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)
https://doi.org/10.4230/LIPIcs.IPEC.2023.22

Abstract

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
Keywords
  • Collective Graph Exploration
  • Parameterized Complexity
  • Approximation Algorithm
  • Vertex Cover
  • Treedepth

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