The Directed Dominating Set Problem: Generalized Leaf Removal and Belief Propagation

  • Yusupjan Habibulla
  • Jin-Hua Zhao
  • Hai-Jun ZhouEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9130)


A minimum dominating set for a digraph (directed graph) is a smallest set of vertices such that each vertex either belongs to this set or has at least one parent vertex in this set. We solve this hard combinatorial optimization problem approximately by a local algorithm of generalized leaf removal and by a message-passing algorithm of belief propagation. These algorithms can construct near-optimal dominating sets or even exact minimum dominating sets for random digraphs and also for real-world digraph instances. We further develop a core percolation theory and a replica-symmetric spin glass theory for this problem. Our algorithmic and theoretical results may facilitate applications of dominating sets to various network problems involving directed interactions.


Directed graph Dominating vertices Graph observation Core percolation Message passing 



This research is partially supported by the National Basic Research Program of China (grant number 2013CB932804) and by the National Natural Science Foundations of China (grant numbers 11121403 and 11225526). HJZ conceived research, JHZ and YH performed research, HJZ and JHZ wrote the paper. Correspondence should be addressed to HJZ ( or to JHZ (


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Yusupjan Habibulla
    • 1
  • Jin-Hua Zhao
    • 1
  • Hai-Jun Zhou
    • 1
    Email author
  1. 1.State Key Laboratory of Theoretical Physics, Institute of Theoretical PhysicsChinese Academy of SciencesBeijingChina

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