@book{f59841aadf3f408680864afa0211ee40,

title = "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.",

keywords = "METIS-300180, IR-88292, EWI-24630, Polyhedral subdivision, Algorithms, Zonotopes, Scheduling polytope, Convex combination, Decomposition, Polyhedral theory",

author = "R.P. Hoeksma and Bodo Manthey and Uetz, {Marc Jochen}",

year = "2013",

month = dec,

day = "4",

language = "Undefined",

series = "CTIT Technical Report Series",

publisher = "Centre for Telematics and Information Technology (CTIT)",

number = "TR-CTIT-13-25",

address = "Netherlands",

}