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Fixed-Parameter Tractability of Maximum Colored Path and Beyond
Abstract
We introduce a general method for obtaining fixed-parameter algorithms for problems about finding paths in undirected graphs, where the length of the path could be unbounded in the parameter. The first application of our method is as follows.
We give a randomized algorithm, that given a colored \(n\) -vertex undirected graph, vertices \(s\) and \(t\) , and an integer \(k\) , finds an \((s,t)\) -path containing at least \(k\) different colors in time \(2^k n^{\mathcal{O}(1)}\) . This is the first FPT algorithm for this problem, and it generalizes the algorithm of Björklund, Husfeldt, and Taslaman [SODA 2012] on finding a path through \(k\) specified vertices. It also implies the first \(2^k n^{\mathcal{O}(1)}\) time algorithm for finding an \((s,t)\) -path of length at least \(k\) .
Our method yields FPT algorithms for even more general problems. For example, we consider the problem where the input consists of an \(n\) -vertex undirected graph \(G\) , a matroid \(M\) whose elements correspond to the vertices of \(G\) and which is represented over a finite field of order \(q\) , a positive integer weight function on the vertices of \(G\) , two sets of vertices \(S,T \subseteq V(G)\) , and integers \(p,k,w\) , and the task is to find \(p\) vertex-disjoint paths from \(S\) to \(T\) so that the union of the vertices of these paths contains an independent set of \(M\) of cardinality \(k\) and weight \(w\) , while minimizing the sum of the lengths of the paths. We give a \(2^{p+\mathcal{O}(k^2 \log (q+k))} n^{\mathcal{O}(1)} w\) time randomized algorithm for this problem.
References
[1]
Noga Alon, Raphael Yuster, and Uri Zwick. 1995. Color-Coding. J. ACM 42, 4 (1995), 844–856. https://doi.org/10.1145/210332.210337
[2]
Andreas Björklund. 2014. Determinant Sums for Undirected Hamiltonicity. SIAM J. Comput. 43, 1 (2014), 280–299. https://doi.org/10.1137/110839229
[3]
Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. 2017. Narrow sieves for parameterized paths and packings. J. Comput. Syst. Sci. 87 (2017), 119–139. https://doi.org/10.1016/j.jcss.2017.03.003
[4]
Andreas Björklund, Thore Husfeldt, and Nina Taslaman. 2012. Shortest cycle through specified elements. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, Yuval Rabani (Ed.). SIAM, 1747–1753. https://doi.org/10.1137/1.9781611973099.139
[5]
Andreas Björklund, Petteri Kaski, and Lukasz Kowalik. 2016. Constrained Multilinear Detection and Generalized Graph Motifs. Algorithmica 74, 2 (2016), 947–967. https://doi.org/10.1007/s00453-015-9981-1
[6]
Hajo Broersma, Xueliang Li, Gerhard J. Woeginger, and Shenggui Zhang. 2005. Paths and cycles in colored graphs. Australas. J Comb. 31 (2005), 299–312. http://ajc.maths.uq.edu.au/pdf/31/ajc_v31_p299.pdf
[7]
Gruia Călinescu, Chandra Chekuri, Martin Pál, and Jan Vondrák. 2011. Maximizing a Monotone Submodular Function Subject to a Matroid Constraint. SIAM J. Comput. 40, 6 (2011), 1740–1766. https://doi.org/10.1137/080733991
[8]
Raffaele Cerulli, Paolo Dell’Olmo, Monica Gentili, and Andrea Raiconi. 2006. Heuristic approaches for the Minimum Labelling Hamiltonian Cycle Problem. Electron. Notes Discret. Math. 25 (2006), 131–138. https://doi.org/10.1016/j.endm.2006.06.080
[9]
Chandra Chekuri and Martin Pál. 2005. A Recursive Greedy Algorithm for Walks in Directed Graphs. In 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS). IEEE Computer Society, 245–253.
[10]
N. Christofides. 1985. Vehicle routing. In The Traveling Salesman Problem, E.L. Lawler, J.K. Lenstra, A.H.G.Rinnooy Kan, and D.B. Shmoys (Eds.) (Eds.). Wiley, New York, 431–448.
[11]
Johanne Cohen, Giuseppe F. Italiano, Yannis Manoussakis, Nguyen Kim Thang, and Hong Phong Pham. 2021. Tropical paths in vertex-colored graphs. J. Comb. Optim. 42, 3 (2021), 476–498. https://doi.org/10.1007/s10878-019-00416-y
[12]
Basile Couëtoux, Elie Nakache, and Yann Vaxès. 2017. The Maximum Labeled Path Problem. Algorithmica 78, 1 (2017), 298–318. https://doi.org/10.1007/s00453-016-0155-6
[13]
Marek Cygan, Holger Dell, Daniel Lokshtanov, Dániel Marx, Jesper Nederlof, Yoshio Okamoto, Ramamohan Paturi, Saket Saurabh, and Magnus Wahlström. 2016. On Problems as Hard as CNF-SAT. ACM Trans. Algorithms 12, 3 (2016), 41:1–41:24. https://doi.org/10.1145/2925416
[14]
Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. 2015. Parameterized Algorithms. Springer.
[15]
Marek Cygan, Stefan Kratsch, and Jesper Nederlof. 2013. Fast hamiltonicity checking via bases of perfect matchings. In Proceedings of the 45th Annual ACM Symposium on Theory of Computing (STOC). ACM, 301–310.
[16]
Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michał Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. 2011. Solving Connectivity Problems Parameterized by Treewidth in Single Exponential Time. In Proceedings of the 52nd Annual Symposium on Foundations of Computer Science (FOCS). IEEE, 150–159.
[17]
Reinhard Diestel. 2012. Graph Theory, 4th Edition. Graduate texts in mathematics, Vol. 173. Springer.
[18]
Eduard Eiben and Iyad Kanj. 2020. A Colored Path Problem and Its Applications. ACM Trans. Algorithms 16, 4 (2020), 47:1–47:48. https://doi.org/10.1145/3396573
[19]
Fedor V. Fomin, Petr A. Golovach, Danil Sagunov, and Kirill Simonov. 2020. Algorithmic Extensions of Dirac’s Theorem. CoRR abs/2011.03619 (2020). arXiv:2011.03619 https://arxiv.org/abs/2011.03619
[20]
Fedor V. Fomin, Petr A. Golovach, Danil Sagunov, and Kirill Simonov. 2022. Algorithmic Extensions of Dirac’s Theorem. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA22). SIAM, 406–416.
[21]
Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. 2016. Efficient Computation of Representative Families with Applications in Parameterized and Exact Algorithms. J. ACM 63, 4 (2016), 29:1–29:60. https://doi.org/10.1145/2886094
[22]
Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. 2018. Long directed (s, t)-path: FPT algorithm. Inf. Process. Lett. 140 (2018), 8–12. https://doi.org/10.1016/j.ipl.2018.04.018
[23]
Steven Fortune, John E. Hopcroft, and James Wyllie. 1980. The Directed Subgraph Homeomorphism Problem. Theor. Comput. Sci. 10 (1980), 111–121. https://doi.org/10.1016/0304-3975(80)90009-2
[24]
Martin Grohe, Ken-ichi Kawarabayashi, Dániel Marx, and Paul Wollan. 2011. Finding topological subgraphs is fixed-parameter tractable. In Proceedings of the 43rd Annual ACM Symposium on Theory of Computing (STOC). ACM, 479–488.
[25]
Refael Hassin, Jérôme Monnot, and Danny Segev. 2007. Approximation algorithms and hardness results for labeled connectivity problems. J. Comb. Optim. 14, 4 (2007), 437–453. https://doi.org/10.1007/s10878-007-9044-x
[26]
Per M. Jensen and Bernhard Korte. 1982. Complexity of Matroid Property Algorithms. SIAM J. Comput. 11, 1 (1982), 184–190. https://doi.org/10.1137/0211014
[27]
Valentine Kabanets and Russell Impagliazzo. 2004. Derandomizing polynomial identity tests means proving circuit lower bounds. computational complexity 13, 1 (2004), 1–46.
[28]
Ken-ichi Kawarabayashi. 2008. An Improved Algorithm for Finding Cycles Through Elements. In 13th International Conference on Integer Programming and Combinatorial Optimization (IPCO) (Lecture Notes in Computer Science, Vol. 5035). Springer, 374–384. https://doi.org/10.1007/978-3-540-68891-4_26
[29]
Ioannis Koutis. 2008. Faster Algebraic Algorithms for Path and Packing Problems. In Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP) (Lecture Notes in Comput. Sci., Vol. 5125). Springer, 575–586.
[30]
Ioannis Koutis and Ryan Williams. 2016. Algebraic fingerprints for faster algorithms. Commun. ACM 59, 1 (2016), 98–105. https://doi.org/10.1145/2742544
[31]
Lukasz Kowalik and Juho Lauri. 2016. On finding rainbow and colorful paths. Theor. Comput. Sci. 628 (2016), 110–114. https://doi.org/10.1016/j.tcs.2016.03.017
[32]
Neeraj Kumar, Daniel Lokshtanov, Saket Saurabh, and Subhash Suri. 2021. A Constant Factor Approximation for Navigating Through Connected Obstacles in the Plane. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, (SODA). SIAM, 822–839.
[33]
Andrzej Lingas and Mia Persson. 2015. A Fast Parallel Algorithm for Minimum-Cost Small Integral Flows. Algorithmica 72, 2 (2015), 607–619. https://doi.org/10.1007/s00453-013-9865-1
[34]
L. Lovász. 1977. Flats in matroids and geometric graphs. In Combinatorial surveys (Proc. Sixth British Combinatorial Conf., Royal Holloway Coll., Egham, 1977). 45–86.
[35]
László Lovász. 2019. Graphs and geometry. American Mathematical Society Colloquium Publications, Vol. 65. American Mathematical Society, Providence, RI. x+444 pages. https://doi.org/10.1090/coll/065
[36]
L. Lovász and M. D. Plummer. 1986. Matching theory. North-Holland Mathematics Studies, Vol. 121. North-Holland Publishing Co., Amsterdam; North-Holland Publishing Co., Amsterdam. xxvii+544 pages. Annals of Discrete Mathematics, 29.
[37]
Dániel Marx. 2009. A parameterized view on matroid optimization problems. Theor. Comput. Sci. 410, 44 (2009), 4471–4479. https://doi.org/10.1016/j.tcs.2009.07.027
[38]
Juan Andrés Montoya and Moritz Müller. 2013. Parameterized Random Complexity. Theory Comput. Syst. 52, 2 (2013), 221–270. https://doi.org/10.1007/s00224-011-9381-0
[39]
Moni Naor, Leonard J. Schulman, and Aravind Srinivasan. 1995. Splitters and Near-Optimal Derandomization. In Proceedings of the 36th Annual Symposium on Foundations of Computer Science (FOCS 1995). IEEE, 182–191.
[40]
George L. Nemhauser, Laurence A. Wolsey, and Marshall L. Fisher. 1978. An analysis of approximations for maximizing submodular set functions - I. Math. Program. 14, 1 (1978), 265–294. https://doi.org/10.1007/BF01588971
[41]
James Oxley. 2011. Matroid theory (second ed.). Oxford Graduate Texts in Mathematics, Vol. 21. Oxford University Press, Oxford. xiv+684 pages. https://doi.org/10.1093/acprof:oso/9780198566946.001.0001
[42]
Fahad Panolan, Saket Saurabh, and Meirav Zehavi. 2019. Parameterized Algorithms for List K-Cycle. Algorithmica 81, 3 (2019), 1267–1287. https://doi.org/10.1007/s00453-018-0469-7
[43]
Neil Robertson and Paul D. Seymour. 1995. Graph Minors XIII. The Disjoint Paths Problem. J. Comb. Theory, Ser. B 63, 1 (1995), 65–110. https://doi.org/10.1006/jctb.1995.1006
[44]
Jacob T. Schwartz. 1980. Fast Probabilistic Algorithms for Verification of Polynomial Identities. J. ACM 27, 4 (1980), 701–717. https://doi.org/10.1145/322217.322225
[45]
Magnus Wahlström. 2013. Abusing the Tutte Matrix: An Algebraic Instance Compression for the K-set-cycle Problem. In 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013, February 27 - March 2, 2013, Kiel, Germany (LIPIcs, Vol. 20), Natacha Portier and Thomas Wilke (Eds.). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 341–352. https://doi.org/10.4230/LIPIcs.STACS.2013.341
[46]
Ryan Williams. 2009. Finding Paths of Length k in \({\mathcal{O}^*(2^k)}\) Time. Inf. Process. Lett. 109, 6 (2009), 315–318.
[47]
Hans-Christoph Wirth. 2001. Multicriteria approximation of network design and network upgrade problems. Ph. D. Dissertation. Universität Würzburg.
[48]
Meirav Zehavi. 2015. Mixing Color Coding-Related Techniques. In Algorithms - ESA 2015, Nikhil Bansal and Irene Finocchi (Eds.). Springer Berlin Heidelberg, Berlin, Heidelberg, 1037–1049.
[49]
Meirav Zehavi. 2016. A randomized algorithm for long directed cycle. Inf. Process. Lett. 116, 6 (2016), 419–422. https://doi.org/10.1016/j.ipl.2016.02.005
[50]
Richard Zippel. 1979. Probabilistic algorithms for sparse polynomials. In Symbolic and Algebraic Computation, EUROSAM ’79, An International Symposiumon Symbolic and Algebraic Computation, Marseille, France, June 1979, Proceedings (Lecture Notes in Computer Science, Vol. 72), Edward W. Ng (Ed.). Springer, 216–226. https://doi.org/10.1007/3-540-09519-5_73
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- Fixed-Parameter Tractability of Maximum Colored Path and Beyond
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Publication History
Online AM: 28 June 2024
Accepted: 18 June 2024
Revised: 18 June 2024
Received: 23 August 2023
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