ISCO 2014: Combinatorial Optimization pp 280-291

# Decomposition Algorithm for the Single Machine Scheduling Polytope

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8596)

## 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 $$x\in C$$, Carathéodory’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 $$O (n^2)$$, 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 Grötschel, Lovász, and Schrijver applied to one of these subpolytopes.

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