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APPROX

**Published in:** LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

The minimum directed feedback vertex set problem consists in finding the minimum set of vertices that should be removed in order to make a directed graph acyclic. This is a well-known NP-hard optimization problem with applications in various fields, such as VLSI chip design, bioinformatics and transaction processing deadlock prevention and node-weighted network design. We show a constant factor approximation for the directed feedback vertex set problem in graphs of bounded genus.

Hao Sun. A Constant Factor Approximation for Directed Feedback Vertex Set in Graphs of Bounded Genus. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 18:1-18:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{sun:LIPIcs.APPROX/RANDOM.2024.18, author = {Sun, Hao}, title = {{A Constant Factor Approximation for Directed Feedback Vertex Set in Graphs of Bounded Genus}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)}, pages = {18:1--18:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-348-5}, ISSN = {1868-8969}, year = {2024}, volume = {317}, editor = {Kumar, Amit and Ron-Zewi, Noga}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.18}, URN = {urn:nbn:de:0030-drops-210112}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.18}, annote = {Keywords: Feedback Vertex Set, Combinatorial Optimization, Approximation Algorithms, min-max relation, linear programming} }

Document

**Published in:** LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

In the classic Directed Steiner Tree problem (DST), we are given an edge-weighted directed graph G = (V,E) with n nodes, a specified root node r ∈ V, and k terminals X ⊆ V-{r}. The goal is to find the cheapest F ⊆ E such that r can reach any terminal using only edges in F.
Designing approximation algorithms for DST is quite challenging, to date the best approximation guarantee of a polynomial-time algorithm for DST is O(k^ε) for any constant ε > 0 [Charikar et al., 1999]. For network design problems like DST, one often relies on natural cut-based linear programming (LP) relaxations to design approximation algorithms. In general, the integrality gap of such an LP for DST is known to have a polynomial integrality gap lower bound [Zosin and Khuller, 2002; Li and Laekhanukit, 2021]. So particular interest has been invested in special cases or in strengthenings of this LP.
In this work, we show the integrality gap is only O(log k) for instances of DST where no Steiner node has both an edge from another Steiner node and an edge to another Steiner node, i.e. the longest path using only Steiner nodes has length at most 1. This generalizes the well-studied case of quasi-bipartite DST where no edge has both endpoints being Steiner nodes. Our result is also optimal in the sense that the integrality gap can be as bad as poly(n) even if the longest path with only Steiner nodes has length 2.

Zachary Friggstad and Hao Sun. A Logarithmic Integrality Gap for Generalizations of Quasi-Bipartite Instances of Directed Steiner Tree. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{friggstad_et_al:LIPIcs.SWAT.2024.23, author = {Friggstad, Zachary and Sun, Hao}, title = {{A Logarithmic Integrality Gap for Generalizations of Quasi-Bipartite Instances of Directed Steiner Tree}}, booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)}, pages = {23:1--23:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-318-8}, ISSN = {1868-8969}, year = {2024}, volume = {294}, editor = {Bodlaender, Hans L.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.23}, URN = {urn:nbn:de:0030-drops-200638}, doi = {10.4230/LIPIcs.SWAT.2024.23}, annote = {Keywords: Steiner Tree, Approximation Algorithms, Linear Programming} }

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APPROX

**Published in:** LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

A classical branch of graph algorithms is graph transversals, where one seeks a minimum-weight subset of nodes in a node-weighted graph G which intersects all copies of subgraphs F from a fixed family F. Many such graph transversal problems have been shown to admit polynomial-time approximation schemes (PTAS) for planar input graphs G, using a variety of techniques like the shifting technique (Baker, J. ACM 1994), bidimensionality (Fomin et al., SODA 2011), or connectivity domination (Cohen-Addad et al., STOC 2016). These techniques do not seem to apply to graph transversals with parity constraints, which have recently received significant attention, but for which no PTASs are known.
In the even-cycle transversal (ECT) problem, the goal is to find a minimum-weight hitting set for the set of even cycles in an undirected graph. For ECT, Fiorini et al. (IPCO 2010) showed that the integrality gap of the standard covering LP relaxation is Θ(log n), and that adding sparsity inequalities reduces the integrality gap to 10.
Our main result is a primal-dual algorithm that yields a 47/7 ≈ 6.71-approximation for ECT on node-weighted planar graphs, and an integrality gap of the same value for the standard LP relaxation on node-weighted planar graphs.

Alexander Göke, Jochen Koenemann, Matthias Mnich, and Hao Sun. Hitting Weighted Even Cycles in Planar Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 25:1-25:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{goke_et_al:LIPIcs.APPROX/RANDOM.2021.25, author = {G\"{o}ke, Alexander and Koenemann, Jochen and Mnich, Matthias and Sun, Hao}, title = {{Hitting Weighted Even Cycles in Planar Graphs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {25:1--25:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-207-5}, ISSN = {1868-8969}, year = {2021}, volume = {207}, editor = {Wootters, Mary and Sanit\`{a}, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.25}, URN = {urn:nbn:de:0030-drops-147186}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.25}, annote = {Keywords: Even cycles, planar graphs, integrality gap} }

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