To handle the remaining cases, particularly instances with only negative mixed agents, the techniques introduced in the paper can be extended or generalized in several ways:
Exploring Different Reduction Rules: One approach could be to develop new reduction rules that are specifically tailored to handle instances with negative mixed agents. These rules could be designed to address the unique challenges posed by negative utility values and the absence of singletons in MMS allocations for such agents.
Adapting Existing Techniques: The existing techniques used in the paper, such as mimicking instances and applying reduction rules, can be adapted to accommodate instances with negative mixed agents. By modifying these techniques to account for the specific characteristics of negative utility functions, it may be possible to find MMS allocations in these cases.
Considering Alternative Fairness Notions: In cases where MMS allocations are not feasible for instances with negative mixed agents, exploring alternative fairness notions or solution concepts in game theory and economics could provide insights into potential allocation mechanisms that satisfy different fairness criteria.
By extending and generalizing the techniques introduced in the paper, researchers and practitioners can work towards finding solutions for instances with negative mixed agents and further advance the understanding of fair division in mixed manna settings.