The Profile Minimization Problem in Trees

Abstract

The profile minimization problem is to find a one-to-one function $f$ from the vertex set $V(G)$ of a graph $G$ to the set of all positive integers such that $\sum_{x \in V(G)} \{f(x) - \min_{y \in N[x]} f(y)\}$ is as small as possible, where $N[x] = \{x\} \cup \{ y:y \mbox{ is adjacent to } x\}$ is the closed neighborhood of $x$ in $G$. This paper gives an $O(n^{1.722})$ time algorithm for the problem in a tree of $n$ vertices.