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Engineering a PTAS for Minimum Feedback Vertex Set in Planar Graphs

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Analysis of Experimental Algorithms (SEA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11544))

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Abstract

We investigate the practicality of approximation schemes for optimization problems in planar graphs based on balanced separators. The first polynomial-time approximation schemes (PTASes) for problems in planar graphs were based on balanced separators, wherein graphs are recursively decomposed into small enough pieces in which optimal solutions can be found by brute force or other methods. However, this technique was supplanted by the more modern and (theoretically) more efficient approach of decomposing a planar graph into graphs of bounded treewidth, in which optimal solutions are found by dynamic programming. While the latter approach has been tested experimentally, the former approach has not.

To test the separator-based method, we examine the minimum feedback vertex set (FVS) problem in planar graphs. We propose a new, simple \(O(n \log n)\)-time approximation scheme for FVS using balanced separators and a linear kernel. The linear kernel reduces the size of the graph to be linear in the size of the optimal solution. In doing so, we correct a reduction rule in Bonamy and Kowalik’s linear kernel [11] for FVS. We implemented this PTAS and evaluated its performance on large synthetic and real-world planar graphs. Unlike earlier planar PTAS engineering results [8, 36], our implementation guarantees the theoretical error bounds on all tested graphs.

This material is based upon work supported by the National Science Foundation under Grant No. CCF-1252833.

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Notes

  1. 1.

    Given a spanning tree for a graph, a fundamental cycle consists of a non-tree edge and a path in the tree connecting the two endpoints of that edge.

  2. 2.

    This is to return a trivial no-instance for the decision problem when the resulting graph has more than 15k vertices.

  3. 3.

    http://jgaa.info/accepted/2004/BoyerMyrvold2004.8.3/planarity.zip.

  4. 4.

    https://github.com/wata-orz/fvs.

References

  1. Abu-Khzam, F.N., Bou Khuzam, M.: An improved kernel for the undirected planar feedback vertex set problem. In: Thilikos, D.M., Woeginger, G.J. (eds.) IPEC 2012. LNCS, vol. 7535, pp. 264–273. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-33293-7_25

    Chapter  Google Scholar 

  2. Aleksandrov, L., Djidjev, H., Guo, H., Maheshwari, A.: Partitioning planar graphs with costs and weights. J. Exp. Algorithm. (JEA) 11, 1–5 (2007)

    MathSciNet  MATH  Google Scholar 

  3. Alon, N., Seymour, P., Thomas, R.: A separator theorem for nonplanar graphs. J. Am. Math. Soc. 3(4), 801–808 (1990)

    Article  MathSciNet  Google Scholar 

  4. Arora, S., Grigni, M., Karger, D.R., Klein, P.N., Woloszyn, A.: A polynomial-time approximation scheme for weighted planar graph TSP. In: SODA, vol. 98, pp. 33–41 (1998)

    Google Scholar 

  5. Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM (JACM) 41(1), 153–180 (1994)

    Article  MathSciNet  Google Scholar 

  6. Bar-Yehuda, R., Geiger, D., Naor, J., Roth, R.M.: Approximation algorithms for the feedback vertex set problem with applications to constraint satisfaction and Bayesian inference. SIAM J. Comput. 27(4), 942–959 (1998)

    Article  MathSciNet  Google Scholar 

  7. Bateni, M.H., Hajiaghayi, M.T., Marx, D.: Approximation schemes for Steiner forest on planar graphs and graphs of bounded treewidth. J. ACM 58(5), 21 (2011)

    Article  MathSciNet  Google Scholar 

  8. Becker, A., Fox-Epstein, E., Klein, P.N., Meierfrankenfeld, D.: Engineering an approximation scheme for traveling salesman in planar graphs. In: LIPIcs-Leibniz International Proceedings in Informatics, vol. 75 (2017)

    Google Scholar 

  9. Becker, A., Geiger, D.: Optimization of Pearl’s method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem. Artif. Intell. 83(1), 167–188 (1996)

    Article  MathSciNet  Google Scholar 

  10. Bodlaender, H.L., Fomin, F.V., Lokshtanov, D., Penninkx, E., Saurabh, S., Thilikos, D.M.: (Meta) kernelization. J. ACM (JACM) 63(5), 44 (2016)

    Article  MathSciNet  Google Scholar 

  11. Bonamy, M., Kowalik, Ł.: A 13k-kernel for planar feedback vertex set via region decomposition. Theoret. Comput. Sci. 645, 25–40 (2016)

    Article  MathSciNet  Google Scholar 

  12. Borradaile, G., Klein, P., Mathieu, C.: An \({O}(n \log n)\) approximation scheme for Steiner tree in planar graphs. ACM Trans. Algorithms (TALG) 5(3), 1–31 (2009)

    Article  MathSciNet  Google Scholar 

  13. Boyer, J.M., Myrvold, W.J.: On the cutting edge: simplified \({O}(n)\) planarity by edge addition. J. Graph Algorithms Appl. 8(2), 241–273 (2004)

    Article  MathSciNet  Google Scholar 

  14. Chiba, N., Nishizeki, T., Saito, N.: Applications of the Lipton and Tarjan’s planar separator theorem. J. Inf. Process. 4(4), 203–207 (1981)

    MathSciNet  MATH  Google Scholar 

  15. Chiba, N., Nishizeki, T., Saito, N.: An approximation algorithm for the maximum independent set problem on planar graphs. SIAM J. Comput. 11(4), 663–675 (1982)

    Article  MathSciNet  Google Scholar 

  16. Cohen-Addad, V., de Verdière, É.C., Klein, P.N., Mathieu, C., Meierfrankenfeld, D.: Approximating connectivity domination in weighted bounded-genus graphs. In: Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, pp. 584–597. ACM (2016)

    Google Scholar 

  17. Demaine, E.D., Hajiaghayi, M.T., Kawarabayashi, K.-i.: Algorithmic graph minor theory: decomposition, approximation, and coloring. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, pp. 637–646 (2005)

    Google Scholar 

  18. Demaine, E.D., Hajiaghayi, M.T.: Bidimensionality: new connections between FPT algorithms and PTASs. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, pp. 590–601 (2005)

    Google Scholar 

  19. Demetrescu, C., Goldberg, A., Johnson, D.: Implementation challenge for shortest paths. In: Kao, M.Y. (ed.) Encyclopedia of Algorithms. Springer, Boston (2008). https://doi.org/10.1007/978-0-387-30162-4

    Chapter  Google Scholar 

  20. Eisenstat, D., Klein, P., Mathieu, C.: An efficient polynomial-time approximation scheme for Steiner forest in planar graphs. In: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 626–638. SIAM (2012)

    Google Scholar 

  21. Fomin, F.V., Lokshtanov, D., Saurabh, S., Thilikos, D.M.: Bidimensionality and kernels. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 503–510 (2010)

    Google Scholar 

  22. Fomin, F.V., Lokshtanov, D., Saurabh, S., Thilikos, D.M.: Linear kernels for (connected) dominating set on H-minor-free graphs. In: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 82–93 (2012)

    Google Scholar 

  23. Fox-Epstein, E., Mozes, S., Phothilimthana, P.M., Sommer, C.: Short and simple cycle separators in planar graphs. J. Exp. Algorithm. (JEA) 21, 2 (2016)

    MathSciNet  MATH  Google Scholar 

  24. Grohe, M.: Local tree-width, excluded minors, and approximation algorithms. Combinatorica 23(4), 613–632 (2003)

    Article  MathSciNet  Google Scholar 

  25. Holzer, M., Schulz, F., Wagner, D., Prasinos, G., Zaroliagis, C.: Engineering planar separator algorithms. J. Exp. Algorithm. (JEA) 14, 5 (2009)

    MathSciNet  MATH  Google Scholar 

  26. Iwata, Y.: Linear-time kernelization for feedback vertex set. CoRR (2016)

    Google Scholar 

  27. Iwata, Y., Wahlström, M., Yoshida, Y.: Half-integrality, LP-branching, and FPT algorithms. SIAM J. Comput. 45(4), 1377–1411 (2016)

    Article  MathSciNet  Google Scholar 

  28. Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations. The IBM Research Symposia Series. Springer, Boston (1972). https://doi.org/10.1007/978-1-4684-2001-2_9

    Chapter  Google Scholar 

  29. Klein, P.N.: A linear-time approximation scheme for TSP in undirected planar graphs with edge-weights. SIAM J. Comput. 37(6), 1926–1952 (2008)

    Article  MathSciNet  Google Scholar 

  30. Kozen, D.C.: The Design and Analysis of Algorithms. Springer, Heidelberg (2012)

    Google Scholar 

  31. Le, H., Zheng, B.: Local search is a PTAS for feedback vertex set in minor-free graphs. CoRR, abs/1804.06428 (2018)

    Google Scholar 

  32. Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Appl. Math. 36(2), 177–189 (1979)

    Article  MathSciNet  Google Scholar 

  33. Lipton, R.J., Tarjan, R.E.: Applications of a planar separator theorem. SIAM J. Comput. 9(3), 615–627 (1980)

    Article  MathSciNet  Google Scholar 

  34. Marzban, M., Qian-Ping, G.: Computational study on a PTAS for planar dominating set problem. Algorithms 6(1), 43–59 (2013)

    Article  MathSciNet  Google Scholar 

  35. Mehlhorn, K., Näher, S., Uhrig, C.: The LEDA platform for combinatorial and geometric computing. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, pp. 7–16. Springer, Heidelberg (1997). https://doi.org/10.1007/3-540-63165-8_161

    Chapter  Google Scholar 

  36. Tazari, S., Müller-Hannemann, M.: Dealing with large hidden constants: engineering a planar Steiner tree PTAS. J. Exp. Algorithm. (JEA) 16, 3–6 (2011)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Oregon State University, Corvallis, OR, USA

    Glencora Borradaile, Hung Le & Baigong Zheng

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  1. Glencora Borradaile
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  2. Hung Le
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  3. Baigong Zheng
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Correspondence to Hung Le .

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Editors and Affiliations

  1. Wilfrid Laurier University, Waterloo, ON, Canada

    Ilias Kotsireas

  2. University of Florida, Gainesville, FL, USA

    Panos Pardalos

  3. University of Ioannina, Ioannina, Greece

    Konstantinos E. Parsopoulos

  4. Delft University of Technology, Delft, The Netherlands

    Dimitris Souravlias

  5. University of Florida, Gainesville, FL, USA

    Arsenis Tsokas

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Borradaile, G., Le, H., Zheng, B. (2019). Engineering a PTAS for Minimum Feedback Vertex Set in Planar Graphs. In: Kotsireas, I., Pardalos, P., Parsopoulos, K., Souravlias, D., Tsokas, A. (eds) Analysis of Experimental Algorithms. SEA 2019. Lecture Notes in Computer Science(), vol 11544. Springer, Cham. https://doi.org/10.1007/978-3-030-34029-2_7

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  • DOI: https://doi.org/10.1007/978-3-030-34029-2_7

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