Collective Graph Exploration Parameterized by Vertex Cover

Authors Siddharth Gupta , Guy Sa'ar, Meirav Zehavi

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Author Details

Siddharth Gupta
  • BITS Pilani, Goa Campus, India
Guy Sa'ar
  • Ben Gurion University of the Negev, Beersheba, Israel
Meirav Zehavi
  • Ben Gurion University of the Negev, Beersheba, Israel

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Siddharth Gupta, Guy Sa'ar, and Meirav Zehavi. Collective Graph Exploration Parameterized by Vertex Cover. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We initiate the study of the parameterized complexity of the Collective Graph Exploration (CGE) problem. In CGE, the input consists of an undirected connected graph G and a collection of k robots, initially placed at the same vertex r of G, and each one of them has an energy budget of B. The objective is to decide whether G can be explored by the k robots in B time steps, i.e., there exist k closed walks in G, one corresponding to each robot, such that every edge is covered by at least one walk, every walk starts and ends at the vertex r, and the maximum length of any walk is at most B. Unfortunately, this problem is NP-hard even on trees [Fraigniaud et al., 2006]. Further, we prove that the problem remains W[1]-hard parameterized by k even for trees of treedepth 3. Due to the para-NP-hardness of the problem parameterized by treedepth, and motivated by real-world scenarios, we study the parameterized complexity of the problem parameterized by the vertex cover number (vc) of the graph, and prove that the problem is fixed-parameter tractable (FPT) parameterized by vc. Additionally, we study the optimization version of CGE, where we want to optimize B, and design an approximation algorithm with an additive approximation factor of O(vc).

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Approximation algorithms analysis
  • Collective Graph Exploration
  • Parameterized Complexity
  • Approximation Algorithm
  • Vertex Cover
  • Treedepth


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  1. Igor Averbakh and Oded Berman. A heuristic with worst-case analysis for minimax routing of two travelling salesmen on a tree. Discret. Appl. Math., 68(1-2):17-32, 1996. URL:
  2. Igor Averbakh and Oded Berman. (p - 1)/(p + 1)-approximate algorithms for p-traveling salesmen problems on a tree with minmax objective. Discret. Appl. Math., 75(3):201-216, 1997. URL:
  3. Béla Bollobás. Modern graph theory, volume 184. Springer Science & Business Media, 1998. Google Scholar
  4. Peter Brass, Flavio Cabrera-Mora, Andrea Gasparri, and Jizhong Xiao. Multirobot tree and graph exploration. IEEE Trans. Robotics, 27(4):707-717, 2011. URL:
  5. Romain Cosson, Laurent Massoulié, and Laurent Viennot. Breadth-first depth-next: Optimal collaborative exploration of trees with low diameter. CoRR, abs/2301.13307, 2023. URL:
  6. Shantanu Das, Dariusz Dereniowski, and Christina Karousatou. Collaborative exploration of trees by energy-constrained mobile robots. Theory Comput. Syst., 62(5):1223-1240, 2018. URL:
  7. Dariusz Dereniowski, Yann Disser, Adrian Kosowski, Dominik Pajak, and Przemyslaw Uznanski. Fast collaborative graph exploration. Inf. Comput., 243:37-49, 2015. URL:
  8. Yann Disser, Frank Mousset, Andreas Noever, Nemanja Skoric, and Angelika Steger. A general lower bound for collaborative tree exploration. Theor. Comput. Sci., 811:70-78, 2020. URL:
  9. Miroslaw Dynia, Miroslaw Korzeniowski, and Christian Schindelhauer. Power-aware collective tree exploration. In Werner Grass, Bernhard Sick, and Klaus Waldschmidt, editors, Architecture of Computing Systems - ARCS 2006, 19th International Conference, Frankfurt/Main, Germany, March 13-16, 2006, Proceedings, volume 3894 of Lecture Notes in Computer Science, pages 341-351. Springer, 2006. URL:
  10. Miroslaw Dynia, Jaroslaw Kutylowski, Friedhelm Meyer auf der Heide, and Christian Schindelhauer. Smart robot teams exploring sparse trees. In Rastislav Kralovic and Pawel Urzyczyn, editors, Mathematical Foundations of Computer Science 2006, 31st International Symposium, MFCS 2006, Stará Lesná, Slovakia, August 28-September 1, 2006, Proceedings, volume 4162 of Lecture Notes in Computer Science, pages 327-338. Springer, 2006. URL:
  11. Pierre Fraigniaud, Leszek Gasieniec, Dariusz R. Kowalski, and Andrzej Pelc. Collective tree exploration. Networks, 48(3):166-177, 2006. URL:
  12. András Frank and Éva Tardos. An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica, 7(1):49-65, 1987. URL:
  13. Siddharth Gupta, Guy Sa'ar, and Meirav Zehavi. Collective graph exploration parameterized by vertex cover, 2023. URL:
  14. Yuya Higashikawa, Naoki Katoh, Stefan Langerman, and Shin-ichi Tanigawa. Online graph exploration algorithms for cycles and trees by multiple searchers. J. Comb. Optim., 28(2):480-495, 2014. URL:
  15. Klaus Jansen, Stefan Kratsch, Dániel Marx, and Ildikó Schlotter. Bin packing with fixed number of bins revisited. J. Comput. Syst. Sci., 79(1):39-49, 2013. URL:
  16. Hendrik W. Lenstra Jr. Integer programming with a fixed number of variables. Math. Oper. Res., 8(4):538-548, 1983. URL:
  17. Ravi Kannan. Minkowski’s convex body theorem and integer programming. Math. Oper. Res., 12(3):415-440, 1987. URL:
  18. Hiroshi Nagamochi and Kohei Okada. A faster 2-approximation algorithm for the minmax p-traveling salesmen problem on a tree. Discret. Appl. Math., 140(1-3):103-114, 2004. URL:
  19. Christian Ortolf and Christian Schindelhauer. A recursive approach to multi-robot exploration of trees. In Magnús M. Halldórsson, editor, Structural Information and Communication Complexity - 21st International Colloquium, SIROCCO 2014, Takayama, Japan, July 23-25, 2014. Proceedings, volume 8576 of Lecture Notes in Computer Science, pages 343-354. Springer, 2014. URL: