The Weighted Sitting Closer to Friends than Enemies Problem in the Line

Abstract

The weighted \emph{Sitting Closer to Friends than Enemies} (SCFE) problem is to find an injection of the vertex set of a given weighted graph into a given metric space so that, for every pair of incident edges with different weight, the end vertices of the heavier edge are closer than the end vertices of the lighter edge. The \emph{Seriation} problem is to find a simultaneous reordering of the rows and columns of a symmetric matrix such that the entries are monotone nondecreasing in rows and columns when moving towards the diagonal. If such a reordering exists, it is called a \emph{Robinson} ordering. In this work, we establish a connection between the SCFE problem and the Seriation problem. We show that if the \emph{extended adjacency matrix} of a given weighted graph $G$ has no Robinson ordering then $G$ has no injection in $\mathbb{R}$ that solves the SCFE problem. On the other hand, if the extended adjacency matrix of $G$ has a Robinson ordering, we construct a polyhedron that is not empty if and only if there is an injection of the vertex set of $G$ in $\mathbb{R}$ that solves the SCFE problem. As a consequence of these results, we conclude that deciding the existence of (and constructing) such an injection in $\mathbb{R}$ for a given \emph{complete} weighted graph can be done in polynomial time. On the other hand, we show that deciding if an \emph{incomplete} weighted graph has such an injection in $\mathbb{R}$ is NP-Complete.