, Volume 70, Issue 3, pp 457–492 | Cite as

Polynomial-Time Algorithms for Energy Games with Special Weight Structures

  • Krishnendu Chatterjee
  • Monika Henzinger
  • Sebastian Krinninger
  • Danupon Nanongkai


Energy games belong to a class of turn-based two-player infinite-duration games played on a weighted directed graph. It is one of the rare and intriguing combinatorial problems that lie in NPco-NP, but are not known to be in P. The existence of polynomial-time algorithms has been a major open problem for decades and apart from pseudopolynomial algorithms there is no algorithm that solves any non-trivial subclass in polynomial time.

In this paper, we give several results based on the weight structures of the graph. First, we identify a notion of penalty and present a polynomial-time algorithm when the penalty is large. Our algorithm is the first polynomial-time algorithm on a large class of weighted graphs. It includes several worst-case instances on which previous algorithms, such as value iteration and random facet algorithms, require at least sub-exponential time. Our main technique is developing the first non-trivial approximation algorithm and showing how to convert it to an exact algorithm. Moreover, we show that in a practical case in verification where weights are clustered around a constant number of values, the energy game problem can be solved in polynomial time. We also show that the problem is still as hard as in general when the clique-width is bounded or the graph is strongly ergodic, suggesting that restricting the graph structure does not necessarily help.


Graph algorithms Polynomial-time algorithms Turn-based infinite duration games Energy games Mean-payoff games 


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Krishnendu Chatterjee
    • 1
  • Monika Henzinger
    • 2
  • Sebastian Krinninger
    • 2
  • Danupon Nanongkai
    • 3
  1. 1.Institute of Science and TechnologyKlosterneuburgAustria
  2. 2.University of Vienna, Faculty of Computer ScienceViennaAustria
  3. 3.Nanyang Technological UniversitySingaporeSingapore

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