  Allocating indivisible items among a set of agents is a frequently studied
discrete optimization problem. In the setting considered in this work, the
agents' preferences over the items are assumed to be identical. We consider a
very recent measure for the overall quality of an allocation which does not
rely on numerical valuations of the items. Instead, it captures the agents'
opinion by a directed acyclic preference graph with vertices representing
items. An arc $(a,b)$ in such a graph means that the agents prefer item $a$
over item $b$. For a given allocation of items the dissatisfaction of an agent
is defined as the number of items which the agent does not receive and for
which no more preferred item is given to the agent. Our goal is to find an
efficient allocation of the items to the agents such that the total
dissatisfaction over all agents is minimized.
  We explore the dichotomy between NP-hard and polynomially solvable instances,
depending on properties of the underlying preference graph. While the problem
is NP-hard already for three agents even on very restricted graph classes, it
is polynomially solvable for two agents on general preference graphs. For an
arbitrary number of agents, we derive polynomial-time algorithms for relevant
restrictions of the underlying undirected graph. These are trees and, among the
graphs of treewidth two, series-parallel graphs and cactus graphs.
