# Approximation algorithms for the Graph Orientation Minimizing the Maximum Weighted Outdegree

CREATED: 200712180727 Speaker: Hirotaka Ono (Kyushu University) ** Background

- art gallery problem, placing guards to observe the gallery
- guards assumed to be omni-directional
- suppose using k-way directional camera
- what is the min k required? = graph orientation (minimizing the max outdegree)
- graph model: camera - vertex, corridor - edge ** Related problem
- job scheduling, where each job can be assigned to one of 2 machines
- job - edge, and processing time does not depend on machine ** Inapproximability
- reduction from at most 3SAT (2 literals) ** Approximation algorithm
- G -> G’ ** replace weighted edge with multiple unweighted edge (2 approx = LP relaxation)
- G’-> F ** find an oriented cycle ** reverse the edges along the cycle (outdegree remains unchanged) ** if all edges from a node oriented in the same direction, then fix the edge
- F ** orient the edges for the forest ** Graph classes
- tree -> cactus -> outer planar -> series-parallel -> planar -> general