Lexico-minimum Replica Placement in Multitrees

In this work, we consider the problem of placing replicas in a data center or storage area network, represented as a digraph, so as to lexico-minimize a previously proposed reliability measure which minimizes the impact of all failure events in the model in decreasing order of severity. Prior work focuses on the special case in which the digraph is an arborescence. In this work, we consider the broader class of multitrees: digraphs in which the subgraph induced by vertices reachable from a fixed node forms a tree. We parameterize multitrees by their number of "roots" (nodes with in-degree zero), and rule out membership in the class of fixed-parameter tractable problems (FPT) by showing that finding optimal replica placements in multitrees with 3 roots is NP-hard. On the positive side, we show that the problem of finding optimal replica placements in the class of \emph{untangled} multitrees is FPT, as parameterized by the replication factor $\rho$ and the number of roots $k$. Our approach combines dynamic programming (DP) with a novel tree decomposition to find an optimal placement of $\rho$ replicas on the leaves of a multitree with $n$ nodes and $k$ roots in $O(n^2\rho^{2k+3})$ time.