Art gallery theorems and algorithms
Art gallery theorems and algorithms
Approximation algorithms for scheduling unrelated parallel machines
Mathematical Programming: Series A and B
A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth
SIAM Journal on Computing
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
Dynamic Representation of Sparse Graphs
WADS '99 Proceedings of the 6th International Workshop on Algorithms and Data Structures
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
Graph classes and the complexity of the graph orientation minimizing the maximum weighted outdegree
CATS '08 Proceedings of the fourteenth symposium on Computing: the Australasian theory - Volume 77
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
Monadic Second Order Logic on Graphs with Local Cardinality Constraints
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
Approximation scheme for lowest outdegree orientation and graph density measures
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
The complexity of the proper orientation number
Information Processing Letters
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Given an undirected graph with edge weights, we are asked to find an orientation, that is, 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 in 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 tighter thresholds of complexity: We show that MMO is (i) in P for cactus graphs, (ii) weakly NP-hard for outerplanar graphs, and also (iii) strongly NP-hard for graphs which are both planar and bipartite. This implies the NP-hardness for P"4-bipartite, diamond-free or house-free graphs, each of which is a superclass of cactus. We also show (iv) the NP-hardness for series-parallel graphs and multi-outerplanar graphs, and (v) present a pseudo-polynomial time algorithm for graphs with bounded treewidth.