The Guarding Problem --- Complexity and Approximation

• Authors: T. V. Reddy;D. Sai Krishna;C. Pandu Rangan
• Venue: Combinatorial Algorithms
• Year: 2009

Let G = (V, E) be the given graph and G R = (V R ,E R ) and G C = (V C ,E C ) be the sub graphs of G such that V R 驴 V C = 驴 and V R 驴 V C = V. G C is referred to as the cops region and G R is called as the robber region. Initially a robber is placed at some vertex of V R and the cops are placed at some vertices of V C . The robber and cops may move from their current vertices to one of their neighbours. While a cop can move only within the cops region, the robber may move to any neighbour. The robber and cops move alternatively. A vertex v 驴 V C is said to be attacked if the current turn is the robber's turn, the robber is at vertex u where u 驴 V R , (u,v) 驴 E and no cop is present at v. The guarding problem is to find the minimum number of cops required to guard the graph G C from the robber's attack. We first prove that the decision version of this problem when G R is an arbitrary undirected graph is PSPACE-hard. We also prove that the complexity of the decision version of the guarding problem when G R is a wheel graph is NP-hard. We then present approximation algorithms if G R is a star graph, a clique and a wheel graph with approximation ratios H(n 1), 2 H(n 1) and $\left( H(n_{1}) + \frac{3}{2} \right)$ respectively, where $H(n_{1}) = 1 + \frac{1}{2} + ... + \frac{1}{n_{1}}$ and n 1 = 驴 V R 驴.