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# Seminar in Discrete Optimization

**Organizers**: René Brandenberg, Michael Ritter, Andreas S. Schulz, Stefan Weltge, Andreas Wiese

## Upcoming talks

## Previous talks

## 21.03.2023 16:00 Andreas Wiese: How to be productive

Instead of a classical research talk, Andreas Wiese will give a short presentation on the topic "How to be productive".
The talk will be followed by an open discussion, where everyone is invited to participate and give their own input, opinions and ideas on the topic.

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## 14.03.2023 16:00 Alexander Armbruster: A PTAS for Minimizing Weighted Flow Time on a Single Machine

An important objective function in the scheduling literature is to minimize the sum of weighted flow times. We are given a set of jobs, where each job is characterized by a release time, a processing time, and a weight. Our goal is to find a preemptive schedule on a single machine that minimizes the sum of the weighted flow times of the jobs, where the flow time of a job is the time between its completion time and its release time. We answer this question in the affirmative and present a polynomial time (1 + 𝜀)-approximation algorithm for weighted flow time on a single machine. We use a reduction of the problem to a geometric covering problem, which was introduced in previous approaches
and which loses only a factor of 1+𝜀 in the approximation ratio. However, unlike the previous algorithm, we solve the resulting instances of the covering problem exactly, rather than losing a factor 2 + 𝜀. Key for this is to identify and exploit structural properties of instances of that problem covering problem which arise in the reduction from weighted flow time.
This talk is based on joint work with Lars Rohwedder and Andreas Wiese.

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## 07.03.2023 16:00 Stefan Kober: Totally Delta-modular IPs on a TU constraint matrix with one additional row

It is a well-known conjecture that integer programs on an integer constraint matrix whose subdeterminants are bounded in absolute value by a constant could be solvable in polynomial time. Despite some recent progress in special cases, the question is still wide open. We consider the special case consisting of a TU constraint matrix with one additional row, which in general encompasses different hard and interesting problems. We present partial progress to the resolution of the question of polynomial solvability of such problems in particular for transposed network matrices. The applied techniques range from Seymour's decomposition of TU matrices over certain types of proximity results for integer programs to graph and minor theory.
This is joint work with Manuel Aprile, Samuel Fiorini, Stefan Weltge and Yelena Yuditsky.

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## 14.02.2023 16:00 Moritz Keuthen: Graphisomorphism issues in the field of gear modeling

As part of our work on a standardized interface for exchanging models for gear calculation (www.rexs.info), we have defined a modeling structure for gears. Different component types are available as building blocks (e.g. shafts, bearings, gears). These can be connected to each other via component-specific relations. For example, there is a stage_relation with the roles "stage", "gear 1", "gear 2" for a gear stage or a side_relation that connects a bearing with two other components. Most relations have 2 or 3 roles, or components that connect them. So the model structure resembles a graph.
Now it is in most cases so that such a model in the interface format can be imported by different tools and thereby into their internal data structure is transferred which deviates from this modeling structure (at least partly). If the model is exported then again thereby in many cases the Ids of components and relations change. After importing and exporting, certain aspects of the model may be more or less detailed than before (depending on what the tool can do and what it requires). This poses a challenge when you want to compare the two model states. The idea is therefore to create a matching between the two models based on the similarity of the model structure, which identifies related components.
In the seminar I would like to describe the REXS modeling system, outline the matching problem and discuss if there are any solutions that could be modified for the concrete problem.

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## 07.02.2023 16:00 Alberto Moreno: Matchings of Students and Seminars: a Linear Programming Approach

The goal of this project was, assuming both a group of seminars and a group of students give preferences over the agents in the other group they are interested in, to find an homogeneous assignment of participants and seminars which addresses the situation of each seminar and each student as best as possible. This problem is a generalization of the classical Hospitals/Residents Problem, where we allowed indifference in the preference relations of each agent, assigned an arbitrary but fixed number of types to each student (each belonging to a category, e.g. study program, exchange student or not, etc) and were interested in allowing each seminar to state the minimal number of participants they would require to be held. The impact of all these changes was studied and the resulting problem formulated as a Linear Program.

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## 31.01.2023 16:00 Mia Runge: Geometric Inequalities Involving Different Diameter Definitions

When considering non-symmetric gauges there are several ways to define the diameter of a convex body. These correspond to different symmetrizations of the gauge, i.e., means of the gauge $C$ and $-C$. We study inequalities involving the inradius, circumradius and diameter and present examples and results that confirm that not only does studying the symmetrizations help to understand the diameters better but also the other way around.

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## 31.01.2023 16:00 Florian Grundbacher: Ellipsoids in convex bodies of a given asymmetry

In „Ellipsoids of maximal volume in convex bodies“ Keith Ball proved a general bound on the volume of k-dimensional ellipsoids in n-dimensional convex bodies in relation to their John ellipsoid. A stronger bound is known in the symmetric case. Our goal was to connect these results by establishing a bound depending on the John asymmetry s_0. We could prove a tight bound for all k and all asymmetry values s_0 not in (1,1+2/n), and characterize the equality cases.

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## 17.01.2023 17:00 Jesús Yepes Nicolas: Discrete analogues for the lattice point enumerator of classical geometric inequalities

The classical Brunn-Minkowski inequality in the $n$-dimensional Euclidean space asserts that the volume (Lebesgue measure) to the power $1/n$ is a concave functional when dealing with convex bodies (non-empty compact convex sets). It quickly yields, among other results, the classical isoperimetric inequality, which can be summarized by saying that the Euclidean balls minimize the surface area measure (Minkowski content) among those convex bodies with prescribed positive volume. Moreover, it implies the so-called Rogers-Shephard inequality, which provides a sharp upper bound for the volume of the difference set in terms of the volume of the original convex body.
There exist various facets of the previous results, due to their different versions, generalizations, and extensions.
In this talk, after recalling the above classical inequalities for the volume, we will discuss and show certain discrete analogues of them for the lattice point enumerator, which gives the number of integer points of a bounded set. Moreover, we will show that these new discrete inequalities imply the corresponding classical results for convex bodies.
This is about joint works with David Alonso-Gutiérrez (Universidad de Zaragoza), David Iglesias (Universidad de Murcia), Eduardo Lucas (Universidad de Murcia), and Artem Zvavitch (Kent State University).

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## 20.12.2022 16:00 Steffen Borgwardt: Short and Separable Clustering Transitions

Clustering is one of the fundamental tasks in data analytics and machine learning. In many situations, different clusterings of the same data set become relevant. For example, different algorithms for the same clustering task may return dramatically different solutions. We are interested in applications in which one clustering has to be transformed into another; such a scenario arises, for example, when a gradual transition from an old solution to a new one is required.
Based on linear programming and network theory, we develop methods for the construction of a sequence of so-called elementary moves that accomplishes such a transition. Specifically, we discuss two types of transitions: short transitions with a low number of steps and transitions that retain separation of clusters throughout.

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## 06.12.2022 16:00 Andreas Wiese: How to find a research topic

After a short presentation by Andreas Wiese, there will be an open discussion where everyone is invited to participate and give their own input, opinions and ideas on the topic.

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## 29.11.2022 16:00 Mirjeta Palloshi: Kombinatorische Matching-Algorithmen für die Kurszuweisungen

Eine bestmögliche Einteilung von Studierenden zu Kursen stellt für viele Algorithmen eine große Herausforderung dar. Dabei sind Eigenschaften wie Strategiesicherheit, Pareto-Effizienz und Fairness schwer einzuhalten. Es haben sich in diesem Zusammenhang zwei bekannte Algorithmen bewährt und sollten bezüglich ihrer Vor- und Nachteile verglichen werden.

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For talks not listed above please have a look at the Munich Mathematical Calendar.