Generalized Permutahedra and Optimal Auctions

Joswig, Michael; Klimm, Max; Spitz, Sylvain

doi:10.1137/21m1441286

Cited by 2 publications

(1 citation statement)

References 28 publications

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“…They were rediscovered by Postnikov in 2009 [31], who initiated the investigation of their rich combinatorial structure. They have since become a widely studied family of polytopes that appears naturally in several areas of mathematics, such as algebraic combinatorics [1,2,34], optimization [17], game theory [14], statistics [27,28], and economic theory [20]. The set of deformed permutahedra can be parametrized by the cone of submodular functions [16,31].…”

Section: Introductionmentioning

confidence: 99%

Deformed Graphical Zonotopes

Padrol,

Pilaud,

Poullot

2023

Discrete Comput Geom

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We study deformations of graphical zonotopes. Deformations of the classical permutahedron (which is the graphical zonotope of the complete graph) have been intensively studied in recent years under the name of generalized permutahedra. We provide an irredundant description of the deformation cone of the graphical zonotope associated to a graph G, consisting of independent equations defining its linear span (in terms of non-cliques of G) and of the inequalities defining its facets (in terms of common neighbors of neighbors in G). In particular, we deduce that the faces of the standard simplex corresponding to induced cliques in G form a linear basis of the deformation cone, and that the deformation cone is simplicial if and only if G is triangle-free.

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Section: Introductionmentioning

confidence: 99%

Deformed Graphical Zonotopes

Padrol,

Pilaud,

Poullot

2023

Discrete Comput Geom

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The polyhedral geometry of truthful auctions

Joswig,

Klimm,

Spitz

2024

Math. Program.

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The difference set of an outcome in an auction is the set of types that the auction mechanism maps to the outcome. We give a complete characterization of the geometry of the difference sets that can appear for a dominant strategy incentive compatible multi-unit auction showing that they correspond to regular subdivisions of the unit cube. Similarly, we describe the geometry for affine maximizers for n players and m items, showing that they correspond to regular subdivisions of the m-fold product of $$(n-1)$$ ( n - 1 ) -dimensional simplices. These observations are then used to construct mechanisms that are robust in the sense that the sets of items allocated to the players change only slightly when the players’ reported types are changed slightly.

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Generalized Permutahedra and Optimal Auctions

Cited by 2 publications

References 28 publications

Deformed Graphical Zonotopes

Deformed Graphical Zonotopes

The polyhedral geometry of truthful auctions

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