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Near-Boolean Optimization: A Continuous Approach to Set Packing and Partitioning

  • Giovanni RossiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10163)

Abstract

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.

Keywords

Pseudo-Boolean function Möbius inversion Set packing Partition function Polynomial multilinear extension Local search 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer Science and Engineering - DISIUniversity of BolognaBolognaItaly

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