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APPROX

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

The problem of routing in graphs using node-disjoint paths has received a lot of attention and a polylogarithmic approximation algorithm with constant congestion is known for undirected graphs [Chuzhoy and Li 2016] and [Chekuri and Ene 2013]. However, the problem is hard to approximate within polynomial factors on directed graphs, for any constant congestion [Chuzhoy, Kim and Li 2016].
Recently, [Chekuri, Ene and Pilipczuk 2016] have obtained a polylogarithmic approximation with constant congestion on directed planar graphs, for the special case of symmetric demands. We extend their result by obtaining a polylogarithmic approximation with constant congestion on arbitrary directed minor-free graphs, for the case of symmetric demands.

Timothy Carpenter, Ario Salmasi, and Anastasios Sidiropoulos. Routing Symmetric Demands in Directed Minor-Free Graphs with Constant Congestion. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{carpenter_et_al:LIPIcs.APPROX-RANDOM.2019.14, author = {Carpenter, Timothy and Salmasi, Ario and Sidiropoulos, Anastasios}, title = {{Routing Symmetric Demands in Directed Minor-Free Graphs with Constant Congestion}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {14:1--14:15}, 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.2019.14}, URN = {urn:nbn:de:0030-drops-112290}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.14}, annote = {Keywords: Routing, Node-disjoint, Symmetric demands, Minor-free graphs} }

Document

**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

In the Asymmetric Traveling Salesperson Problem (ATSP) the goal is to find a closed walk of minimum cost in a directed graph visiting every vertex. We consider the approximability of ATSP on topologically restricted graphs. It has been shown by Oveis Gharan and Saberi [SODA, 2011] that there exists polynomial-time constant-factor approximations on planar graphs and more generally graphs of constant orientable genus. This result was extended to non-orientable genus by Erickson and Sidiropoulos [SoCG, 2014].
We show that for any class of nearly-embeddable graphs, ATSP admits a polynomial-time constant-factor approximation. More precisely, we show that for any fixed non-negative k, there exist positive alpha and beta, such that ATSP on n-vertex k-nearly-embeddable graphs admits an alpha-approximation in time O(n^beta). The class of k-nearly-embeddable graphs contains graphs with at most k apices, k vortices of width at most k, and an underlying surface of either orientable or non-orientable genus at most k. Prior to our work, even the case of graphs with a single apex was open. Our algorithm combines tools from rounding the Held-Karp LP via thin trees with dynamic programming.
We complement our upper bounds by showing that solving ATSP exactly on graphs of pathwidth k (and hence on k-nearly embeddable graphs) requires time n^{Omega(k)}, assuming the Exponential-Time Hypothesis (ETH). This is surprising in light of the fact that both TSP on undirected graphs and Minimum Cost Hamiltonian Cycle on directed graphs are FPT parameterized by treewidth.

Dániel Marx, Ario Salmasi, and Anastasios Sidiropoulos. Constant-Factor Approximations for Asymmetric TSP on Nearly-Embeddable Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 16:1-16:54, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{marx_et_al:LIPIcs.APPROX-RANDOM.2016.16, author = {Marx, D\'{a}niel and Salmasi, Ario and Sidiropoulos, Anastasios}, title = {{Constant-Factor Approximations for Asymmetric TSP on Nearly-Embeddable Graphs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {16:1--16:54}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.16}, URN = {urn:nbn:de:0030-drops-66391}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.16}, annote = {Keywords: asymmetric TSP, approximation algorithms, nearly-embeddable graphs, Held-Karp LP, exponential time hypothesis} }

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