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**Published in:** LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

We study the prize-collecting version of the node-weighted Steiner tree problem (NWPCST) restricted to planar graphs. We give a new primal-dual Lagrangian-multiplier-preserving (LMP) 3-approximation algorithm for planar NWPCST. We then show a 2.88-approximation which establishes a new best approximation guarantee for planar NWPCST. This is done by combining our LMP algorithm with a threshold rounding technique and utilizing the 2.4-approximation of Berman and Yaroslavtsev [6] for the version without penalties. We also give a primal-dual 4-approximation algorithm for the more general forest version using techniques introduced by Hajiaghay and Jain [17].

Jaroslaw Byrka, Mateusz Lewandowski, and Carsten Moldenhauer. Approximation Algorithms for Node-Weighted Prize-Collecting Steiner Tree Problems on Planar Graphs. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 2:1-2:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{byrka_et_al:LIPIcs.SWAT.2016.2, author = {Byrka, Jaroslaw and Lewandowski, Mateusz and Moldenhauer, Carsten}, title = {{Approximation Algorithms for Node-Weighted Prize-Collecting Steiner Tree Problems on Planar Graphs}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {2:1--2:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-011-8}, ISSN = {1868-8969}, year = {2016}, volume = {53}, editor = {Pagh, Rasmus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.2}, URN = {urn:nbn:de:0030-drops-60313}, doi = {10.4230/LIPIcs.SWAT.2016.2}, annote = {Keywords: approximation algorithms, Node-Weighted Steiner Tree, primal-dual algorithm, LMP, planar graphs} }

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**Published in:** LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

The classic stable set problem asks to find a maximum cardinality set of pairwise non-adjacent vertices in an undirected graph G. This problem is NP-hard to approximate with factor n^{1-epsilon} for any constant epsilon>0 [Hastad/Acta Mathematica/1996; Zuckerman/STOC/2006], where n is the number of vertices, and therefore there is no hope for good approximations in the general case. We study the stable set problem when restricted to graphs with bounded odd cycle packing number ocp(G), possibly by a function of n. This is the largest number of vertex-disjoint odd cycles in G. Equivalently, it is the logarithm of the largest absolute value of a sub-determinant of the edge-node incidence matrix A_G of G. Hence, if A_G is totally unimodular, then ocp(G)=0. Therefore, ocp(G) is a natural distance measure of A_G to the set of totally unimodular matrices on a scale from 1 to n/3.
When ocp(G)=0, the graph is bipartite and it is well known that stable set can be solved in polynomial time. Our results imply that the odd cycle packing number indeed strongly influences the approximability of stable set. More precisely, we obtain a polynomial-time approximation scheme for graphs with ocp(G)=o(n/log(n)), and an alpha-approximation algorithm for any graph where alpha smoothly increases from a constant to n as ocp(G) grows from O(n/log(n)) to n/3. On the hardness side, we show that, assuming the exponential-time hypothesis, stable set cannot be solved in polynomial time if ocp(G)=Omega(log^{1+epsilon}(n)) for some epsilon>0. Finally, we generalize a theorem by Györi et al. [Györi et al./Discrete Mathematics/1997] and show that graphs without odd cycles of small weight can be made bipartite by removing a small number of vertices. This allows us to extend some of our above results to the weighted stable set problem.

Adrian Bock, Yuri Faenza, Carsten Moldenhauer, and Andres Jacinto Ruiz-Vargas. Solving the Stable Set Problem in Terms of the Odd Cycle Packing Number. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 187-198, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{bock_et_al:LIPIcs.FSTTCS.2014.187, author = {Bock, Adrian and Faenza, Yuri and Moldenhauer, Carsten and Ruiz-Vargas, Andres Jacinto}, title = {{Solving the Stable Set Problem in Terms of the Odd Cycle Packing Number}}, booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)}, pages = {187--198}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-77-4}, ISSN = {1868-8969}, year = {2014}, volume = {29}, editor = {Raman, Venkatesh and Suresh, S. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.187}, URN = {urn:nbn:de:0030-drops-48422}, doi = {10.4230/LIPIcs.FSTTCS.2014.187}, annote = {Keywords: stable set problem, independent set problem, approximation algorithms, odd cycle packing number, maximum subdeterminants} }

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**Published in:** LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

We consider the discrepancy problem of coloring n intervals with k colors such that at each point on the line, the maximal difference between the number of intervals of any two colors is minimal. Somewhat surprisingly, a coloring with maximal difference at most one always exists. Furthermore, we give an algorithm with running time O(n log n + kn log k) for its construction. This is in particular interesting because many known results for discrepancy problems are non-constructive. This problem naturally models a load balancing scenario, where $n$~tasks with given start- and endtimes have to be distributed among $k$~servers. Our results imply that this can be done ideally balanced.
When generalizing to $d$-dimensional boxes (instead of intervals), a solution with difference at most one is not always possible. We show that for any d >= 2 and any k >= 2 it is NP-complete to decide if such a solution exists, which implies also NP-hardness of the respective minimization problem.
In an online scenario, where intervals arrive over time and the color has to be decided upon arrival, the maximal difference in the size of color classes can become arbitrarily high for any online algorithm.

Antonios Antoniadis, Falk Hueffner, Pascal Lenzner, Carsten Moldenhauer, and Alexander Souza. Balanced Interval Coloring. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 531-542, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{antoniadis_et_al:LIPIcs.STACS.2011.531, author = {Antoniadis, Antonios and Hueffner, Falk and Lenzner, Pascal and Moldenhauer, Carsten and Souza, Alexander}, title = {{Balanced Interval Coloring}}, booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)}, pages = {531--542}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-25-5}, ISSN = {1868-8969}, year = {2011}, volume = {9}, editor = {Schwentick, Thomas and D\"{u}rr, Christoph}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.531}, URN = {urn:nbn:de:0030-drops-30413}, doi = {10.4230/LIPIcs.STACS.2011.531}, annote = {Keywords: Load balancing, discrepancy theory, NP-hardness} }

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