Art gallery theorems and algorithms
Art gallery theorems and algorithms
Approximation algorithms for scheduling unrelated parallel machines
Mathematical Programming: Series A and B
Journal of the ACM (JACM)
Finding and counting small induced subgraphs efficiently
Information Processing Letters
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computing Nash equilibria for scheduling on restricted parallel links
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Discrete Applied Mathematics
Approximation Algorithms for the Graph Orientation Minimizing the Maximum Weighted Outdegree
AAIM '07 Proceedings of the 3rd international conference on Algorithmic Aspects in Information and Management
Approximation scheme for lowest outdegree orientation and graph density measures
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
Monadic Second Order Logic on Graphs with Local Cardinality Constraints
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
A note on graph balancing problems with restrictions
Information Processing Letters
Monadic second order logic on graphs with local cardinality constraints
ACM Transactions on Computational Logic (TOCL)
Graph classes and the complexity of the graph orientation minimizing the maximum weighted outdegree
Discrete Applied Mathematics
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Given an undirected graph with edge weights, we are asked to find an orientation, i.e., an assignment of a direction to each edge, so as to minimize the weighted maximum outdegree in the resulted directed graph. The problem is called MMO, and is a restricted variant of the well-known minimum makespan problem. As previous studies, it is shown that MMO is in P for trees, weak NP-hard for planar bipartite graphs, and strong NP-hard for general graphs. There are still gaps between those graph classes. The objective of this paper is to show tight thresholds of complexity: We show that MMO is (i) in P for cactuses, (ii) weakly NP-hard for outerplanar graphs, and also (iii) strongly NP-hard for P4-bipartite graphs. The latter two are minimal superclasses of the former. Also, we show the NP-hardness for the other related graph classes, diamond-free, house-free, series-parallel, bipartite and planar.