Mobile agents on chordal graphs: Maximum independent set and beyond

Abstract

We consider the problem of finding Maximum Independent Set (MaxIS) for chordal graphs using mobile agents. Suppose n agents are initially placed arbitrarily on the nodes of an n-node chordal graph $G=(V,E)$. Agents need to find a Maximum Independent Set M of G such that each node of M is occupied by at least one agent. Also, each of the n agents must know whether its occupied node is a part of M or not. We provide distributed algorithms for n mobile agents, each having $O(\log n)$ memory, to compute MaxIS of G in $O(mn\log \Delta )$ time, where m denotes the number of edges in G, n denotes the number of nodes in G, and Δ is the maximum degree of the graph. At first, we design an algorithm considering the case where all agents are initially placed at the same node (i.e., rooted initial configuration). To run this algorithm, the agents do not require prior knowledge of any global parameter. Then we propose an algorithm for the case where agents are initially distributed arbitrarily across the graph (i.e., arbitrary initial configuration). To run this algorithm, the agents require prior knowledge of certain global parameters. Further, we provide faster algorithms for finding MaxIS in chordal graphs either by increasing the memory available to each agent or by employing more agents. We report that by using a similar approach, it is possible to find the maximum clique in chordal graphs and color any chordal graph with the minimum number of colors. We also provide a dynamic programming-based distributed algorithm to find a Maximum Independent Set for trees in $O\left(n\right)$ time.