## 8 Conclusions  
In this paper, we proposed several models for novel resource allocation problems where agents express their preferences over conflicting bundles of items as edge-weighted DAGs (DPAP). We particularly focused on conflicts on vertices (V-DPAP) and conflicts on resources (R-DPAP). We introduced and analyzed several solution methods (utilitarian, leximin, approximate leximin, greedy) against the classically used round-robin allocations from the utilitarianism and fairness perspectives. We evaluated these methods on large randomly generated instances of orbit slot allocation problems, where requests could be fully or partially fulfilled. We showed that when requests must be fully fulfilled, allowing resource sharing via R-DPAP encoding improves the performance of the system compared to V-DPAP with respect to normalized utility and global reward, while the computation times are equivalent or lower. When considering the request full satisfaction mode, problems encoded as V-DPAP are much more constrained with respect to the number of agents that can receive a non-empty allocation. Therefore, algorithm util is a relevant approach. In the case of R-DPAP, algorithm a-lex provides good results with respect to utility and is much fairer than other approaches, even if it requires a longer computation time. In the case of partial request satisfaction mode and V-DPAP problems, there is no clear winner on all metrics for small constellations: lex clearly returns fair allocations with a good global utility but requires a long computation time. On the other hand, algorithm util is faster but not as fair. For large constellations, algorithm util allows us to reach the fairest allocations and is, therefore, the most suitable. Finally, when offering even more flexibility, i.e., allowing partial request fulfilling, the performances become even better, to a point where, for larger constellations, all the algorithms reach the same optimal and fair allocations. This highlights that adding request flexibility eases the allocation process, whilst the problems remain NP-hard in general. In such a case, non-exact algorithms such as greedy offer the best trade-off with respect to utility, fairness, and computation time. We identify several tracks for future investigations. First, as DPAPs are strongly constrained by conflicts, we aim to explore minimum conflict heuristics to improve our algorithms. Secondly, we believe DPAP and its variants have great potential to be used in a variety of domains, and we thus aim to evaluate the proposed techniques on problems coming from other application fields, such as the NFV domain (function chains modeled as graphs and incompatibilities controlling the access to nodes) or the multi-agent path finding domain (path preferences modeled as graphs and incompatibilities, imposing that two agents cannot occupy the same position at the same time). Depending on the targeted application, other ways for expressing conflicting bundles could be explored. For instance, one could consider that items can consume resources with capacity. Finally, in the Earth observation domain, once the slots have been allocated, the agents have to plan their own observations within the allocated slots, and may have to interact to accept external observations. Such a coordination scheme has been investigated [3], but we aim to evaluate the whole chain (slot allocation followed by coordinated observation scheduling) on realistic data. Conceptualization, S.R., G.P., C.P. and S.M. ; data curation, S.R., G.P., C.P. and S.M. ; formal analysis, S.R., G.P. and C.P. ; funding acquisition, S.R. and C.P. ; investigation, S.R., G.P. and S.M. ; methodology, S.R., G.P., C.P. and S.M. ; project administration, S.R., G.P. and C.P. ; resources, S.R., G.P., C.P. and S.M. ; software, S.R., G.P., C.P. and S.M. ; supervision, S.R., G.P. and S.M. ; validation, S.R., G.P., C.P. and S.M. ; visualization, S.R., G.P. and C.P. ; writing--original draft, S.R., G.P. and C.P. ; writing--review and editing, S.R., G.P., C.P. and S.M.