We study the fair division of indivisible items. In the general model, the goal is to allocate $m$ indivisible items to $n$ agents while satisfying fairness criteria such as MMS, EF1, and EFX. We also study a recently-introduced graphical model that represents the fair division problem as a multigraph, in which vertices correspond to agents and edges to items. The graphical model stipulates that an item can have non-zero marginal utility to an agent only if its corresponding edge is incident to the agent's corresponding vertex. We study orientations (allocations that allocate each edge to an endpoint) in this model, as they are particularly desirable. Our first contribution concerns MMS allocations of mixed manna (i.e. a mixture of goods and chores) in the general model. It is known that MMS allocations of goods exist when $m \leq n+5$. We generalize this and show that when $m \leq n+5$, MMS allocations of mixed manna exist as long as $n \leq 3$, there is an agent whose MMS threshold is non-negative, or every item is a chore. Remarkably, our result leaves only the case where every agent has a negative MMS threshold unanswered. Our second contribution concerns EFX orientations of multigraphs of goods. We show that deciding whether EFX orientations exist for multigraphs is NP-complete, even for symmetric bi-valued multigraphs. Complementarily, we show symmetric bi-valued multigraphs that do not contain non-trivial odd multitrees have EFX orientations that can be found in polynomial time. Our third contribution concerns EF1 and EFX orientations of graphs and multigraphs of chores. We obtain polynomial-time algorithms for deciding whether such graphs have EF1 and EFX orientations, resolving a previous conjecture and showing a fundamental difference between goods and chores division. In addition, we show that the analogous problems for multigraphs are NP-hard.

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