Near-Boolean Optimization: A Continuous Approach to Set Packing and Partitioning
Near-Boolean functions essentially associate with every partition of a finite ground set the sum of the real values taken on blocks by an underlying set function. Given a family of feasible subsets of the ground set, the packing problem is to find a largest subfamily of pairwise disjoint family members. Through a multilinear polynomial whose variables are indexed by the elements of the ground set and correspond each to a unit membership distribution over those feasible subsets where their index is included, the problem is translated into a continuous version with the objective function taking values on peculiar collections of points in a unit hypercube. Extremizers are shown to include feasible solutions for the original combinatorial optimization problem, allowing for a gradient-based local search. Least-squares approximations with bounded degree and coalition formation games are also finally discussed.
KeywordsPseudo-Boolean function Möbius inversion Set packing Partition function Polynomial multilinear extension Local search
- 4.Rossi, G.: Continuous set packing and near-Boolean functions. In: Proceedings of 5th International Conference on Pattern Recognition Applications and Methods, pp. 84–96 (2016)Google Scholar
- 6.Trevisan, L.: Non-approximability results for optimization problems on bounded degree instances. In: Proceedings 33rd ACM Symposium on Theory of Computing, pp. 453–461 (2001)Google Scholar
- 26.Schrijver, A.: A Course in Combinatorial Optimization. Alexander Schrijver (2013). http://homepages.cwi.nl/~lex/files/dict.pdf
- 29.Bowles, S.: Microeconomics: Behavior, Institutions, and Evolution. Princeton University Press, Princeton (2004)Google Scholar
- 33.Rossi, G.: Multilinear objective function-based clustering. In: Proceedings of 7th International Joint Conference on Computational Intelligence, vol. 2(FCTA), pp. 141–149 (2015)Google Scholar