Graduiertenkolleg: Methods for Discrete Structures

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Monday Lecture and Colloquium

Monday, February 9, 2015

Technische Universität Berlin
Institut für Mathematik
Straße des 17. Juni 136
10623 Berlin
room MA 041

Lecture - 14:15

Friedrich Eisenbrand - EPFL, Lausanne

Geometric Random Edge

We show that a variant of the random-edge pivoting rule results in a strongly polynomial time simplex algorithm for linear programs max{cTx:Ax≤b}, whose constraint matrix A satisfies a geometric property introduced by Brunsch and Röglin: The sine of the angle of a row of A to a hyperplane spanned by n-1 other rows of A is at least δ. This property is a geometric generalization of A being integral and all sub-determinants of A being bounded by Δ in absolute value (since δ≥1/(Δ2n)). In particular, linear programs defined by totally unimodular matrices are captured in this famework (δ≥1/n) for which Dyer and Frieze previously described a strongly polynomial-time randomized algorithm.

The number of pivots of the simplex algorithm is polynomial in the dimension and 1/δ and independent of the number of constraints of the linear program. Our main result can be viewed as an algorithmic realization of the proof of small diameter for such polytopes by Bonifas et al., using the ideas of Dyer and Frieze.

Colloquium - 16:00

Max Klimm - Technische Universität Berlin

Optimal Impartial Selection

We study the fundamental problem of selecting a member of a set of agents based on impartial nominations by agents from that set. This problem arises, e.g., when representatives are selected from within a group or where publishing or funding decisions are made based on a process of peer review. A mechanism is called impartial if the selection probability it assigns to each agent is independent of that agent's nominations. We give an impartial randomized mechanism that in expectation selects an agent with at least half the maximum number of nominations. This is best possible subject to impartiality and resolves a conjecture of Alon et al. Further results are given for the case where some agent receives many nominations and the case where each agent casts at least one nomination.

Letzte Aktualisierung: 30.01.2015