Server Placements, Roman Domination and other Dominating Set Variants

Dominating sets in their many variations model a wealth of optimization problems like facility location or distributed le sharing. For instance, when a request can occur at any node in a graph and requires a server at that node, a minimumdominating set represents a minimum set of servers that serve an arbitrary single request by moving a server along at most one edge. This paper studies domination problems for two requests. For the problem of placing a minimum number of servers such that two requests at diierent nodes can be served with two diier-ent servers (called win-win), we present a logarithmic approximation, and we prove that nothing better is possible. We show that the same is true for Roman domination, the well studied problem variant that asks for each vertex to either possess its own server or to have a neighbor with two servers. Still the same is true if each idle server can move along one edge while the rst of both requests is being served. For planar graphs, we propose a PTAS for Roman domination (and show that nothing better exists), and we get a constant approximation for win-win.