在未知環境中進行地圖探勘是具有挑戰性的，因為最大化覆蓋率
是個NP-hard的問題，這項研究提出了基於動態更新樹狀結構路徑而
最大化覆蓋率之邊際效益的方法。由於缺少環境資訊，樹狀結構路徑
需於不同時間段進行更新。因此，此項研究證明出每個時間段的理論
保證值。實驗結果說明此項研究提出的方法效能更勝於其他方法。

Map exploration in unknown environments is challenging since finding the optimal solution to maximize the environmental coverage is NPhard. This research proposes a method that maximizes the dynamic marginal gain of the coverage with the tree-Structured routing. Since the map is unknown, the tree-structured routing is dynamically updated at each time step. Thus, the theoretical guarantees at each time step are proved. The experiments show that the proposed method outperforms benchmark methods.

[1] Sanjeev Arora. Polynomial time approximation schemes for euclidean traveling salesman and other geometric problems. Journal
of the ACM, 45(5):753–782, 1998.
[2] Jonathan Binney, Andreas Krause, and Gaurav S. Sukhatme. Informative path planning for an autonomous underwater vehicle. IEEE
International Conference on Robotics and Automation, pages 4791–
4796, 2010.
[3] Jonathan Binney and Gaurav S. Sukhatme. Branch and bound
for informative path planning. IEEE International Conference on
Robotics and Automation, pages 2147–2154, 2012.
[4] Andreas Bircher, Mina Kamel, Kostas Alexis, Helen Oleynikova,
and Roland Siegwart. Receding horizon "next-best-view" planner
for 3d exploration. IEEE International Conference on Robotics and
Automation, pages 1462–1468, 2016.
[5] Andreas Bircher, Mina Samir Kamel, Kostas Alexis, Helen
Oleynikova, and Roland Siegwart. Receding horizon path planning for 3d exploration and surface inspection. Autonomous Robots,
42(2):291–306, 2018.
[6] Cesar Cadena, Luca Carlone, Henry Carrillo, Yasir Latif, Davide
Scaramuzza, José Neira, Ian Reid, and John J. Leonard. Past,
present, and future of simultaneous localization and mapping: Toward the robust-perception age. IEEE Transactions on Robotics,
32(6):1309–1332, 2016.
[7] Wei-Hsiang Chiu. Informative path planning using cost-benefit
spanning tree. Master’s thesis, National Central University, 2023.
30￾￾￾￾￾￾￾￾￾￾￾￾

[8] Sanjiban Choudhury, Mohak Bhardwaj, Sankalp Arora, Ashish
Kapoor, Gireeja Ranade, Sebastian Scherer, and Debadeepta Dey.
Data-driven planning via imitation learning. The International
Journal of Robotics Research, 37(13-14):1632–1672, 2018.
[9] Nicos Christofides. Worst-case analysis of a new heuristic for the
travelling salesman problem. Operations Research Forum, 3(1):1–4,
2022.
[10] Michele Conforti and Gérard Cornuéjols. Submodular set functions,
matroids and the greedy algorithm: Tight worst-case bounds and
some generalizations of the rado-edmonds theorem. Discret. Appl.
Math., 7(3):251–274, 1984.
[11] Chen Feng, Haojia Li, Fei Gao, Boyu Zhou, and Shaojie Shen.
Predrecon: A prediction-boosted planning framework for fast and
high-quality autonomous aerial reconstruction. IEEE International
Conference on Robotics and Automation, pages 1207–1213, 2023.
[12] Héctor H. González-Baños and Jean-Claude Latombe. Navigation
strategies for exploring indoor environments. The International
Journal of Robotics Research, 21(10-11):829 – 848, 2002.
[13] Tal Grossman and Avishai Wool. Computational experience with
approximation algorithms for the set covering problem. European
Journal of Operational Research, 101(1):81–92, 1997.
[14] Michael Held and Richard M. Karp. The traveling-salesman problem and minimum spanning trees. Operations Research, 18(6):1138–
1162, 1970.
[15] Lionel Heng, Alkis Gotovos, Andreas Krause, and Marc Pollefeys.
Efficient visual exploration and coverage with a micro aerial vehicle in unknown environments. IEEE International Conference on
Robotics and Automation, pages 1071–1078, 2015.
[16] Yale T. Herer. Submodularity and the traveling salesman problem.
European Journal of Operational Research, 114(3):489–508, 1999.
31

[17] Geoffrey Hollinger and Sanjiv Singh. Proofs and experiments in
scalable, near-optimal search by multiple robots. Proceedings of
Robotics: Science and Systems, pages 206 – 213, 2008.
[18] Geoffrey A. Hollinger, Chiranjib Choudhuri, Urbashi Mitra, and
Gaurav S. Sukhatme. Squared error distortion metrics for motion
planning in robotic sensor networks. IEEE Globecom Workshops,
pages 1426–1431, 2013.
[19] Samir Khuller, Anna Moss, and Joseph Naor. The budgeted maximum coverage problem. Inf. Process. Lett., 70(1):39–45, 1999.
[20] A. Krause, C. Guestrin, A. Gupta, and J. Kleinberg. Near-optimal
sensor placements: maximizing information while minimizing communication cost. 5th International Conference on Information Processing in Sensor Networks, pages 2–10, 2006.
[21] Andreas Krause and Carlos Guestrin. A note on the budgeted maximization of submodular functions. 2005.
[22] Zhan Wei Lim, David Hsu, and Wee Sun Lee. Adaptive informative path planning in metric spaces. The International Journal of
Robotics Research, 35(5):585–598, 2015.
[23] Pao-Te Lin. Maximal coverage problems with routing constraints
using cross-entropy monte carlo tree search. Master’s thesis, National Central University, 2022.
[24] Pao-Te Lin and Kuo-Shih Tseng. Improvement of submodular maximization problems with routing constraints via submodularity and
fourier sparsity. IEEE Robotics and Automation Letters, 8(4):1927–
1934, 2023.
[25] Shen Lin and Brian W. Kernighan. An effective heuristic algorithm for the traveling-salesman problem. Oper. Res., 21(2):498–
516, 1973.
[26] Bing-Xian Lu and Kuo-Shih Tseng. 3d map exploration via learning
submodular functions in the fourier domain. International Conference on Unmanned Aircraft Systems, pages 1199–1205, 2020.
32

[27] Bing Xian Lu and Kuo Shih Tseng. 3d map exploration using topological fourier sparse set. Journal of Intelligent and Robotic Systems, 104(4), 2022.
[28] Zehui Meng, Hailong Qin, Ziyue Chen, Xudong Chen, Hao Sun,
Feng Lin, and Marcelo H. Ang. A two-stage optimized next-view
planning framework for 3-d unknown environment exploration, and
structural reconstruction. IEEE Robotics and Automation Letters,
2(3):1680–1687, 2017.
[29] George L. Nemhauser, Laurence A. Wolsey, and Marshall L. Fisher.
An analysis of approximations for maximizing submodular set functions—i. Mathematical Programming, 14(1):265–294, 1978.
[30] Mike Roberts, Debadeepta Dey, Anh Truong, Sudipta Sinha, Shital
Shah, Ashish Kapoor, Pat Hanrahan, and Neel Joshi. Submodular trajectory optimization for aerial 3d scanning. International
Conference on Computer Vision, pages 5334–5343, 2017.
[31] L. M. Schmid, M. Pantic, R. Khanna, L. Ott, and J. Nieto. An efficient sampling-based method for online informative path planning
in unknown environments. IEEE Robotics and Automation Letters,
5(2):1500–1507, 2019.
[32] Amarjeet Singh, Andreas Krause, Carlos Guestrin, William J.
Kaiser, and Maxim A. Batalin. Efficient planning of informative
paths for multiple robots. International Joint Conference on Artificial Intelligence, page 2204–2211, 2007.
[33] Amarjeet Singh, Andreas Krause, and William Kaiser. Nonmyopic
adaptive informative path planning for multiple robots. Proceedings
of the 21st International Joint Conference on Artificial Intelligence,
pages 1843–1850, 2009.
[34] Peter Stobbe and Andreas Krause. Learning fourier sparse set functions. Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics, 22:1125–1133, 2012.
33

[35] Maxim Sviridenko. A note on maximizing a submodular set function subject to a knapsack constraint. Operations Research Letters,
32(1):41–43, 2004.
[36] Yu-Chung Tsai and Kuo-Shih Tseng. Deep compressed sensing for
learning submodular functions. Sensors, 20(9):2591, 2020.
[37] Ji Jie Wu and Kuo Shih Tseng. Adaptive submodular inverse reinforcement learning for spatial search and map exploration. Autonomous Robots, 46(2):321–347, 2022.
[38] B. Yamauchi. A frontier-based approach for autonomous exploration. Proceedings IEEE International Symposium on Computational Intelligence in Robotics and Automation, pages 146–151,
1997.
[39] Leonardo Zambito. The traveling salesman problem: A comprehensive survey. Project for CSE, 2006.
[40] Haifeng Zhang and Yevgeniy Vorobeychik. Submodular optimization with routing constraints. Proceedings of the AAAI conference
on Artificial Intelligence, pages 819–825, 2016.
[41] Boyu Zhou, Yichen Zhang, Xinyi Chen, and Shaojie Shen. Fuel:
Fast uav exploration using incremental frontier structure and hierarchical planning. IEEE Robotics and Automation Letters, 6(2):779–
786, 2021.