On the most imbalanced orientation of a graph

We study the problem of orienting the edges of a graph such that the minimum over all the vertices of the absolute difference between the outdegree and the indegree of a vertex is maximized. We call this minimum the imbalance of the orientation, i.e. the higher it gets, the more imbalanced the orientation is. The studied problem is denoted by $${{\mathrm{\textsc {MaxIm}}}}$$ M A X I M . We first characterize graphs for which the optimal objective value of $${{\mathrm{\textsc {MaxIm}}}}$$ M A X I M is zero. Next we show that $${{\mathrm{\textsc {MaxIm}}}}$$ M A X I M is generally NP-hard and cannot be approximated within a ratio of $$\frac{1}{2}+\varepsilon $$ 1 2 + ε for any constant $$\varepsilon >0$$ ε > 0 in polynomial time unless $$\texttt {P}=\texttt {NP}$$ P = NP even if the minimum degree of the graph $$\delta $$ δ equals 2. Then we describe a polynomial-time approximation algorithm whose ratio is almost equal to $$\frac{1}{2}$$ 1 2 . An exact polynomial-time algorithm is also derived for cacti. Finally, two mixed integer linear programming formulations are presented. Several valid inequalities are exhibited with the related separation algorithms. The performance of the strengthened formulations is assessed through several numerical experiments.