Graph Traversal via Connected Mobile Agents

This paper considers the Hamiltonian walk problem in the multi-agent coordination framework, referred to as $k$-agents Hamiltonian walk problem ($k$-HWP). In this problem, a set of $k$ connected agents collectively compute a spanning walk of a given undirected graph in the minimum steps. At each step, the agents are at $k$ distinct vertices and the induced subgraph made by the occupied vertices remains connected. In the next consecutive steps, each agent may remain stationary or move to one of its neighbours.To the best of our knowledge, this problem has not been previously explored in the context of multi-agent systems with connectivity. As a generalization of the well-known Hamiltonian walk problem (when $k=1$), $k$-HWP is NP-hard. We propose a $(3-\frac{1}{21})$-approximation algorithm for 2-HWP on arbitrary graphs. For the tree, we define a restricted version of the problem and present an optimal algorithm for arbitrary values of $k$. Finally, we formalize the problem for $k$-uniform hypergraphs and present a $2(1+\ln k)$-approximation algorithm. This result is also adapted to design an approximation algorithm for $k$-HWP on general graphs when $k = O(1)$.