Abstract
Given a set of edge pairs in a bipartite graph, we want to find a bipartite matching that includes a maximum number of those edge pairs. While the problem has many applications to wireless localization, to the best of our knowledge, there is no theoretical work for the problem. In this work, unless \(P = NP\), we show that there is no constant approximation for the problem. Suppose that k denotes the maximum number of input edge pairs such that a particular node can be in. Inspired by experimental results, we consider the case that k is small. While there is a simple polynomial-time algorithm for the problem when k is one, we show that the problem is NP-hard when k is greater than one. We also devise an efficient O(k)-approximation algorithm for the problem.
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References
Arora, S., Babai, L., Stern, J., Sweedyk, Z.: The hardness of approximate optima in lattices, codes, and systems of linear equations. J. Comput. Syst. Sci. 54(2), 317–331 (1997)
Biswas, P., Lian, T.C., Wang, T.C., Ye, Y.: Semidefinite programming based algorithms for sensor network localization. ACM Trans. Sens. Netw. (TOSN) 2(2), 188–220 (2006)
Charikar, M., Hajiaghayi, M., Karloff, H.: Improved approximation algorithms for label cover problems. ESA 2009, 23–34 (2009)
Fleischner, H.: X. 1 algorithms for Eulerian trails. In: Eulerian Graphs and Related Topics: Part 1. Annals of Discrete Mathematics, vol. 50, pp. 1–13 (1991)
Fleischner, H., Sabidussi, G., Sarvanov, V.I.: Maximum independent sets in 3-and 4-regular Hamiltonian graphs. Discrete Math. 310(20), 2742–2749 (2010)
Fleury, M.: Deux problemes de geometrie de situation. Journal de mathematiques elementaires 2(2), 257–261 (1883)
Ghafourian, A., Georgiou, O., Barter, E., Gross, T.: Wireless localization with diffusion maps. arXiv preprint arXiv:1908.05216 (2019)
Halldórsson, M.M., Radhakrishnan, J.: Greed is good: approximating independent sets in sparse and bounded-degree graphs. Algorithmica 18(1), 145–163 (1997)
Hopcroft, J.E., Karp, R.M.: An \(n^{5/2}\) algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2(4), 225–231 (1973)
Hu, L., Evans, D.: Localization for mobile sensor networks. In: MobiCom 2004, pp. 45–57 (2004)
Nguyen, C.L., Georgiou, O., Yonezawa, Y., Doi, Y.: The wireless localisation matching problem and a maximum likelihood based solution. In: ICC 2017, pp. 1–7 (2017)
Nguyen, C.L., Georgiou, O., Yonezawa, Y., Doi, Y.: The wireless localization matching problem. IEEE Internet Things J. 4(5), 1312–1326 (2017)
Nguyen, L., Georgiou, O., Suppakitpaisarn, V.: Improved localization accuracy using machine learning and refining RSS measurements. In: GLOBECOM Workshops 2018 (2018)
Skiena, S.: Coloring bipartite graphs, chap. 5.5.2. In: Skiena, S. (ed.) Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica, p. 213. Addison-Wesley, Reading (1990)
Suppakitpaisarn, V., Dai, W., Baffier, J.F.: Robust network flow against attackers with knowledge of routing method. In: HPSR 2015, pp. 40–47 (2015)
Acknowledgement
Parts of the work had been done when Phanu Vajanopath was conducting an internship at The University of Tokyo. The internship was supported by the GSI Internship program, Faculty of Science, The University of Tokyo. He was hosted by Prof. Reiji Suda during the program. Also, the authors want to thank reviewers of WALCOM 2020 who kindly gave comments that significantly improve this paper.
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Nguyen, C.L., Suppakitpaisarn, V., Surarerks, A., Vajanopath, P. (2020). On the Maximum Edge-Pair Embedding Bipartite Matching. In: Rahman, M., Sadakane, K., Sung, WK. (eds) WALCOM: Algorithms and Computation. WALCOM 2020. Lecture Notes in Computer Science(), vol 12049. Springer, Cham. https://doi.org/10.1007/978-3-030-39881-1_20
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DOI: https://doi.org/10.1007/978-3-030-39881-1_20
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