Continuous Set Packing and Near-Boolean Functions

Giovanni Rossi


Given a family of feasible subsets of a ground set, the packing problem is to find a largest subfamily of pairwise disjoint family members. Non-approximability renders heuristics attractive viable options, while efficient methods with worst-case guarantee are a key concern in computational complexity. This work proposes a novel near-Boolean optimization method relying on a polynomial multilinear form with variables ranging each in a high-dimensional unit simplex. These variables are the elements of the ground set, and distribute each a unit membership over those feasible subsets where they are included. The given problem is thus translated into a continuous version where the objective is to maximize a function taking values on collections of points in a unit hypercube. Maximizers are shown to always include collections of hypercube disjoint vertices, i.e. partitions of the ground set, which constitute feasible solutions for the original discrete version of the problem. A gradient-based local search in the expanded continuous domain is designed. Approximations with polynomials of bounded degree and near-Boolean coalition formation games are also finally discussed.


  1. Aigner, M. (1997). Combinatorial Theory. Springer. Reprint of the 1979 Edition.
  2. and Boros, E. and Hammer, P. (2002). Pseudo-Boolean optimization. Discrete App. Math., 123:155-225.
  3. Bowles, S. (2004). Microeconomics: Behavior, Institutions, and Evolution. Princeton University Press.
  4. Chandra, B. and Halldorsson, M. M. (2001). Greedy local improvement and weighted set packing. Journal of Algorithms, (39):223-240.
  5. Conitzer, V. and Sandholm, T. (2006). Complexity of constructing solutions in the core based on synergies among coalitions. Artificial Intel,. 170:607-619.
  6. Crama, Y. and Hammer, P. L. (2011). Boolean Functions: Theory, Algorithms, and Applications. Cambridge University Press.
  7. Gilboa, I. and Lehrer, E. (1990). Global games. International Journal of Game Theory, (20):120-147.
  8. Gilboa, I. and Lehrer, E. (1991). The value of information - an axiomatic approach. Journal of Mathematical Economics, 20(5):443-459.
  9. Grünbaum, B. (2001). Convex Polytopes 2nd ed. Springer.
  10. Hammer, P. and Holzman, R. (1992). Approximations of pseudo-Boolean functions; applications to game theory. Math. Methods of Op. Res. - ZOR, 36(1):3-21.
  11. Hazan, E., Safra, S., and Schwartz, O. (2006). On the complexity of approximating k-set packing. Computational Complexity, (15):20-39.
  12. Korte, B. and Vygen, J. (2002). Combinatorial Optimization. Theory and Algorithms. Springer.
  13. Mas-Colell, A., Whinston, M. D., and Green, J. R. (1995). Microeconomic Theory. Oxford University Press.
  14. Milgrom, P. (2004). Putting Auction Theory to Work. Cambridge University Press.
  15. Monderer, D. and Shapley, L. S. (1996). Potential games. Games and Economic Behavior, 14(1):124-143.
  16. Papadimitriou, C. (1994). Computational Complexity. Addison Wesley.
  17. Rahwan, T. and Jenning, N. (2007). An algorithm for distributing coalitional value calculations among cooperating agents. Artificial Intelligence, 171:535-567.
  18. Rossi, G. (2015). Multilinear objective function-based clustering. In Proc. 7th Int. J. Conf. on Computational Intelligence, volume 2 (FCTA), pages 141-149.
  19. Rota, G.-C. (1964a). The number of partitions of a set. American Mathematical Monthly, 71:499-504.
  20. Rota, G.-C. (1964b). On the foundations of combinatorial theory I: theory of Möbius functions. Z. Wahrscheinlichkeitsrechnung u. verw. Geb., 2:340-368.
  21. Roth, A. (1988). The Shapley value. Cambridge Univ. Press.
  22. Sandholm, T. (2002). Algorithm for optimal winner determination in combinatorial auctions. Artificial Intelligence, (135):1-54.
  23. Schaeffer, S. E. (2007). Graph clustering. Computer Science Review, 1(1):27-64.
  24. Slikker, M. (2001). Coalition formation and potential games. Games and Econ. Behavior, 37(2):436 - 448.
  25. Stanley, R. (1971). Modular elements of geometric lattices. Algebra Universalis, (1):214-217.
  26. Stern, M. (1999). Semimodular Lattices. Theory and Applications. Cambridge University Press.
  27. Trevisan, L. (2001). Non-approximability results for optimization problems on bounded degree instances. In Proc. 33rd ACM Symp. on Theory of Computing, pages 453-461.

Paper Citation

in Harvard Style

Rossi G. (2016). Continuous Set Packing and Near-Boolean Functions . In Proceedings of the 5th International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM, ISBN 978-989-758-173-1, pages 84-96. DOI: 10.5220/0005697800840096

in Bibtex Style

author={Giovanni Rossi},
title={Continuous Set Packing and Near-Boolean Functions},
booktitle={Proceedings of the 5th International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,},

in EndNote Style

JO - Proceedings of the 5th International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,
TI - Continuous Set Packing and Near-Boolean Functions
SN - 978-989-758-173-1
AU - Rossi G.
PY - 2016
SP - 84
EP - 96
DO - 10.5220/0005697800840096