On the Complexity of Finding Justifications for Collective Decisions

In a collective decision-making process, having the possibility to provide non-expert agents with a justification for why a target outcome is a good compromise given their individual preferences, is an appealing idea. Such questions have recently been addressed in the computational social choice community at large---whether it was to explain the outcomes of a specific rule in voting theory or to seek transparency and accountability in multi-criteria decision making. Ultimately, the development of real-life applications based on these notions depends on their practical feasibility and on the scalability of the approach taken. In this paper, we provide computational complexity results that address the problem of finding and verifying justifications for collective decisions. In particular, we focus on the recent development of a general notion of justification for outcomes in voting theory. Such a justification consists of a step-by-step explanation, grounded in a normative basis, showing how the selection of the target outcome follows from the normative principles considered. We consider a language in which normative principles can be encoded---either as an explicit list of instances of the principles (by means of quantifier-free sentences), or in a succinct fashion (using quantifiers). We then analyse the computational complexity of identifying and checking justifications. For the case where the normative principles are given in the form of a list of instances, verifying the correctness of a justification is DP-complete and deciding on the existence of such a justification is complete for Sigma 2 P. For the case where the normative principles are given succinctly, deciding whether a justification is correct is in NEXP wedge coNEXP, and NEXP-hard, and deciding whether a justification exists is in EXP with access to an NP oracle and is NEXP-hard.