On the most imbalanced orientation of a graph

Walid Ben-Ameur (1, 2, 3) , Antoine Glorieux (1, 2, 3) , José Neto (1, 2, 3)

1 METHODES-SAMOVAR - Méthodes et modèles pour les réseaux
2 TSP - RS2M - Département Réseaux et Services Multimédia Mobiles
3 CNRS - Centre National de la Recherche Scientifique

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 MAXIM. We first characterize graphs for which the optimal objective value of MAXIM is zero. Next we show that MAXIM is generally NP-hard and cannot be approximated within a ratio of 1 2 + ε for any constant ε > 0 in polynomial time unless P = NP even if the minimum degree of the graph δ equals 2. Then we describe a polynomial-time approximation algorithm whose ratio is almost equal to 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.

Mots clés: Polynomial separation; NP-complete; Approximation algorithm; Complexity; Exact algorithm; Graph orientation; Cacti; Cutting plane algorithm; Valid inequalities; Imbalance; Combinatorial optimization; Graph theory; Separation problem; Mixed Integer programming

Domaines: Combinatoire [math.CO]; Informatique [cs]; Mathématique discrète [cs.DM]

Walid Ben-Ameur, Antoine Glorieux, José Neto. On the most imbalanced orientation of a graph. Journal of Combinatorial Optimization, 2018, 36 (2), pp.637-669. ⟨10.1007/s10878-017-0117-1⟩. ⟨hal-01497902⟩