The Subset Sum Game Revisited

  • Astrid Pieterse
  • Gerhard J. WoegingerEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10576)


We discuss a game theoretic variant of the subset sum problem, in which two players compete for a common resource represented by a knapsack. Each player owns a private set of items, players pack items alternately, and each player either wants to maximize the total weight of his own items packed into the knapsack or to minimize the total weight of the items of the other player.

We show that finding the best packing strategy against a hostile or a selfish adversary is PSPACE-complete, and that against these adversaries the optimal reachable item weight for a player cannot be approximated within any constant factor (unless P=NP). The game becomes easier when the adversary is short-sighted and plays greedily: finding the best packing strategy against a greedy adversary is NP-complete in the weak sense. This variant forms one of the rare examples of pseudo-polynomially solvable problems that have a PTAS, but do not allow an FPTAS.


  1. 1.
    Caprara, A., Carvalho, M., Lodi, A., Woeginger, G.J.: A complexity and approximability study of the bilevel knapsack problem. SIAM J. Optim. 24, 823–838 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Darmann, A., Nicosia, G., Pferschy, U., Schauer, J.: The subset sum game. Euro. J. Oper. Res. 233, 539–549 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  4. 4.
    Schaefer, T.J.: On the complexity of some two-person perfect-information games. J. Comput. Syst. Sci. 16, 185–225 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Woeginger, G.J.: When does a dynamic programming formulation guarantee the existence of a fully polynomial time approximation scheme (FPTAS)? INFORMS J. Comput. 12, 57–75 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceEindhoven University of TechnologyEindhovenNetherlands
  2. 2.Department of Computer ScienceRWTH AachenAachenGermany

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