Continuous set packing and near-Boolean functions

TL;DR

This paper introduces a novel continuous optimization approach for the set packing problem using near-Boolean functions, enabling efficient approximation of solutions through gradient-based local search.

Contribution

It proposes a new polynomial multilinear form with variables in a high-dimensional simplex, translating the discrete problem into a continuous domain for improved solution methods.

Findings

Maximizers include disjoint hypercube vertices representing feasible solutions.

The method provides a continuous relaxation that can be optimized with gradient-based techniques.

Feasible solutions correspond to partitions of the ground set.

Abstract

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, rather than in the unit interval as usual. The 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 polynomial multilinear form of bounded degree and near-Boolean games are also finally discussed.