TY - GEN

T1 - Collective Graph Exploration Parameterized by Vertex Cover

AU - Gupta, Siddharth

AU - Sa'ar, Guy

AU - Zehavi, Meirav

N1 - Publisher Copyright:
© 2023 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.

PY - 2023/12/1

Y1 - 2023/12/1

N2 - 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).

AB - 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).

KW - Approximation Algorithm

KW - Collective Graph Exploration

KW - Parameterized Complexity

KW - Treedepth

KW - Vertex Cover

UR - http://www.scopus.com/inward/record.url?scp=85180554897&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.IPEC.2023.22

DO - 10.4230/LIPIcs.IPEC.2023.22

M3 - Conference contribution

AN - SCOPUS:85180554897

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 18th International Symposium on Parameterized and Exact Computation, IPEC 2023

A2 - Misra, Neeldhara

A2 - Wahlstrom, Magnus

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 18th International Symposium on Parameterized and Exact Computation, IPEC 2023

Y2 - 6 September 2023 through 8 September 2023

ER -