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**Published in:** LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

The Edge-disjoint s-t Paths Problem (s-t EDP) is a classical network design problem whose goal is to connect for some k ≥ 1 two given vertices of a graph under the condition that any k-1 edges of the graph may fail. We extend the simple uniform failure model of the s-t EDP as follows: the edge set of the graph is partitioned into vulnerable, and safe edges, and a set of at most k vulnerable edges may fail, while safe edges do not fail. In particular we study the Fault-Tolerant Path (FTP) problem, the counterpart of the Shortest s-t Path problem in this non-uniform failure model as well as the Fault-Tolerant Flow (FTF) problem, the counterpart of s-t EDP. We present complexity results alongside exact and approximation algorithms for both problems. We emphasize the vast increase in complexity of the problems compared to s-t EDP.

David Adjiashvili, Felix Hommelsheim, Moritz Mühlenthaler, and Oliver Schaudt. Fault-Tolerant Edge-Disjoint s-t Paths - Beyond Uniform Faults. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 5:1-5:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{adjiashvili_et_al:LIPIcs.SWAT.2022.5, author = {Adjiashvili, David and Hommelsheim, Felix and M\"{u}hlenthaler, Moritz and Schaudt, Oliver}, title = {{Fault-Tolerant Edge-Disjoint s-t Paths - Beyond Uniform Faults}}, booktitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)}, pages = {5:1--5:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-236-5}, ISSN = {1868-8969}, year = {2022}, volume = {227}, editor = {Czumaj, Artur and Xin, Qin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.5}, URN = {urn:nbn:de:0030-drops-161659}, doi = {10.4230/LIPIcs.SWAT.2022.5}, annote = {Keywords: graph algorithms, network design, fault tolerance, approximation algorithms} }

Document

**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

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 and transshipment problems, where a set of jobs needs to be assigned to the available set of machines or personnel (resources), in a way that all jobs have assigned resources, and no two jobs share the same resource. In its nominal form, the resulting computational problem becomes the assignment problem.
This paper deals with the Robust Assignment Problem (RAP) which models situations in which certain assignments are vulnerable and may become unavailable after the solution has been chosen. The goal is to choose a minimum-cost collection of assignments (edges in the corresponding bipartite graph) so that if any vulnerable edge becomes unavailable, the remaining part of the solution contains an assignment of all jobs.
We develop algorithms and hardness results for RAP and establish several connections to well-known concepts from matching theory, robust optimization, LP-based techniques and combinations
thereof.

David Adjiashvili, Viktor Bindewald, and Dennis Michaels. Robust Assignments via Ear Decompositions and Randomized Rounding. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 71:1-71:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{adjiashvili_et_al:LIPIcs.ICALP.2016.71, author = {Adjiashvili, David and Bindewald, Viktor and Michaels, Dennis}, title = {{Robust Assignments via Ear Decompositions and Randomized Rounding}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {71:1--71:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.71}, URN = {urn:nbn:de:0030-drops-62133}, doi = {10.4230/LIPIcs.ICALP.2016.71}, annote = {Keywords: robust optimization, matching theory, ear decomposition, randomized rounding, approximation algorithm} }

Document

**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

Robust optimization is concerned with constructing solutions that remain feasible also when a limited number of resources is removed from the solution. Most studies of robust combinatorial optimization to date made the assumption that every resource is equally vulnerable, and that the set of scenarios is implicitly given by a single budget constraint. This paper studies a robustness model of a different kind. We focus on Bulk-Robustness, a model recently introduced (Adjiashvili, Stiller, Zenklusen 2015) for addressing the need to model non-uniform failure patterns in systems.
We significantly extend the techniques used by Adjiashvili et al. to design approximation algorithm for bulk-robust network design problems in planar graphs. Our techniques use an augmentation framework, combined with linear programming (LP) rounding that depends on a planar embedding of the input graph. A connection to cut covering problems and the dominating set problem in circle graphs is established. Our methods use few of the specifics of bulk-robust optimization, hence it is conceivable that they can be adapted to solve other robust network design problems.

David Adjiashvili. Non-Uniform Robust Network Design in Planar Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 61-77, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{adjiashvili:LIPIcs.APPROX-RANDOM.2015.61, author = {Adjiashvili, David}, title = {{Non-Uniform Robust Network Design in Planar Graphs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {61--77}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.61}, URN = {urn:nbn:de:0030-drops-52948}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.61}, annote = {Keywords: Robust optimization, Network design, Planar graph, Approximation algorithm, LP rounding} }

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