Feasibility criteria for high-multiplicity partitioning problems

TL;DR

This paper establishes near-optimal feasibility criteria for high-multiplicity partitioning problems, providing conditions that are both sufficient and necessary in many cases, with applications in algebraic geometry.

Contribution

It introduces explicit feasibility criteria for partitioning balls into bins with capacity constraints, matching necessary conditions up to constants, and proves their optimality for distinct weights.

Findings

Criteria are sufficient and nearly necessary for large d

Constants are proven optimal for distinct weights

Applications to asymptotic algebraic geometry results

Abstract

For fixed weights w_1,...,w_n, and for d>0, we let B denote a collection of d*n balls, with d balls of weight w_i for each i=1,...,n. We consider the problem of assigning the balls to n bins with capacities C_1,...,C_n, in such a way that each bin is assigned d balls, without exceeding its capacity. When d>>0, we give sufficient criteria for the feasibility of this problem, which coincide up to explicit constants with the natural set of necessary conditions. Furthermore, we show that our constants are optimal when the weights w_i are distinct. The feasibility criteria that we present here are used elsewhere (in commutative algebra) to study the asymptotic behavior of the Castelnuovo-Mumford regularity of symmetric monomial ideals.