On the star decomposition of a graph: Hardness results and approximation for the max-min optimization problem

We study the problem of decomposing a graph into stars so that the minimum size star in the decomposition is as large as possible. Problems of graph decomposition have been actively investigated since the 70's. The question we consider here also combines the structure of a facility location problem (choosing the centres of the stars) with a max-min fairness optimization criterion that has recently received attention in resource allocation problems, e.g., the Santa Claus problem. We focus on computational and algorithmic questions: We show that the problem is hard even in the case of planar graphs of maximum degree not larger than four, and already for decompositions into stars of size at least three. We are able to tightly characterize the boundaries between efficiently solvable instances and hard ones: we show that relaxing any of the conditions in our hardness result (minimum size of the stars or degree of the input graph) makes the problem polynomially solvable. Our complexity result implies also the APX hardness of the problem ruling out any approximation guarantee better than 2/3. We complement this inapproximability result with a 1/2-approximation algorithm. Finally, we give a polynomial time algorithm for trees. A nice property of our algorithms is that they can all be implemented to run in time linear in the size of the input graph. (C) 2020 Published by Elsevier B.V.