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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

A map is a 2-cell decomposition of a closed compact surface, i.e., an embedding of a graph such that every face is homeomorphic to an open disc. An automorphism of a map can be thought of as a permutation of the vertices which preserves the vertex-edge-face incidences in the embedding. When the underlying surface is orientable, every automorphism of a map determines an angle-preserving homeomorphism of the surface. While it is conjectured that there is no "truly subquadratic" algorithm for testing map isomorphism for unconstrained genus, we present a linear-time algorithm for computing the generators of the automorphism group of a map, parametrized by the genus of the underlying surface. The algorithm applies a sequence of local reductions and produces a uniform map, while preserving the automorphism group. The automorphism group of the original map can be reconstructed from the automorphism group of the uniform map in linear time. We also extend the algorithm to non-orientable surfaces by making use of the antipodal double-cover.

Ken-ichi Kawarabayashi, Bojan Mohar, Roman Nedela, and Peter Zeman. Automorphisms and Isomorphisms of Maps in Linear Time. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 86:1-86:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kawarabayashi_et_al:LIPIcs.ICALP.2021.86, author = {Kawarabayashi, Ken-ichi and Mohar, Bojan and Nedela, Roman and Zeman, Peter}, title = {{Automorphisms and Isomorphisms of Maps in Linear Time}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {86:1--86:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.86}, URN = {urn:nbn:de:0030-drops-141558}, doi = {10.4230/LIPIcs.ICALP.2021.86}, annote = {Keywords: maps on surfaces, automorphisms, isomorphisms, algorithm} }

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**Published in:** LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)

We present improved results for approximating maximum-weight independent set (MaxIS) in the CONGEST and LOCAL models of distributed computing. Given an input graph, let n and Δ be the number of nodes and maximum degree, respectively, and let MIS(n,Δ) be the running time of finding a maximal independent set (MIS) in the CONGEST model. Bar-Yehuda et al. [PODC 2017] showed that there is an algorithm in the CONGEST model that finds a Δ-approximation for MaxIS in O(MIS(n,Δ)log W) rounds, where W is the maximum weight of a node in the graph, which can be as large as poly (n). Whether their algorithm is deterministic or randomized that succeeds with high probability depends on the MIS algorithm that is used as a black-box. Our results:
1) A deterministic O(MIS(n,Δ)/ε)-round algorithm that finds a (1+ε)Δ-approximation for MaxIS in the CONGEST model.
2) A randomized (poly(log log n)/ε)-round algorithm that finds, with high probability, a (1+ε)Δ-approximation for MaxIS in the CONGEST model. That is, by sacrificing only a tiny fraction of the approximation guarantee, we achieve an exponential speed-up in the running time over the previous best known result.
3) A randomized O(log n⋅ poly(log log n)/ε)-round algorithm that finds, with high probability, a 8(1+ε)α-approximation for MaxIS in the CONGEST model, where α is the arboricity of the graph. For graphs of arboricity α < Δ/(8(1+ε)), this result improves upon the previous best known result in both the approximation factor and the running time.
One may wonder whether it is possible to approximate MaxIS with high probability in fewer than poly(log log n) rounds. Interestingly, a folklore randomized ranking algorithm by Boppana implies a single round algorithm that gives an expected Δ-approximation in the CONGEST model. However, it is unclear how to convert this algorithm to one that succeeds with high probability without sacrificing a large number of rounds. For unweighted graphs of maximum degree Δ ≤ n/log n, we show a new analysis of the randomized ranking algorithm, which we combine with the local-ratio technique, to provide a O(1/ε)-round algorithm in the CONGEST model that, with high probability, finds an independent set of size at least n/((1+ε)(Δ+1)). This result cannot be extended to very high degree graphs, as we show a lower bound of Ω(log^*n) rounds for any randomized algorithm that with probability at least 1-1/log n finds an independent set of size Ω(n/Δ). This lower bound holds even for the LOCAL model. The hard instances that we use to prove our lower bound are graphs of maximum degree Δ = Ω(n/log^*n).

Ken-ichi Kawarabayashi, Seri Khoury, Aaron Schild, and Gregory Schwartzman. Improved Distributed Approximations for Maximum Independent Set. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 35:1-35:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kawarabayashi_et_al:LIPIcs.DISC.2020.35, author = {Kawarabayashi, Ken-ichi and Khoury, Seri and Schild, Aaron and Schwartzman, Gregory}, title = {{Improved Distributed Approximations for Maximum Independent Set}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {35:1--35:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.35}, URN = {urn:nbn:de:0030-drops-131135}, doi = {10.4230/LIPIcs.DISC.2020.35}, annote = {Keywords: Distributed graph algorithms, Approximation algorithms, Lower bounds} }

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**Published in:** LIPIcs, Volume 146, 33rd International Symposium on Distributed Computing (DISC 2019)

We present a time-optimal deterministic distributed algorithm for approximating a minimum weight vertex cover in hypergraphs of rank f. This problem is equivalent to the Minimum Weight Set Cover problem in which the frequency of every element is bounded by f. The approximation factor of our algorithm is (f+epsilon). Let Delta denote the maximum degree in the hypergraph. Our algorithm runs in the congest model and requires O(log{Delta} / log log Delta) rounds, for constants epsilon in (0,1] and f in N^+. This is the first distributed algorithm for this problem whose running time does not depend on the vertex weights nor the number of vertices. Thus adding another member to the exclusive family of provably optimal distributed algorithms.
For constant values of f and epsilon, our algorithm improves over the (f+epsilon)-approximation algorithm of [Fabian Kuhn et al., 2006] whose running time is O(log Delta + log W), where W is the ratio between the largest and smallest vertex weights in the graph. Our algorithm also achieves an f-approximation for the problem in O(f log n) rounds, improving over the classical result of [Samir Khuller et al., 1994] that achieves a running time of O(f log^2 n). Finally, for weighted vertex cover (f=2) our algorithm achieves a deterministic running time of O(log n), matching the randomized previously best result of [Koufogiannakis and Young, 2011].
We also show that integer covering-programs can be reduced to the Minimum Weight Set Cover problem in the distributed setting. This allows us to achieve an (f+epsilon)-approximate integral solution in O((1+f/log n)* ((log Delta)/(log log Delta) + (f * log M)^{1.01}* log epsilon^{-1}* (log Delta)^{0.01})) rounds, where f bounds the number of variables in a constraint, Delta bounds the number of constraints a variable appears in, and M=max {1, ceil[1/a_{min}]}, where a_{min} is the smallest normalized constraint coefficient. This improves over the results of [Fabian Kuhn et al., 2006] for the integral case, which combined with rounding achieves the same guarantees in O(epsilon^{-4}* f^4 * log f * log(M * Delta)) rounds.

Ran Ben-Basat, Guy Even, Ken-ichi Kawarabayashi, and Gregory Schwartzman. Optimal Distributed Covering Algorithms. In 33rd International Symposium on Distributed Computing (DISC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 146, pp. 5:1-5:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{benbasat_et_al:LIPIcs.DISC.2019.5, author = {Ben-Basat, Ran and Even, Guy and Kawarabayashi, Ken-ichi and Schwartzman, Gregory}, title = {{Optimal Distributed Covering Algorithms}}, booktitle = {33rd International Symposium on Distributed Computing (DISC 2019)}, pages = {5:1--5:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-126-9}, ISSN = {1868-8969}, year = {2019}, volume = {146}, editor = {Suomela, Jukka}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2019.5}, URN = {urn:nbn:de:0030-drops-113129}, doi = {10.4230/LIPIcs.DISC.2019.5}, annote = {Keywords: Distributed Algorithms, Approximation Algorithms, Vertex Cover, Set Cover} }

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**Published in:** LIPIcs, Volume 146, 33rd International Symposium on Distributed Computing (DISC 2019)

In this work, we initiate a thorough study of graph optimization problems parameterized by the output size in the distributed setting. In such a problem, an algorithm decides whether a solution of size bounded by k exists and if so, it finds one. We study fundamental problems, including Minimum Vertex Cover (MVC), Maximum Independent Set (MaxIS), Maximum Matching (MaxM), and many others, in both the LOCAL and CONGEST distributed computation models. We present lower bounds for the round complexity of solving parameterized problems in both models, together with optimal and near-optimal upper bounds.
Our results extend beyond the scope of parameterized problems. We show that any LOCAL (1+epsilon)-approximation algorithm for the above problems must take Omega(epsilon^{-1}) rounds. Joined with the (epsilon^{-1}log n)^{O(1)} rounds algorithm of [Ghaffari et al., 2017] and the Omega (sqrt{(log n)/(log log n)}) lower bound of [Fabian Kuhn et al., 2016], the lower bounds match the upper bound up to polynomial factors in both parameters. We also show that our parameterized approach reduces the runtime of exact and approximate CONGEST algorithms for MVC and MaxM if the optimal solution is small, without knowing its size beforehand. Finally, we propose the first o(n^2) rounds CONGEST algorithms that approximate MVC within a factor strictly smaller than 2.

Ran Ben-Basat, Ken-ichi Kawarabayashi, and Gregory Schwartzman. Parameterized Distributed Algorithms. In 33rd International Symposium on Distributed Computing (DISC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 146, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{benbasat_et_al:LIPIcs.DISC.2019.6, author = {Ben-Basat, Ran and Kawarabayashi, Ken-ichi and Schwartzman, Gregory}, title = {{Parameterized Distributed Algorithms}}, booktitle = {33rd International Symposium on Distributed Computing (DISC 2019)}, pages = {6:1--6:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-126-9}, ISSN = {1868-8969}, year = {2019}, volume = {146}, editor = {Suomela, Jukka}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2019.6}, URN = {urn:nbn:de:0030-drops-113135}, doi = {10.4230/LIPIcs.DISC.2019.6}, annote = {Keywords: Distributed Algorithms, Approximation Algorithms, Parameterized Algorithms} }

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**Published in:** LIPIcs, Volume 121, 32nd International Symposium on Distributed Computing (DISC 2018)

It is a well known fact that sequential algorithms which exhibit a strong "local" nature can be adapted to the distributed setting given a legal graph coloring. The running time of the distributed algorithm will then be at least the number of colors. Surprisingly, this well known idea was never formally stated as a unified framework. In this paper we aim to define a robust family of local sequential algorithms which can be easily adapted to the distributed setting. We then develop new tools to further enhance these algorithms, achieving state of the art results for fundamental problems.
We define a simple class of greedy-like algorithms which we call orderless-local algorithms. We show that given a legal c-coloring of the graph, every algorithm in this family can be converted into a distributed algorithm running in O(c) communication rounds in the CONGEST model. We show that this family is indeed robust as both the method of conditional expectations and the unconstrained submodular maximization algorithm of Buchbinder et al. [Niv Buchbinder et al., 2015] can be expressed as orderless-local algorithms for local utility functions - Utility functions which have a strong local nature to them.
We use the above algorithms as a base for new distributed approximation algorithms for the weighted variants of some fundamental problems: Max k-Cut, Max-DiCut, Max 2-SAT and correlation clustering. We develop algorithms which have the same approximation guarantees as their sequential counterparts, up to a constant additive epsilon factor, while achieving an O(log^* n) running time for deterministic algorithms and O(epsilon^{-1}) running time for randomized ones. This improves exponentially upon the currently best known algorithms.

Ken-ichi Kawarabayashi and Gregory Schwartzman. Adapting Local Sequential Algorithms to the Distributed Setting. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 35:1-35:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kawarabayashi_et_al:LIPIcs.DISC.2018.35, author = {Kawarabayashi, Ken-ichi and Schwartzman, Gregory}, title = {{Adapting Local Sequential Algorithms to the Distributed Setting}}, booktitle = {32nd International Symposium on Distributed Computing (DISC 2018)}, pages = {35:1--35:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-092-7}, ISSN = {1868-8969}, year = {2018}, volume = {121}, editor = {Schmid, Ulrich and Widder, Josef}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2018.35}, URN = {urn:nbn:de:0030-drops-98245}, doi = {10.4230/LIPIcs.DISC.2018.35}, annote = {Keywords: Distributed, Approximation Algorithms, Derandomization, Max-Cut} }

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

Robin Thomas asked whether for every proper minor-closed class G, there exists a polynomial-time algorithm approximating the chromatic number of graphs from G up to a constant additive error independent on the class G. We show this is not the case: unless P=NP, for every integer k >= 1, there is no polynomial-time algorithm to color a K_{4k+1}-minor-free graph G using at most chi(G)+k-1 colors. More generally, for every k >= 1 and 1 <=beta <=4/3, there is no polynomial-time algorithm to color a K_{4k+1}-minor-free graph G using less than beta chi(G)+(4-3 beta)k colors. As far as we know, this is the first non-trivial non-approximability result regarding the chromatic number in proper minor-closed classes.
We also give somewhat weaker non-approximability bound for K_{4k+1}-minor-free graphs with no cliques of size 4. On the positive side, we present an additive approximation algorithm whose error depends on the apex number of the forbidden minor, and an algorithm with additive error 6 under the additional assumption that the graph has no 4-cycles.

Zdenek Dvorák and Ken-ichi Kawarabayashi. Additive Non-Approximability of Chromatic Number in Proper Minor-Closed Classes. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 47:1-47:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dvorak_et_al:LIPIcs.ICALP.2018.47, author = {Dvor\'{a}k, Zdenek and Kawarabayashi, Ken-ichi}, title = {{Additive Non-Approximability of Chromatic Number in Proper Minor-Closed Classes}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {47:1--47:12}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.47}, URN = {urn:nbn:de:0030-drops-90510}, doi = {10.4230/LIPIcs.ICALP.2018.47}, annote = {Keywords: non-approximability, chromatic number, minor-closed classes} }

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**Published in:** LIPIcs, Volume 62, 25th EACSL Annual Conference on Computer Science Logic (CSL 2016)

We show that the model-checking problem for successor-invariant first-order logic is fixed-parameter tractable on graphs with excluded topological subgraphs when parameterised by both the size of the input formula and the size of the exluded topological subgraph. Furthermore, we show that model-checking for order-invariant first-order logic is tractable on coloured posets of bounded width, parameterised by both the size of the input formula and the width of the poset.
Results of this form, i.e. showing that model-checking for a certain logic is tractable on a certain class of structures, are often referred to as algorithmic meta-theorems since they give a unified proof for the tractability of a whole range of problems. First-order logic is arguably one of the most important logics in this context since it is powerful enough to express many computational problems (e.g. the existence of cliques, dominating sets etc.) and yet its model-checking problem is tractable on rich classes of graphs. In fact, Grohe et al have shown that model-checking for FO is tractable on all nowhere dense classes of graphs.
Successor-invariant FO is a semantic extension of FO by allowing the use of an additional binary relation which is interpreted as a directed Hamiltonian cycle, restricted to formulae whose truth value does not depend on the specific choice of a Hamiltonian cycle. While this is very natural in the context of model-checking (after all, storing a structure in computer memory usually brings with it a linear order on the structure), the question of how the computational complexity of the model-checking problem for this richer logic compares to that of plain FO is still open.
Our result for successor-invariant FO extends previous results for this logic on planar graphs and graphs with excluded minors, further narrowing the gap between what is known for FO and what is known for successor-invariant FO. The proof uses Grohe and Marx's structure theorem for graphs with excluded topological subgraphs. For order-invariant FO we show that Gajarský et al.'s recent result for FO carries over to order-invariant FO.

Kord Eickmeyer and Ken-ichi Kawarabayashi. Successor-Invariant First-Order Logic on Graphs with Excluded Topological Subgraphs. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{eickmeyer_et_al:LIPIcs.CSL.2016.18, author = {Eickmeyer, Kord and Kawarabayashi, Ken-ichi}, title = {{Successor-Invariant First-Order Logic on Graphs with Excluded Topological Subgraphs}}, booktitle = {25th EACSL Annual Conference on Computer Science Logic (CSL 2016)}, pages = {18:1--18:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-022-4}, ISSN = {1868-8969}, year = {2016}, volume = {62}, editor = {Talbot, Jean-Marc and Regnier, Laurent}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2016.18}, URN = {urn:nbn:de:0030-drops-65583}, doi = {10.4230/LIPIcs.CSL.2016.18}, annote = {Keywords: model-checking, algorithmic meta-theorem, successor-invariant, first-order logic, topological subgraphs, parameterised complexity} }

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**Published in:** LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

Recognizing 3-colorable graphs is one of the most famous NP-complete problems [Garey, Johnson, and Stockmeyer, STOC'74]. The problem of coloring 3-colorable graphs in polynomial time with as few colors as possible has been intensively studied: O(n^{1/2}) colors [Wigderson, STOC'82], O(n^{2/5}) colors [Blum, STOC'89], O(n^{3/8}) colors [Blum, FOCS'90], O(n^{1/4}) colors [Karger, Motwani and Sudan, FOCS'94], O(n^{3/14})=O(n^0.2142) colors [Blum and Karger, IPL'97], O(n^{0.2111}) colors [Arora, Chlamtac, and Charikar, STOC'06], and O(n^{0.2072}) colors [Chlamtac, FOCS'07]. Recently the authors got down to O(n^{0.2049}) colors [FOCS'12]. In this paper we get down to O(n^{0.19996})=o(n^{1/5}) colors.
Since 1994, the best bounds have all been obtained balancing between combinatorial and semi-definite approaches. We present a new combinatorial recursion that only makes sense in collaboration with semi-definite programming. We specifically target the worst-case for semi-definite programming: high degrees. By focusing on the interplay, we obtained the biggest improvement in the exponent since 1997.

Ken-ichi Kawarabayashi and Mikkel Thorup. Coloring 3-colorable graphs with o(n^{1/5}) colors. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 458-469, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{kawarabayashi_et_al:LIPIcs.STACS.2014.458, author = {Kawarabayashi, Ken-ichi and Thorup, Mikkel}, title = {{Coloring 3-colorable graphs with o(n^\{1/5\}) colors}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {458--469}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.458}, URN = {urn:nbn:de:0030-drops-44797}, doi = {10.4230/LIPIcs.STACS.2014.458}, annote = {Keywords: Approximation Algorithms, Graph Coloring} }

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**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

Finding edge-disjoint odd cycles is one of the most important problems
in graph theory, graph algorithm and combinatorial optimization. In
fact, it is closely related to the well-known max-cut problem. One of
the difficulties of this problem is that the Erdös-Pósa property does not hold for odd cycles in general. Motivated by this fact, we prove that for any positive integer k, there exists an integer f(k) satisfying the following: For any 4-edge-connected graph G=(V,E),
either G has edge-disjoint k odd cycles or there exists an edge set F
subseteq E with |F| <= f(k) such that G-F is bipartite. We note that
the 4-edge-connectivity is best possible in this statement.
Similar approach can be applied to an algorithmic question. Suppose
that the input graph G is a 4-edge-connected graph with n vertices.
We show that, for any epsilon > 0, if k = O ((log log log n)^{1/2-epsilon}), then the edge-disjoint k odd cycle packing in G can be solved in polynomial time of n.

Ken-ichi Kawarabayashi and Yusuke Kobayashi. Edge-disjoint Odd Cycles in 4-edge-connected Graphs. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 206-217, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{kawarabayashi_et_al:LIPIcs.STACS.2012.206, author = {Kawarabayashi, Ken-ichi and Kobayashi, Yusuke}, title = {{Edge-disjoint Odd Cycles in 4-edge-connected Graphs}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {206--217}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.206}, URN = {urn:nbn:de:0030-drops-34173}, doi = {10.4230/LIPIcs.STACS.2012.206}, annote = {Keywords: odd-cycles, disjoint paths problem, Erd\"{o}s-Posa property, packing algorithm, 4-edge-connectivity} }

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**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

A key theorem in algorithmic graph-minor theory is a min-max relation
between the treewidth of a graph and its largest grid minor. This min-max relation is a keystone of the Graph Minor Theory of Robertson
and Seymour, which ultimately proves Wagner's Conjecture about the structure of minor-closed graph properties.
In 2008, Demaine and Hajiaghayi proved a remarkable linear min-max
relation for graphs excluding any fixed minor H: every H-minor-free
graph of treewidth at least c_H r has an r times r-grid minor for some
constant c_H. However, as they pointed out, there is still a major
problem left in this theorem. The problem is that their proof heavily
depends on Graph Minor Theory, most of which lacks explicit bounds and
is believed to have very large bounds. Hence c_H is not explicitly
given in the paper and therefore this result is usually not strong
enough to derive efficient algorithms.
Motivated by this problem, we give another (relatively short and
simple) proof of this result without using big machinery of Graph
Minor Theory. Hence we can give an explicit bound for c_H (an exponential function of a polynomial of |H|). Furthermore, our result
gives a constant w=2^O(r^2 log r) such that every graph of treewidth
at least w has an r times r-grid minor, which improves the previously
known best bound 2^Theta(r^5)$ given by Robertson, Seymour, and Thomas
in 1994.

Ken-ichi Kawarabayashi and Yusuke Kobayashi. Linear min-max relation between the treewidth of H-minor-free graphs and its largest grid. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 278-289, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{kawarabayashi_et_al:LIPIcs.STACS.2012.278, author = {Kawarabayashi, Ken-ichi and Kobayashi, Yusuke}, title = {{Linear min-max relation between the treewidth of H-minor-free graphs and its largest grid}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {278--289}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.278}, URN = {urn:nbn:de:0030-drops-34165}, doi = {10.4230/LIPIcs.STACS.2012.278}, annote = {Keywords: grid minor, treewidth, graph minor} }

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