Efficient implementation of Carathéodory’s theorem for the single machine scheduling polytope

Abstract

In a fundamental paper in polyhedral combinatorics, Queyranne describes the complete facial structure of a classical object in combinatorial optimization, the single machine scheduling polytope. In the same paper, he answers essentially all relevant algorithmic questions with respect to optimization and separation. In the present paper, motivated by recent applications in the design of optimal incentive compatible mechanisms, we address an algorithmic question that was apparently not addressed before. Namely, we turn Carathéodory's theorem into an algorithm, and ask to write an arbitrary point in the scheduling polytope as a convex combination of the vertices of the polytope. We give a combinatorial O(n 2 ) time algorithm, which is linear in the naive encoding of the output size. We obtain this result by exploiting the fact that the scheduling polytope is a zonotope, and by the observation that its barycentric subdivision has a simple, linear description. The actual decomposition algorithm is an implementation of a method proposed by Grötschel, Lovász and Schrijver, applied to one of the subpolytopes of the barycentric subdivision. We thereby also shed new light on an algorithm recently proposed for a special case, namely the permutahedron.

Introducing the Expohedron for Efficient Pareto-optimal Fairness-Utility Amortizations in Repeated Rankings

“…This is already a significant improvement over the approach using BvN decomposition, as BvN decompositions use at most ( − 1) 2 + 1 permutation matrices [15]. There exist already some published Carathéodory decomposition algorithms for permutohedra [13,41], but they cannot naturally be extended to expohedra. Expohedra are not always zonotopes, which is a key property used in [13], and they are generally not defined with = (1, .…”

“…There exist already some published Carathéodory decomposition algorithms for permutohedra [13,41], but they cannot naturally be extended to expohedra. Expohedra are not always zonotopes, which is a key property used in [13], and they are generally not defined with 𝜸 = (1, . .…”

“…Figure 2: GLS procedure, inspired from an illustration in [13]. The barycenter x of the expohedron is decomposed into a convex sum of vertices v 1 , v 2 , v 3 .…”