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APPROX

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

In malleable job scheduling, jobs can be executed simultaneously on multiple machines with the processing time depending on the number of allocated machines. Jobs are required to be executed non-preemptively and in unison, in the sense that they occupy, during their execution, the same time interval over all the machines of the allocated set. In this work, we study generalizations of malleable job scheduling inspired by standard scheduling on unrelated machines. Specifically, we introduce a general model of malleable job scheduling, where each machine has a (possibly different) speed for each job, and the processing time of a job j on a set of allocated machines S depends on the total speed of S for j. For machines with unrelated speeds, we show that the optimal makespan cannot be approximated within a factor less than e/(e-1), unless P = NP. On the positive side, we present polynomial-time algorithms with approximation ratios 2e/(e-1) for machines with unrelated speeds, 3 for machines with uniform speeds, and 7/3 for restricted assignments on identical machines. Our algorithms are based on deterministic LP rounding and result in sparse schedules, in the sense that each machine shares at most one job with other machines. We also prove lower bounds on the integrality gap of 1+phi for unrelated speeds (phi is the golden ratio) and 2 for uniform speeds and restricted assignments. To indicate the generality of our approach, we show that it also yields constant factor approximation algorithms (i) for minimizing the sum of weighted completion times; and (ii) a variant where we determine the effective speed of a set of allocated machines based on the L_p norm of their speeds.

Dimitris Fotakis, Jannik Matuschke, and Orestis Papadigenopoulos. Malleable Scheduling Beyond Identical Machines. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{fotakis_et_al:LIPIcs.APPROX-RANDOM.2019.17, author = {Fotakis, Dimitris and Matuschke, Jannik and Papadigenopoulos, Orestis}, title = {{Malleable Scheduling Beyond Identical Machines}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {17:1--17:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.17}, URN = {urn:nbn:de:0030-drops-112324}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.17}, annote = {Keywords: malleable, jobs, moldable, machines, unrelated, uniform, parallel, speeds, approximation, scheduling} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

The min-cost matching problem suffers from being very sensitive to small changes of the input. Even in a simple setting, e.g., when the costs come from the metric on the line, adding two nodes to the input might change the optimal solution completely. On the other hand, one expects that small changes in the input should incur only small changes on the constructed solutions, measured as the number of modified edges. We introduce a two-stage model where we study the trade-off between quality and robustness of solutions. In the first stage we are given a set of nodes in a metric space and we must compute a perfect matching. In the second stage 2k new nodes appear and we must adapt the solution to a perfect matching for the new instance.
We say that an algorithm is (alpha,beta)-robust if the solutions constructed in both stages are alpha-approximate with respect to min-cost perfect matchings, and if the number of edges deleted from the first stage matching is at most beta k. Hence, alpha measures the quality of the algorithm and beta its robustness. In this setting we aim to balance both measures by deriving algorithms for constant alpha and beta. We show that there exists an algorithm that is (3,1)-robust for any metric if one knows the number 2k of arriving nodes in advance. For the case that k is unknown the situation is significantly more involved. We study this setting under the metric on the line and devise a (10,2)-robust algorithm that constructs a solution with a recursive structure that carefully balances cost and redundancy.

Jannik Matuschke, Ulrike Schmidt-Kraepelin, and José Verschae. Maintaining Perfect Matchings at Low Cost. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 82:1-82:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{matuschke_et_al:LIPIcs.ICALP.2019.82, author = {Matuschke, Jannik and Schmidt-Kraepelin, Ulrike and Verschae, Jos\'{e}}, title = {{Maintaining Perfect Matchings at Low Cost}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {82:1--82:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.82}, URN = {urn:nbn:de:0030-drops-106582}, doi = {10.4230/LIPIcs.ICALP.2019.82}, annote = {Keywords: matchings, robust optimization, approximation algorithms} }

Document

**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

In the Steiner Forest problem, we are given a graph and a collection of source-sink pairs, and the goal is to find a subgraph of minimum total length such that all pairs are connected. The problem is APX-Hard and can be 2-approximated by, e.g., the elegant primal-dual algorithm of Agrawal, Klein, and Ravi from 1995.
We give a local-search-based constant-factor approximation for the problem. Local search brings in new techniques to an area that has for long not seen any improvements and might be a step towards a combinatorial algorithm for the more general survivable network design problem. Moreover, local search was an essential tool to tackle the dynamic MST/Steiner Tree problem, whereas dynamic Steiner Forest is still wide open.
It is easy to see that any constant factor local search algorithm requires steps that add/drop many edges together. We propose natural local moves which, at each step, either (a) add a shortest path in the current graph and then drop a bunch of inessential edges, or (b) add a set of edges to the current solution. This second type of moves is motivated by the potential function we use to measure progress, combining the cost of the solution with a penalty for each connected component. Our carefully-chosen local moves and potential function work in tandem to eliminate bad local minima that arise when using more traditional local moves.
Our analysis first considers the case where the local optimum is a single tree, and shows optimality w.r.t. moves that add a single edge (and drop a set of edges) is enough to bound the locality gap. For the general case, we show how to "project" the optimal solution onto the different trees of the local optimum without incurring too much cost (and this argument uses optimality w.r.t. both kinds of moves), followed by a tree-by-tree argument. We hope both the potential function, and our analysis techniques will be useful to develop and analyze local-search algorithms in other contexts.

Martin Groß, Anupam Gupta, Amit Kumar, Jannik Matuschke, Daniel R. Schmidt, Melanie Schmidt, and José Verschae. A Local-Search Algorithm for Steiner Forest. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gro_et_al:LIPIcs.ITCS.2018.31, author = {Gro{\ss}, Martin and Gupta, Anupam and Kumar, Amit and Matuschke, Jannik and Schmidt, Daniel R. and Schmidt, Melanie and Verschae, Jos\'{e}}, title = {{A Local-Search Algorithm for Steiner Forest}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {31:1--31:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.31}, URN = {urn:nbn:de:0030-drops-83134}, doi = {10.4230/LIPIcs.ITCS.2018.31}, annote = {Keywords: Local Search, Steiner Forest, Approximation Algorithms, Network Design} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We introduce and investigate reroutable flows, a robust version of network flows in which link failures can be mitigated by rerouting the affected flow. Given a capacitated network, a path flow is reroutable if after failure of an arbitrary arc, we can reroute the interrupted flow from the tail of that arc to the sink, without modifying the flow that is not affected by the failure. Similar types of restoration, which are often termed "local", were previously investigated in the context of network design, such as min-cost capacity planning. In this paper, our interest is in computing maximum flows under this robustness assumption. An important new feature of our model, distinguishing it from existing max robust flow models, is that no flow can get lost in the network.
We also study a tightening of reroutable flows, called strictly reroutable flows, making more restrictive assumptions on the capacities available for rerouting. For both variants, we devise a reroutable-flow equivalent of an s-t-cut and show that the corresponding max flow/min cut gap is bounded by 2. It turns out that a strictly reroutable flow of maximum value can be found using a compact LP formulation, whereas the problem of finding a maximum reroutable flow is NP-hard, even when all capacities are in {1, 2}. However, the tightening can be used to get a 2-approximation for reroutable flows. This ratio is tight in general networks, but we show that in the case of unit capacities, every reroutable flow can be transformed into a strictly reroutable flow of same value. While it is NP-hard to compute a maximal integral flow even for unit capacities, we devise a surprisingly simple combinatorial algorithm that finds a half-integral strictly reroutable flow of value 1, or certifies that no such solutions exits. Finally, we also give a hardness result for the case of multiple arc failures.

Jannik Matuschke, S. Thomas McCormick, and Gianpaolo Oriolo. Rerouting Flows When Links Fail. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 89:1-89:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{matuschke_et_al:LIPIcs.ICALP.2017.89, author = {Matuschke, Jannik and McCormick, S. Thomas and Oriolo, Gianpaolo}, title = {{Rerouting Flows When Links Fail}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {89:1--89:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.89}, URN = {urn:nbn:de:0030-drops-74466}, doi = {10.4230/LIPIcs.ICALP.2017.89}, annote = {Keywords: network flows, network interdiction, robust optimization} }

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**Published in:** OASIcs, Volume 25, 12th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (2012)

In this paper, we study a multi-dimensional commodity covering problem, which we encountered as a subproblem in optimizing large scale transportation networks in logistics. The problem asks for a selection of containers for transporting a given set of commodities, each commodity having different extensions of properties such as weight or volume. Each container can be selected multiple times and is specified by a fixed charge and capacities in the relevant properties. The task is to find a cost minimal collection of containers and a feasible assignment of the demand to all selected containers.
From theoretical point of view, by exploring similarities to the well known SetCover problem, we derive NP-hardness and see that the non-approximability result known for set cover also carries over to our problem. For practical applications we need very fast heuristics to be integrated into a meta-heuristic framework that - depending on the context - either provide feasible near optimal solutions or only estimate the cost value of an optimal solution. We develop and analyze a flexible family of greedy algorithms that meet these challenges. In order to find best-performing configurations for different requirements of the meta-heuristic framework, we provide an extensive computational study on random and real world instance sets obtained from our project partner 4flow AG.
We outline a trade-off between running times and solution quality and conclude that the proposed methods achieve the accuracy and efficiency necessary for serving as a key ingredient in more complex meta-heuristics enabling the optimization of large-scale networks.

Felix G. König, Jannik Matuschke, and Alexander Richter. Multi-Dimensional Commodity Covering for Tariff Selection in Transportation. In 12th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems. Open Access Series in Informatics (OASIcs), Volume 25, pp. 58-70, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{konig_et_al:OASIcs.ATMOS.2012.58, author = {K\"{o}nig, Felix G. and Matuschke, Jannik and Richter, Alexander}, title = {{Multi-Dimensional Commodity Covering for Tariff Selection in Transportation}}, booktitle = {12th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems}, pages = {58--70}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-939897-45-3}, ISSN = {2190-6807}, year = {2012}, volume = {25}, editor = {Delling, Daniel and Liberti, Leo}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2012.58}, URN = {urn:nbn:de:0030-drops-37034}, doi = {10.4230/OASIcs.ATMOS.2012.58}, annote = {Keywords: Covering, Heuristics, Transportation, Tariff Selection} }

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