Robust Assignments via Ear Decompositions and Randomized Rounding

Many real-life planning problems require making a priori decisions before all parameters of the problem have been revealed. An important special case of such problem arises in scheduling problems, where a set of tasks needs to be assigned to the available set of machines or personnel (resources), in a way that all tasks have assigned resources, and no two tasks share the same resource. In its nominal form, the resulting computational problem becomes the \emph{assignment problem} on general bipartite graphs. This paper deals with a robust variant of the assignment problem modeling situations where certain edges in the corresponding graph are \emph{vulnerable} and may become unavailable after a solution has been chosen. The goal is to choose a minimum-cost collection of edges such that if any vulnerable edge becomes unavailable, the remaining part of the solution contains an assignment of all tasks. We present approximation results and hardness proofs for this type of problems, and establish several connections to well-known concepts from matching theory, robust optimization and LP-based techniques.Comment: Full version of ICALP 2016 pape

An Improved Approximation Algorithm for the Matching Augmentation Problem

We present a 5/3-approximation algorithm for the matching augmentation problem (MAP): given a multi-graph with edges of cost either zero or one such that the edges of cost zero form a matching, find a 2-edge connected spanning subgraph (2-ECSS) of minimum cost. A 7/4-approximation algorithm for the same problem was presented recently, see Cheriyan, et al., "The matching augmentation problem: a 7/4-approximation algorithm," Math. Program., 182(1):315-354, 2020. Our improvement is based on new algorithmic techniques, and some of these may lead to advances on related problems

Spectral Relaxations and Fair Densest Subgraphs

Reducing hidden bias in the data and ensuring fairness in algorithmic data analysis has recently received significant attention. In this paper, we address the problem of identifying a densest subgraph, while ensuring that none of one binary protected attribute is disparately impacted. Unfortunately, the underlying algorithmic problem is NP-hard, even in its approximation version: approximating the densest fair subgraph with a polynomial-time algorithm is at least as hard as the densest subgraph problem of at most k vertices, for which no constant approximation algorithms are known. Despite such negative premises, we are able to provide approximation results in two important cases. In particular, we are able to prove that a suitable spectral embedding allows recovery of an almost optimal, fair, dense subgraph hidden in the input data, whenever one is present, a result that is further supported by experimental evidence. We also show a polynomial-time, $2$-approximation algorithm, whenever the underlying graph is itself fair. We finally prove that, under the small set expansion hypothesis, this result is tight for fair graphs. The above theoretical findings drive the design of heuristics, which we experimentally evaluate on a scenario based on real data, in which our aim is to strike a good balance between diversity and highly correlated items from Amazon co-purchasing graphs

Dynamic operation, efficient calibration, and advanced data analysis of gas sensors : from modelling to real-world operation

This thesis demonstrates the use of dynamic operation, efficient calibration and advanced data analysis using metal oxide semiconductor (MOS) gas sensors as an example – from modeling to real-world operation. The necessary steps for an applicationspecific, selective indoor volatile organic compound (VOC) measurement system are addressed, analyzed and improved. Factors such as sensors, operation, electronics and calibration are considered. The developed methods and tools are universally transferable to other gas sensors and applications. The basis for selective measurement is temperature cyclic operation (TCO). The model-based understanding of a semiconductor gas sensor in TCO for the optimized development of operating modes and data evaluation is addressed and, for example, the tailored and stable detection of short gas pulses is developed. Two successful interlaboratory tests for the measurement of VOCs in independent laboratories are described. Selective measurements of VOCs in the laboratory and in the field are successfully demonstrated. Calibrations using the proposed techniques of randomized design of experiment (DoE), model-based data evaluation and calibration with machine learning methods are employed. The calibrated models are compared with analytical measurements using release tests. The high agreement of the results is unique in current research.Diese Thesis zeigt den Einsatz von dynamischem Betrieb, effizienter Kalibrierung, und fortschrittlicher Datenanalyse am Beispiel von Metalloxid Halbleiter (MOS) Gassensoren – von der Modellierung bis zum realen Betrieb. Die notwendigen Schritte für ein anwendungsspezifisches, selektives Messystem für flüchtige organische Verbindungen (VOC) im Innenraum werden adressiert, analysiert und verbessert. Faktoren wie z.B. Sensoren, Funktionsweise, Elektronik und Kalibrierung werden berücksichtigt. Die entwickelten Methoden und Tools sind universell auf andere Gassensoren und Anwendungen übertragbar. Grundlage für die selektive Messung ist der temperaturzyklische Betrieb (TCO). Auf das modellbasierte Verständnis eines Halbleitergassensors im TCO für die optimierte Entwicklung von Betriebsmodi und Datenauswertung wird eingegangen und z.B. die maßgeschneiderte und stabile Detektion von kurzen Gaspulsen entwickelt. Zwei erfolgreiche Ringversuche zur Messung von VOCs in unabhängigen Laboren werden beschrieben. Selektive Messungen verschiedener VOCs im Labor und im Feld werden erfolgreich demonstriert. Dabei kommen Kalibrierungen mit den vorgeschlagenen Techniken des randomisierten Design of Experiment (DoE), der modellbasierten Datenauswertung und Kalibrierung mit Methoden des maschinellen Lernens zum Einsatz. Die kalibrierten Modelle werden anhand von Freisetzungstests mit analytischen Messungen verglichen. Die hohe Übereinstimmung der Ergebnisse ist einzigartig in der aktuellen Forschung

Recent Advances in Fully Dynamic Graph Algorithms

In recent years, significant advances have been made in the design and analysis of fully dynamic algorithms. However, these theoretical results have received very little attention from the practical perspective. Few of the algorithms are implemented and tested on real datasets, and their practical potential is far from understood. Here, we present a quick reference guide to recent engineering and theory results in the area of fully dynamic graph algorithms

Faster Streaming and Scalable Algorithms for Finding Directed Dense Subgraphs in Large Graphs

Finding dense subgraphs is a fundamental algorithmic tool in data mining, community detection, and clustering. In this problem, one aims to find an induced subgraph whose edge-to-vertex ratio is maximized. We study the directed case of this question in the context of semi-streaming and massively parallel algorithms. In particular, we show that it is possible to find a $(2+\epsilon)$ approximation on randomized streams even in a single pass by using $O(n \cdot {\rm poly} \log n)$ memory on $n$-vertex graphs. Our result improves over prior works, which were designed for arbitrary-ordered streams: the algorithm by Bahmani et al. (VLDB 2012) which uses $O(\log n)$ passes, and the work by Esfandiari et al. (2015) which makes one pass but uses $O(n^{3/2})$ memory. Moreover, our techniques extend to the Massively Parallel Computation model yielding $O(1)$ rounds in the super-linear and $O(\sqrt{\log n})$ rounds in the nearly-linear memory regime. This constitutes a quadratic improvement over state-of-the-art bounds by Bahmani et al. (VLDB 2012 and WAW 2014), which require $O(\log n)$ rounds even in the super-linear memory regime. Finally, we empirically evaluate our single-pass semi-streaming algorithm on $6$ benchmarks and show that, even on non-randomly ordered streams, the quality of its output is essentially the same as that of Bahmani et al. (VLDB 2012) while it is $2$ times faster on large graphs

Approximate Fully Dynamic Directed Densest Subgraph

We give a fully dynamic algorithm maintaining a $(1-\varepsilon)$-approximate directed densest subgraph in $\tilde{O}(\log^3(n)/\varepsilon^6)$ amortized time or $\tilde{O}(\log^4(n)/\varepsilon^7)$ worst-case time per edge update (where $\tilde{O}$ hides $\log\log$ factors), based on earlier work by Chekuri and Quanrud [arXiv:2210.02611, arXiv:2310.18146]. This result improves on earlier work done by Sawlani and Wang [arXiv:1907.03037], which guarantees $O(\log^5(n)/\varepsilon^7)$ worst case time for edge insertions and deletions