Decomposition Algorithm for the Single Machine Scheduling Polytope

Abstract

Given an $n$-vector $p$ of processing times of jobs, the single machine scheduling polytope $C$ arises as the convex hull of completion times of jobs when these are scheduled without idle time on a single machine. Given a point $x\in C$, Carath\'eodory's theorem implies that $x$ can be written as convex combination of at most $n$ vertices of $C$. We show that this convex combination can be computed from $x$ and $p$ in time \bigO{n^2}, which is linear in the naive encoding of the output. We obtain this result using essentially two ingredients. First, we build on the fact that the scheduling polytope is a zonotope. Therefore, all of its faces are centrally symmetric. Second, instead of $C$, we consider the polytope $Q$ of half times and its barycentric subdivision. We show that the subpolytopes of this barycentric subdivison of $Q$ have a simple, linear description. The final decomposition algorithm is in fact an implementation of an algorithm proposed by Gr{\o}tschel, Lov{\'a}sz, and Schrijver applied to one of these subpolytopes.