Scheduling Problems

Abstract

We introduce the notion of a scheduling problem which is a boolean function $S$ over atomic formulas of the form $x_i \leq x_j$. Considering the $x_i$ as jobs to be performed, an integer assignment satisfying $S$ schedules the jobs subject to the constraints of the atomic formulas. The scheduling counting function counts the number of solutions to $S$. We prove that this counting function is a polynomial in the number of time slots allowed. Scheduling polynomials include the chromatic polynomial of a graph, the zeta polynomial of a lattice, the Billera-Jia-Reiner polynomial of a matroid. To any scheduling problem, we associate not only a counting function for solutions, but also a quasisymmetric function and a quasisymmetric function in non-commuting variables. These scheduling functions include the chromatic symmetric functions of Sagan, Gebhard, and Stanley, and a close variant of Ehrenborg's quasisymmetric function for posets. Geometrically, we consider the space of all solutions to a given scheduling problem. We extend a result of Steingr\'immson by proving that the $h$-vector of the space of solutions is given by a shift of the scheduling polynomial. Furthermore, under certain niceness conditions on the defining boolean function, we prove partitionability of the space of solutions and positivity of fundamental expansions of the scheduling quasisymmetric functions and of the $h$-vector of the scheduling polynomial.