Using hierarchies to efficiently combine evidence with Dempster's rule of combination

Dempster's rule of combination allows us to combine various independent pieces of evidence that each have a certain degree of uncertainty. This provides a useful way for dealing with uncertain evidence, but the rule is computationally intractable. In this paper, we analyze the complexity of this rule for differently structured bodies of evidence and we consider a known algorithm by Shafer and Logan to compute this rule efficiently over a hierarchical set of evidence. We show that one can check in polynomial time whether an arbitrary set of evidence has a hierarchical shape, enabling the use of Shafer and Logan's algorithm. Moreover, we consider two different approaches to deal with non-hierarchical sets of evidence: (i) considering hierarchical subsets and (ii) taking advantage of internal hierarchical structures in the overall set. For the former case, we conclude that getting different hierarchies from an arbitrary set of pieces of evidence corresponds to the VERTEX COVER problem and we present algorithms for obtaining these hierarchies based on this correspondence. For the latter case, we present a fixed-parameter tractable algorithm which computes the belief function of any piece of evidence included in the set.