Minkowski Addition of Polytopes: Computational Complexity and Applications to Gröbner Bases

Gritzmann, Peter; Sturmfels, Bernd

doi:10.1137/0406019

Cited by 179 publications

(146 citation statements)

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“…A direction of an edge [u, v] of a polytope P is any nonzero scalar multiple of v − u. We provide a simple proof of the following fact (see [23]) which is quite central to our considerations. Lemma 2.3.…”

Section: Edge-directions and Zonotopesmentioning

confidence: 99%

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Convex Combinatorial Optimization

Onn

Rothblum

2004

Discrete Comput Geom

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Section: Edge-directions and Zonotopesmentioning

confidence: 99%

“…The following bound on the number of vertices of zonotopes has been rediscovered many times over the years; see, e.g., [11] and [28] for early references and [23] and [48] for recent extensions and refinements. The collection of normal cones of a polytope P at all faces is called the normal fan of P (see [26]).…”

Section: Edge-directions and Zonotopesmentioning

confidence: 99%

Convex Combinatorial Optimization

Onn

Rothblum

2004

Discrete Comput Geom

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“…Since the worst case size of the output for all the three operations can be exponential in the size of input (see [7,8]), it is natural to talk of output sensitive algorithms. The complexity of an output-sensitive algorithm is measured in terms of the size of both the input and the output.…”

Section: Introductionmentioning

confidence: 99%

On the Hardness of Computing Intersection, Union and Minkowski Sum of Polytopes

Tiwary

2008

Discrete Comput Geom

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For polytopes P 1 , P 2 ⊂ R d we consider the intersection P 1 ∩ P 2 , the convex hull of the union CH(P1 ∪ P2), and the Minkowski sum P1 + P2.For Minkowski sum we prove that enumerating the facets of P1 +P2 is NPhard if P 1 and P 2 are specified by facets, or if P 1 is specified by vertices and P2 is a polyhedral cone specified by facets. For intersection we prove that computing the facets or the vertices of the intersection of two polytopes is NP-hard if one of them is given by vertices and the other by facets.Also, computing the vertices of the intersection of two polytopes given by vertices is shown to be NP-hard. Analogous results for computing the convex hull of the union of two polytopes follow from polar duality. All of the hardness results are established by showing that the appropriate decision version, for each of these problems, is NP-complete .

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“…Let the normal vector at u ′ be n. Clearly, the point on δS ∞ LL with the same normal vector n is u ′′ = δu ′ . By the properties of Minkowski sum (Gritzmann and Sturmfels 1993), u ′ = u ′′ + t ′ , where t ′ is the point on T with the same normal vector n. Thus, we have u ′ = δu ′ + t ′ , or, u ′ = (1 − δ) −1 t ′ , and consequently…”

Section: A5 Proof Of Theoremmentioning

confidence: 99%

“…This equation can be solved through the properties of the Minkowski sum (Gritzmann and Sturmfels, 1993;Zhang, 2010). Second, if we can show that…”

Section: Benchmark Contract: Inducing (L L) Forevermentioning

confidence: 99%

Dynamic Business Share Allocation in a Supply Chain with Competing Suppliers

Zhang

Fine

2013

Operations Research

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This paper studies a repeated game between a manufacturer and two competing suppliers with imperfect monitoring. We present a principal-agent model for managing long-term supplier relationships using a unique form of measurement and incentive scheme. We measure a supplier's overall performance with a rating equivalent to its continuation utility (the expected total discounted utility of its future payoffs), and incentivize supplier effort with larger allocations of future business. We obtain the vector of the two suppliers' ratings as the state of a Markov decision process, and solve an infinite horizon contracting problem in which the manufacturer allocates business volume between the two suppliers and updates their ratings dynamically based on their current ratings and the current performance outcome.Our contributions are both theoretical and managerial: We propose a repeated principal-agent model with a novel incentive scheme to tackle a common, but challenging incentive problem in a multi-period supply chain setting. Assuming binary effort choices and performance outcomes by the suppliers, we characterize the structure of the optimal contract through a novel fixed-point analysis.Our results provide a theoretical foundation for the emergence of "business-as-usual" (low effort) trapping states and tournament competition (high effort) recurrent states as the long-run incentive drivers for motivating critical suppliers.

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Minkowski Addition of Polytopes: Computational Complexity and Applications to Gröbner Bases

Cited by 179 publications

References 31 publications

Convex Combinatorial Optimization

Convex Combinatorial Optimization

On the Hardness of Computing Intersection, Union and Minkowski Sum of Polytopes

Dynamic Business Share Allocation in a Supply Chain with Competing Suppliers

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