Acyclic colorings of planar graphs
Discrete Mathematics
Network flows: theory, algorithms, and applications
Network flows: theory, algorithms, and applications
Efficient algorithms for acyclic colorings of graphs
Theoretical Computer Science
Introduction to algorithms
The Approximation of Maximum Subgraph Problems
ICALP '93 Proceedings of the 20th International Colloquium on Automata, Languages and Programming
Efficient Algorithms for Vertex Arboricity of Planar Graphs
Proceedings of the 15th Conference on Foundations of Software Technology and Theoretical Computer Science
CP '01 Proceedings of the 7th International Conference on Principles and Practice of Constraint Programming
On the vertex-arboricity of planar graphs
European Journal of Combinatorics
2-List-coloring planar graphs without monochromatic triangles
Journal of Combinatorial Theory Series B
Computer Aided Geometric Design
Exact algorithms for coloring graphs while avoiding monochromatic cycles
AAIM'10 Proceedings of the 6th international conference on Algorithmic aspects in information and management
Heuristics for Deciding Collectively Rational Consumption Behavior
Computational Economics
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We consider the problem of deciding whether a given directed graph can be vertex partitioned into two acyclic subgraphs. Applications of this problem include testing rationality of collective consumption behavior, a subject in microeconomics. We prove that the problem is NP-complete even for oriented graphs and argue that the existence of a constant-factor approximation algorithm is unlikely for an optimization version that maximizes the number of vertices that can be colored using two colors while avoiding monochromatic cycles. We present three exact algorithms---namely, an integer-programming algorithm based on cycle identification, a backtracking algorithm, and a branch-and-check algorithm. We compare these three algorithms both on real-life instances and on randomly generated graphs. We find that for the latter set of graphs, every algorithm solves instances of considerable size within a few seconds; however, the CPU time of the integer-programming algorithm increases with the number of vertices in the graph more clearly than the CPU time of the two other procedures. For real-life instances, the integer-programming algorithm solves the largest instance in about a half hour, whereas the branch-and-check algorithm takes approximately 10 minutes and the backtracking algorithm less than 5 minutes. Finally, for every algorithm, we also study empirically the transition from a high to a low probability of a YES answer as a function of the number of arcs divided by the number of vertices.