This paper addresses the problem of finding EFX orientations of graphs of chores, in which each vertex corresponds to an agent, each edge corresponds to a chore, and a chore has zero marginal utility to an agent if its corresponding edge is not incident to the vertex corresponding to the agent. Recently, Zhou~et~al.~(IJCAI,~2024) analyzed the complexity of deciding whether graphs containing a mixture of goods and chores admit EFX orientations, and conjectured that deciding whether graphs containing only chores admit EFX orientations is NP-complete. In this paper, we resolve this conjecture by exhibiting a polynomial-time algorithm that finds an EFX orientation of a graph containing only chores if one exists, even if the graph contains self-loops. Remarkably, our first result demonstrates a surprising separation between the case of goods and the case of chores, because deciding whether graphs containing only goods admit EFX orientations of goods was shown to be NP-complete by Christodoulou et al.~(EC,~2023). In addition, we show the analogous decision problem for multigraphs to be NP-complete.

We study the problem of finding fair and efficient allocations of a set of indivisible items to a set of agents, where each item may be a good (positively valued) for some agents and a bad (negatively valued) for others, i.e., a mixed manna. As fairness notions, we consider arguably the strongest possible relaxations of envy-freeness and proportionality, namely envy-free up to any item (EFX and EFX$_0$), and proportional up to the maximin good or any bad (PropMX and PropMX$_0$). Our efficiency notion is Pareto-optimality (PO). We study two types of instances: (i) Separable, where the item set can be partitioned into goods and bads, and (ii) Restricted mixed goods (RMG), where for each item $j$, every agent has either a non-positive value for $j$, or values $j$ at the same $v_j>0$. We obtain polynomial-time algorithms for the following: (i) Separable instances: PropMX$_0$ allocation. (ii) RMG instances: Let pure bads be the set of items that everyone values negatively. - PropMX allocation for general pure bads. - EFX+PropMX allocation for identically-ordered pure bads. - EFX+PropMX+PO allocation for identical pure bads. Finally, if the RMG instances are further restricted to binary mixed goods where all the $v_j$'s are the same, we strengthen the results to guarantee EFX$_0$ and PropMX$_0$ respectively.

We consider a fair division setting where indivisible items are allocated to agents. Each agent in the setting has strictly negative, zero or strictly positive utility for each item. We, thus, make a distinction between items that are good for some agents and bad for other agents (i.e. mixed), good for everyone (i.e. goods) or bad for everyone (i.e. bads). For this model, we study axiomatic concepts of allocations such as jealousy-freeness up to one item, envy-freeness up to one item and Pareto-optimality. We obtain many new possibility and impossibility results in regard to combinations of these properties. We also investigate new computational tasks related to such combinations. Thus, we advance the state-of-the-art in fair division of mixed manna.