Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
125 tokens/sec
GPT-4o
47 tokens/sec
Gemini 2.5 Pro Pro
43 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
47 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Balanced Substructures in Bicolored Graphs (2403.06608v2)

Published 11 Mar 2024 in cs.DS and cs.CC

Abstract: An edge-colored graph is said to be balanced if it has an equal number of edges of each color. Given a graph $G$ whose edges are colored using two colors and a positive integer $k$, the objective in the Edge Balanced Connected Subgraph problem is to determine if $G$ has a balanced connected subgraph containing at least $k$ edges. We first show that this problem is NP-complete and remains so even if the solution is required to be a tree or a path. Then, we focus on the parameterized complexity of Edge Balanced Connected Subgraph and its variants (where the balanced subgraph is required to be a path/tree) with respect to $k$ as the parameter. Towards this, we show that if a graph has a balanced connected subgraph/tree/path of size at least $k$, then it has one of size at least $k$ and at most $f(k)$ where $f$ is a linear function. We use this result combined with dynamic programming algorithms based on color coding and representative sets to show that Edge Balanced Connected Subgraph and its variants are FPT. Further, using polynomial-time reductions to the Multilinear Monomial Detection problem, we give faster randomized FPT algorithms for the problems. In order to describe these reductions, we define a combinatorial object called relaxed-subgraph. We define this object in such a way that balanced connected subgraphs, trees and paths are relaxed-subgraphs with certain properties. This object is defined in the spirit of branching walks known for the Steiner Tree problem and may be of independent interest.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (27)
  1. Color-coding. J. ACM, 42(4):844–856, 1995. doi:10.1145/210332.210337.
  2. The balanced connected subgraph problem. In Sudebkumar Prasant Pal and Ambat Vijayakumar, editors, Algorithms and Discrete Applied Mathematics - 5th International Conference, CALDAM 2019, Kharagpur, India, February 14-16, 2019, Proceedings, volume 11394 of Lecture Notes in Computer Science, pages 201–215. Springer, 2019. doi:10.1007/978-3-030-11509-8\_17.
  3. Balanced connected subgraph problem in geometric intersection graphs. In Yingshu Li, Mihaela Cardei, and Yan Huang, editors, Combinatorial Optimization and Applications - 13th International Conference, COCOA 2019, Xiamen, China, December 13-15, 2019, Proceedings, volume 11949 of Lecture Notes in Computer Science, pages 56–68. Springer, 2019. doi:10.1007/978-3-030-36412-0\_5.
  4. On problems without polynomial kernels. J. Comput. Syst. Sci., 75(8):423–434, 2009. doi:10.1016/j.jcss.2009.04.001.
  5. Béla Bollobás. On generalized graphs. Acta Mathematica Hungarica, 16(3-4):447–452, 1965. doi:10.1007/BF01904851.
  6. Highly connected monochromatic subgraphs. Discret. Math., 308(9):1722–1725, 2008. doi:10.1016/j.disc.2006.01.030.
  7. The recognition of bound quivers using edge-coloured homomorphisms. Discret. Math., 297(1-3):13–25, 2005. doi:10.1016/j.disc.2004.10.026.
  8. Large monochromatic components of small diameter. J. Graph Theory, 99(2):247–250, 2022. doi:10.1002/jgt.22739.
  9. On zero-sum spanning trees and zero-sum connectivity. Electron. J. Comb., 29(1), 2022. doi:10.37236/10289.
  10. Parameterized Algorithms. Springer, 2015. doi:10.1007/978-3-319-21275-3.
  11. The balanced connected subgraph problem: Complexity results in bounded-degree and bounded-diameter graphs. In Yingshu Li, Mihaela Cardei, and Yan Huang, editors, Combinatorial Optimization and Applications - 13th International Conference, COCOA 2019, Xiamen, China, December 13-15, 2019, Proceedings, volume 11949 of Lecture Notes in Computer Science, pages 449–460. Springer, 2019. doi:10.1007/978-3-030-36412-0\_36.
  12. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012.
  13. Kernelization lower bounds through colors and ids. ACM Trans. Algorithms, 11(2):13:1–13:20, 2014. doi:10.1145/2650261.
  14. Efficient computation of representative families with applications in parameterized and exact algorithms. J. ACM, 63(4):29:1–29:60, 2016. doi:10.1145/2886094.
  15. Zoltán Füredi. Maximum degree and fractional matchings in uniform hypergraphs. Comb., 1(2):155–162, 1981. doi:10.1007/BF02579271.
  16. The rectilinear steiner tree problem in NP complete. SIAM Journal of Applied Mathematics, 32:826–834, 1977. doi:10.1137/0132071.
  17. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979.
  18. András Gyárfás. Partition coverings and blocking sets of hypergraphs. Comm. Comput. Automat. Inst. Hungar. Acad. Sci., 71:62, 1977.
  19. Algorithms and hardness results for the maximum balanced connected subgraph problem. In Yingshu Li, Mihaela Cardei, and Yan Huang, editors, Combinatorial Optimization and Applications - 13th International Conference, COCOA 2019, Xiamen, China, December 13-15, 2019, Proceedings, volume 11949 of Lecture Notes in Computer Science, pages 303–315. Springer, 2019. doi:10.1007/978-3-030-36412-0\_24.
  20. Ioannis Koutis. Faster algebraic algorithms for path and packing problems. In Luca Aceto, Ivan Damgård, Leslie Ann Goldberg, Magnús M. Halldórsson, Anna Ingólfsdóttir, and Igor Walukiewicz, editors, Automata, Languages and Programming, 35th International Colloquium, ICALP 2008, Reykjavik, Iceland, July 7-11, 2008, Proceedings, Part I: Tack A: Algorithms, Automata, Complexity, and Games, volume 5125 of Lecture Notes in Computer Science, pages 575–586. Springer, 2008. doi:10.1007/978-3-540-70575-8\_47.
  21. LIMITS and applications of group algebras for parameterized problems. ACM Trans. Algorithms, 12(3):31:1–31:18, 2016. doi:10.1145/2885499.
  22. Complexity and inapproximability results for balanced connected subgraph problem. Theor. Comput. Sci., 886:69–83, 2021. doi:10.1016/j.tcs.2021.07.010.
  23. Splitters and near-optimal derandomization. In 36th Annual Symposium on Foundations of Computer Science, Milwaukee, Wisconsin, USA, 23-25 October 1995, pages 182–191. IEEE Computer Society, 1995. doi:10.1109/SFCS.1995.492475.
  24. Jesper Nederlof. Fast polynomial-space algorithms using inclusion-exclusion. Algorithmica, 65(4):868–884, 2013. doi:10.1007/s00453-012-9630-x.
  25. Representative families: A unified tradeoff-based approach. J. Comput. Syst. Sci., 82(3):488–502, 2016. doi:10.1016/j.jcss.2015.11.008.
  26. Steiner trees, connected domination and strongly chordal graphs. Networks, 15(1):109–124, 1985. doi:10.1002/net.3230150109.
  27. Ryan Williams. Finding paths of length k𝑘kitalic_k in O∗⁢(2k)superscript𝑂superscript2𝑘O^{*}(2^{k})italic_O start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( 2 start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) time. Inf. Process. Lett., 109(6):315–318, 2009. doi:10.1016/j.ipl.2008.11.004.

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now