Strongly Polynomial Algorithm for the Intersection of a Line with a Polymatroid

Fonlupt, Jean; Skoda, Alexandre

doi:10.1007/978-3-540-76796-1_5

Cited by 10 publications

(8 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We will define the strength of a not necessarily connected graph by this last formula because this last definition naturally extends to the case of a matroid or a polymatroid (cf [7,6,13]). Let us define the auxiliary function f by, ∀A ⊆ E:…”

mentioning

confidence: 99%

Games induced by the partitioning of a graph

Grabisch

Skoda

2012

Ann Oper Res

View full text Add to dashboard Cite

The paper aims at generalizing the notion of restricted game on a communication graph, introduced by Myerson. We consider communication graphs with weighted edges, and we define arbitrary ways of partitioning any subset of a graph, which we call correspondences. A particularly useful way to partition a graph is obtained by computing the strength of the graph. The strength of a graph is a measure introduced in graph theory to evaluate the resistance of networks under attacks, and it provides a natural partition of the graph (called the Gusfield correspondence) into resistant components. We perform a general study of the inheritance of superadditivity and convexity for the restricted game associated with a given correspondence. Our main result is to give for cycle-free graphs necessary and sufficient conditions for the inheritance of convexity of the restricted game associated with the Gusfield correspondence.

show abstract

mentioning

confidence: 99%

Games induced by the partitioning of a graph

Grabisch

Skoda

2012

Ann Oper Res

View full text Add to dashboard Cite

show abstract

“…To efficiently use the GLS method, we only need a way to efficiently compute the intersection point s and a facet f on which it lies. For general polymatroids, this can be done with an algorithm described by Fonlupt and Skoda (2009). For the scheduling polytope, a direct application of their result leads to an algorithm that runs in time O n 8 .…”

Section: Implementation Of the Optimal Mechanismmentioning

confidence: 99%

Optimal Mechanism Design for a Sequencing Problem with Two-Dimensional Types

Hoeksma

Uetz

2016

Operations Research

View full text Add to dashboard Cite

We study the design of mechanisms for a sequencing problem where the types of job-agents consist of processing times and waiting costs that are private to the jobs. In the Bayes-Nash setting, we seek to find a sequencing rule and incentive compatible payments that minimize the total expected payments that have to be made to the agents. It is known that the problem can be efficiently solved when jobs have single-dimensional types. Here, we address the problem with two-dimensional types. We show that the problem can be solved in polynomial time by linear programming techniques, answering an open problem formulated by Heydenreich et al. Our implementation is randomized and truthful in expectation. Remarkably, it also works when types are correlated across jobs. The main steps are a compactification of an exponential size linear programming formulation, and a convex decomposition algorithm that allows us to implement the optimal linear programming solution. In addition, by means of computational experiments, we generate some new insights into the implementability in different equilibriaMillennium Nucleus Information and Coordination in Networks ICM-FIC RC13000

show abstract

“…Let us briefly sketch the state-of-the-art of combinatorial algorithms for the decomposition problem of the single machine scheduling polytope. An O(n 9 ) algorithm follows directly from work by Fonlupt and Skoda [4] on the intersection of a line with an arbitrary polymatroid and using the GLS method. However, a closer look reveals that an O(n 3 log n) implementation is possible for the scheduling polytope [7].…”

Section: Introductionmentioning

confidence: 99%

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

Hoeksma

Manthey

Uetz

2016

Discrete Applied Mathematics

View full text Add to dashboard Cite

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.

show abstract

Strongly Polynomial Algorithm for the Intersection of a Line with a Polymatroid

Cited by 10 publications

References 17 publications

Games induced by the partitioning of a graph

Games induced by the partitioning of a graph

Optimal Mechanism Design for a Sequencing Problem with Two-Dimensional Types

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

Contact Info

Product

Resources

About