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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We study the classical Node-Disjoint Paths (NDP) problem: given an undirected n-vertex graph G, together with a set {(s_1,t_1),...,(s_k,t_k)} of pairs of its vertices, called source-destination, or demand pairs, find a maximum-cardinality set {P} of mutually node-disjoint paths that connect the demand pairs. The best current approximation for the problem is achieved by a simple greedy O(sqrt{n})-approximation algorithm. Until recently, the best negative result was an Omega(log^{1/2-epsilon}n)-hardness of approximation, for any fixed epsilon, under standard complexity assumptions.
A special case of the problem, where the underlying graph is a grid, has been studied extensively. The best current approximation algorithm for this special case achieves an O~(n^{1/4})-approximation factor. On the negative side, a recent result by the authors shows that NDP is hard to approximate to within factor 2^{Omega(sqrt{log n})}, even if the underlying graph is a subgraph of a grid, and all source vertices lie on the grid boundary. In a very recent follow-up work, the authors further show that NDP in grid graphs is hard to approximate to within factor Omega(2^{log^{1-epsilon}n}) for any constant epsilon under standard complexity assumptions, and to within factor n^{Omega(1/(log log n)^2)} under randomized ETH.
In this paper we study the NDP problem in grid graphs, where all source vertices {s_1,...,s_k} appear on the grid boundary. Our main result is an efficient randomized 2^{O(sqrt{log n}* log log n)}-approximation algorithm for this problem. Our result in a sense complements the 2^{Omega(sqrt{log n})}-hardness of approximation for sub-graphs of grids with sources lying on the grid boundary, and should be contrasted with the above-mentioned almost polynomial hardness of approximation of NDP in grid graphs (where the sources and the destinations may lie anywhere in the grid).
Much of the work on approximation algorithms for NDP relies on the multicommodity flow relaxation of the problem, which is known to have an Omega(sqrt n) integrality gap, even in grid graphs, with all source and destination vertices lying on the grid boundary. Our work departs from this paradigm, and uses a (completely different) linear program only to select the pairs to be routed, while the routing itself is computed by other methods.

Julia Chuzhoy, David H. K. Kim, and Rachit Nimavat. Improved Approximation for Node-Disjoint Paths in Grids with Sources on the Boundary. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 38:1-38:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chuzhoy_et_al:LIPIcs.ICALP.2018.38, author = {Chuzhoy, Julia and Kim, David H. K. and Nimavat, Rachit}, title = {{Improved Approximation for Node-Disjoint Paths in Grids with Sources on the Boundary}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {38:1--38:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.38}, URN = {urn:nbn:de:0030-drops-90423}, doi = {10.4230/LIPIcs.ICALP.2018.38}, annote = {Keywords: Node-disjoint paths, approximation algorithms, routing and layout} }

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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

We study non-preemptive scheduling problems on identical parallel machines and uniformly related machines under both resource constraints and general precedence constraints between jobs. Our first result is an O(logn)-approximation algorithm for the objective of minimizing the makespan on parallel identical machines under resource and general precedence constraints. We then use this result as a subroutine to obtain an O(logn)-approximation algorithm for the
more general objective of minimizing the total weighted completion time on parallel identical machines under both constraints. Finally, we present an O(logm logn)-approximation algorithm for scheduling under these constraints on uniformly related machines. We show that these results can all be generalized to include the case where each job has a release time. This is the first upper bound on the approximability of this class of scheduling problems where both resource and general precedence constraints must be satisfied simultaneously.

Gökalp Demirci, Henry Hoffmann, and David H. K. Kim. Approximation Algorithms for Scheduling with Resource and Precedence Constraints. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{demirci_et_al:LIPIcs.STACS.2018.25, author = {Demirci, G\"{o}kalp and Hoffmann, Henry and Kim, David H. K.}, title = {{Approximation Algorithms for Scheduling with Resource and Precedence Constraints}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {25:1--25:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.25}, URN = {urn:nbn:de:0030-drops-85319}, doi = {10.4230/LIPIcs.STACS.2018.25}, annote = {Keywords: scheduling, resource, precedence, weighted completion time} }

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

In the Node-Disjoint Paths (NDP) problem, the input is an undirected n-vertex graph G, and a collection {(s_1,t_1),...,(s_k,t_k)} of pairs of vertices called demand pairs. The goal is to route the largest possible number of the demand pairs (s_i,t_i), by selecting a path connecting each such pair, so that the resulting paths are node-disjoint. NDP is one of the most basic and extensively studied routing problems. Unfortunately, its approximability is far from being well-understood: the best current upper bound of O(sqrt(n)) is achieved via a simple greedy algorithm, while the best current lower bound on its approximability is Omega(log^{1/2-\delta}(n)) for any constant delta. Even for seemingly simpler special cases, such as planar graphs, and even grid graphs, no better approximation algorithms are currently known. A major reason for this impasse is that the standard technique for designing approximation algorithms for routing problems is LP-rounding of the standard multicommodity flow relaxation of the problem, whose integrality gap for NDP is Omega(sqrt(n)) even on grid graphs.
Our main result is an O(n^{1/4} * log(n))-approximation algorithm for NDP on grids. We distinguish between demand pairs with both vertices close to the grid boundary, and pairs where at least one of the two vertices is far from the grid boundary. Our algorithm shows that when all demand pairs are of the latter type, the integrality gap of the multicommodity flow LP-relaxation is at most O(n^{1/4} * log(n)), and we deal with demand pairs of the former type by other methods. We complement our upper bounds by proving that NDP is APX-hard on grid graphs.

Julia Chuzhoy and David H. K. Kim. On Approximating Node-Disjoint Paths in Grids. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 187-211, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{chuzhoy_et_al:LIPIcs.APPROX-RANDOM.2015.187, author = {Chuzhoy, Julia and Kim, David H. K.}, title = {{On Approximating Node-Disjoint Paths in Grids}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {187--211}, 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.2015.187}, URN = {urn:nbn:de:0030-drops-53032}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.187}, annote = {Keywords: Node-disjoint paths, approximation algorithms, routing and layout} }

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