI linear programs. For a 2VPI system with $n$ variables and $m$ constraints, our algorithm runs in $O(mn)$ iterations. Every iteration takes $O(m + n\log n)$ time for DMDPs, and $O(mn)$ time for general 2VPI systems. The key technical idea is a new analysis of the Newton-Dinkelbach method exploiting gauge symmetries of the algorithm. This also leads to an acceleration of the Newton-Dinkelbach method for general fractional combinatorial optimization problems. For the special case of linear fractional combinatorial optimization, our method converges in $O(m\log m)$ iterations, improving upon the previous best bound of $O(m^2\log m)$ by Wang et al. (2006).