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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

A k-coloring of a tournament is a partition of its vertices into k acyclic sets. Deciding if a tournament is 2-colorable is NP-hard. A natural problem, akin to that of coloring a 3-colorable graph with few colors, is to color a 2-colorable tournament with few colors. This problem does not seem to have been addressed before, although it is a special case of coloring a 2-colorable 3-uniform hypergraph with few colors, which is a well-studied problem with super-constant lower bounds.
We present an efficient decomposition lemma for tournaments and show that it can be used to design polynomial-time algorithms to color various classes of tournaments with few colors, including an algorithm to color a 2-colorable tournament with ten colors. For the classes of tournaments considered, we complement our upper bounds with strengthened lower bounds, painting a comprehensive picture of the algorithmic and complexity aspects of coloring tournaments.

Felix Klingelhoefer and Alantha Newman. Coloring Tournaments with Few Colors: Algorithms and Complexity. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 71:1-71:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{klingelhoefer_et_al:LIPIcs.ESA.2023.71, author = {Klingelhoefer, Felix and Newman, Alantha}, title = {{Coloring Tournaments with Few Colors: Algorithms and Complexity}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {71:1--71:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.71}, URN = {urn:nbn:de:0030-drops-187247}, doi = {10.4230/LIPIcs.ESA.2023.71}, annote = {Keywords: Tournaments, Graph Coloring, Algorithms, Complexity} }

Document

**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

We study the traveling salesman problem (TSP) in the case when the objective function of the subtour linear programming relaxation is minimized by a half-cycle point: x_e in {0,1/2,1} where the half-edges form a 2-factor and the 1-edges form a perfect matching. Such points are sufficient to resolve half-integer TSP in general and they have been conjectured to demonstrate the largest integrality gap for the subtour relaxation.
For half-cycle points, the best-known approximation guarantee is 3/2 due to Christofides' famous algorithm. Proving an integrality gap of alpha for the subtour relaxation is equivalent to showing that alpha x can be written as a convex combination of tours, where x is any feasible solution for this relaxation. To beat Christofides' bound, our goal is to show that (3/2 - epsilon)x can be written as a convex combination of tours for some positive constant epsilon. Let y_e = 3/2-epsilon when x_e = 1 and y_e = 3/4 when x_e = 1/2. As a first step towards this goal, our main result is to show that y can be written as a convex combination of tours. In other words, we show that we can save on 1-edges, which has several applications. Among them, it gives an alternative algorithm for the recently studied uniform cover problem. Our main new technique is a procedure to glue tours over proper 3-edge cuts that are tight with respect to x, thus reducing the problem to a base case in which such cuts do not occur.

Arash Haddadan and Alantha Newman. Towards Improving Christofides Algorithm for Half-Integer TSP. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 56:1-56:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{haddadan_et_al:LIPIcs.ESA.2019.56, author = {Haddadan, Arash and Newman, Alantha}, title = {{Towards Improving Christofides Algorithm for Half-Integer TSP}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {56:1--56:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.56}, URN = {urn:nbn:de:0030-drops-111772}, doi = {10.4230/LIPIcs.ESA.2019.56}, annote = {Keywords: Traveling salesman problem, subtour elimination relaxation, integrality gap, gluing subtours} }

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**Published in:** OASIcs, Volume 61, 1st Symposium on Simplicity in Algorithms (SOSA 2018)

In a second seminal paper on the application of semidefinite
programming to graph partitioning problems, Goemans and Williamson
showed in 2004 how to formulate and round a complex semidefinite program to give what is to date still the best-known approximation guarantee of .836008 for Max-3-Cut. (This approximation ratio was also achieved independently around the same time by De Klerk et
al..) Goemans and Williamson left open the problem of how to apply their techniques to Max-k-Cut for general k. They point out that it does not seem straightforward or even possible to formulate a good quality complex semidefinite program for the general Max-k-Cut problem, which presents a barrier for the further application of their techniques.
We present a simple rounding algorithm for the standard semidefinite
programmming relaxation of Max-k-Cut and show that it is equivalent to the rounding of Goemans and Williamson in the case of Max-3-Cut. This allows us to transfer the elegant analysis of Goemans and Williamson for Max-3-Cut to Max-k-Cut. For k > 3, the resulting approximation ratios are about .01 worse than the best known guarantees. Finally, we present a generalization of our rounding algorithm and conjecture (based on computational observations) that it matches the best-known guarantees of De Klerk et al.

Alantha Newman. Complex Semidefinite Programming and Max-k-Cut. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 13:1-13:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{newman:OASIcs.SOSA.2018.13, author = {Newman, Alantha}, title = {{Complex Semidefinite Programming and Max-k-Cut}}, booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)}, pages = {13:1--13:11}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-064-4}, ISSN = {2190-6807}, year = {2018}, volume = {61}, editor = {Seidel, Raimund}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.13}, URN = {urn:nbn:de:0030-drops-83098}, doi = {10.4230/OASIcs.SOSA.2018.13}, annote = {Keywords: Graph Partitioning, Max-k-Cut, Semidefinite Programming} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

Given a set S of integers whose sum is zero, consider the problem of finding a permutation of these integers such that: (i) all prefixes of the ordering are non-negative, and (ii) the maximum value of a prefix sum is minimized. Kellerer et al. referred to this problem as the stock size problem and showed that it can be approximated to within 3/2. They also showed that an approximation ratio of 2 can be achieved via several simple algorithms.
We consider a related problem, which we call the alternating stock size problem, where the number of positive and negative integers in the input set S are equal. The problem is the same as above, but we are additionally required to alternate the positive and negative numbers in the output ordering. This problem also has several simple 2-approximations. We show that it can be approximated to within 1.79.
Then we show that this problem is closely related to an optimization version of the gasoline puzzle due to Lovász, in which we want to minimize the size of the gas tank necessary to go around the track. We present a 2-approximation for this problem, using a natural linear programming relaxation whose feasible solutions are doubly stochastic matrices. Our novel rounding algorithm is based on a transformation that yields another doubly stochastic matrix with special properties, from which we can extract a suitable permutation.

Alantha Newman, Heiko Röglin, and Johanna Seif. The Alternating Stock Size Problem and the Gasoline Puzzle. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 71:1-71:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{newman_et_al:LIPIcs.ESA.2016.71, author = {Newman, Alantha and R\"{o}glin, Heiko and Seif, Johanna}, title = {{The Alternating Stock Size Problem and the Gasoline Puzzle}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {71:1--71:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.71}, URN = {urn:nbn:de:0030-drops-64134}, doi = {10.4230/LIPIcs.ESA.2016.71}, annote = {Keywords: approximation algorithms, stock size problem, scheduling with non-renewable resources} }

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