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**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Given a graph with colors on its vertices, a path is called a rainbow vertex path if all its internal vertices have distinct colors. We say that the graph is rainbow vertex-connected if there is a rainbow vertex path between every pair of its vertices. We study the problem of deciding whether the vertices of a given graph can be colored with at most k colors so that the graph becomes rainbow vertex-connected. Although edge-colorings have been studied extensively under similar constraints, there are significantly fewer results on the vertex variant that we consider. In particular, its complexity on structured graph classes was explicitly posed as an open question.
We show that the problem remains NP-complete even on bipartite apex graphs and on split graphs. The former can be seen as a first step in the direction of studying the complexity of rainbow coloring on sparse graphs, an open problem which has attracted attention but limited progress. We also give hardness of approximation results for both bipartite and split graphs. To complement the negative results, we show that bipartite permutation graphs, interval graphs, and block graphs can be rainbow vertex-connected optimally in polynomial time.

Pinar Heggernes, Davis Issac, Juho Lauri, Paloma T. Lima, and Erik Jan van Leeuwen. Rainbow Vertex Coloring Bipartite Graphs and Chordal Graphs. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 83:1-83:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{heggernes_et_al:LIPIcs.MFCS.2018.83, author = {Heggernes, Pinar and Issac, Davis and Lauri, Juho and Lima, Paloma T. and van Leeuwen, Erik Jan}, title = {{Rainbow Vertex Coloring Bipartite Graphs and Chordal Graphs}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {83:1--83:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.83}, URN = {urn:nbn:de:0030-drops-96657}, doi = {10.4230/LIPIcs.MFCS.2018.83}, annote = {Keywords: Rainbow coloring, graph classes, polynomial-time algorithms, approximation algorithms} }

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**Published in:** LIPIcs, Volume 103, 17th International Symposium on Experimental Algorithms (SEA 2018)

Given a vertex-colored graph H and a multiset M of colors as input, the graph motif problem asks us to decide whether H has a connected induced subgraph whose multiset of colors agrees with M. The graph motif problem is NP-complete but known to admit randomized algorithms based on constrained multilinear sieving over GF(2^b) that run in time O(2^kk^2m {M({2^b})}) and with a false-negative probability of at most k/2^{b-1} for a connected m-edge input and a motif of size k. On modern CPU microarchitectures such algorithms have practical edge-linear scalability to inputs with billions of edges for small motif sizes, as demonstrated by Björklund, Kaski, Kowalik, and Lauri [ALENEX'15]. This scalability to large graphs prompts the dual question whether it is possible to scale to large motif sizes.
We present a vertex-localized variant of the constrained multilinear sieve that enables us to obtain, in time O(2^kk^2m{M({2^b})}) and for every vertex simultaneously, whether the vertex participates in at least one match with the motif, with a per-vertex probability of at most k/2^{b-1} for a false negative. Furthermore, the algorithm is easily vector-parallelizable for up to 2^k threads, and parallelizable for up to 2^kn threads, where n is the number of vertices in H. Here {M({2^b})} is the time complexity to multiply in GF(2^b).
We demonstrate with an open-source implementation that our variant of constrained multilinear sieving can be engineered for vector-parallel microarchitectures to yield hardware utilization that is bound by the available memory bandwidth.
Our main engineering contributions are (a) a version of the recurrence for tightly labeled arborescences that can be executed as a sequence of memory-and-arithmetic coalescent parallel workloads on multiple GPUs, and (b) a bit-sliced low-level implementation for arithmetic in characteristic 2 to support (a).

Petteri Kaski, Juho Lauri, and Suhas Thejaswi. Engineering Motif Search for Large Motifs. In 17th International Symposium on Experimental Algorithms (SEA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 103, pp. 28:1-28:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kaski_et_al:LIPIcs.SEA.2018.28, author = {Kaski, Petteri and Lauri, Juho and Thejaswi, Suhas}, title = {{Engineering Motif Search for Large Motifs}}, booktitle = {17th International Symposium on Experimental Algorithms (SEA 2018)}, pages = {28:1--28:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-070-5}, ISSN = {1868-8969}, year = {2018}, volume = {103}, editor = {D'Angelo, Gianlorenzo}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2018.28}, URN = {urn:nbn:de:0030-drops-89631}, doi = {10.4230/LIPIcs.SEA.2018.28}, annote = {Keywords: algorithm engineering, constrained multilinear sieving, graph motif problem, multi-GPU, vector-parallel, vertex-localization} }

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

The Rainbow k-Coloring problem asks whether the edges of a given graph can be colored in k colors so that every pair of vertices is connected by a rainbow path, i.e., a path with all edges of different colors. Our main result states that for any k >= 2, there is no algorithm for Rainbow k-Coloring running in time 2^{o(n^{3/2})}, unless ETH fails. Motivated by this negative result we consider two parameterized variants of the problem. In the Subset Rainbow k-Coloring problem, introduced by Chakraborty et al. [STACS 2009, J. Comb. Opt. 2009], we are additionally given a set S of pairs of vertices and we ask if there is a coloring in which all the pairs in S are connected by rainbow paths. We show that Subset Rainbow k-Coloring is FPT when parameterized by |S|. We also study Subset Rainbow k-Coloring problem, where we are additionally given an integer q and we ask if there is a coloring in which at least q anti-edges are connected by rainbow paths. We show that the problem is FPT when parameterized by q and has a kernel of size O(q) for every k >= 2, extending the result of Ananth et al. [FSTTCS 2011]. We believe that our techniques used for the lower bounds may shed some light on the complexity of the classical Edge Coloring problem, where it is a major open question if a 2^{O(n)}-time algorithm exists.

Lukasz Kowalik, Juho Lauri, and Arkadiusz Socala. On the Fine-Grained Complexity of Rainbow Coloring. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 58:1-58:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kowalik_et_al:LIPIcs.ESA.2016.58, author = {Kowalik, Lukasz and Lauri, Juho and Socala, Arkadiusz}, title = {{On the Fine-Grained Complexity of Rainbow Coloring}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {58:1--58:16}, 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.2016.58}, URN = {urn:nbn:de:0030-drops-64001}, doi = {10.4230/LIPIcs.ESA.2016.58}, annote = {Keywords: graph coloring, computational complexity, lower bounds, exponential time hypothesis, FPT algorithms} }