Abstract
Aggregating preferences over combinatorial domains has many applications in artificial intelligence (AI). Given the inherent exponential nature of preferences over combinatorial domains, compact representation languages are needed to represent them, and (m)CP-nets are among the most studied ones. Sequential and global voting are two different ways of aggregating preferences represented via CP-nets. In sequential voting, agents' preferences are aggregated feature-by-feature. For this reason, sequential voting may exhibit voting paradoxes, i.e., the possibility to select sub-optimal outcomes when preferences have specific feature dependencies. To avoid paradoxes in sequential voting, one has often assumed the (quite) restrictive constraint of O-legality, which imposes a shared common topological order among all the agents' CP-nets. On the contrary, in global voting, CP-nets are considered as a whole during the preference aggregation process. For this reason, global voting is immune from the voting paradoxes of sequential voting, and hence there is no need to impose restrictions over the CP-nets' structure when preferences are aggregated via global voting. Sequential voting over O-legal CP-nets received much attention, and O-legality of CP-nets has often been required in other studies. On the other hand, global voting over non-O-legal CP-nets has not carefully been analyzed, despite it was explicitly stated in the literature that a theoretical comparison between global and sequential voting was highly promising and a precise complexity analysis for global voting has been asked for multiple times. In quite a few works, only very partial results on the complexity of global voting over CP-nets have been given. In this paper, we start to fill this gap by carrying out a thorough computational complexity analysis of global voting tasks, for Pareto and majority voting, over not necessarily O-legal acyclic binary polynomially connected (m)CP-nets. We show that all these problems belong to various levels of the polynomial hierarchy, and some of them are even in P or LOGSPACE. Our results are a notable achievement, given that the previously known upper bound for most of these problems was the complexity class EXPTIME. We provide various exact complexity results showing tight lower bounds and matching upper bounds for problems that (up to now) did not have any explicit non-obvious lower bound.
Highlights
We focus on voting procedures based on comparisons of pairs of outcomes
The distinctive element of conditional preference nets (CP-nets) is that a directed graph, whose vertices represent the features of a combinatorial domain, is used to intuitively model the conditional part of conditional ceteris paribus preference statements
Given two disjoint sets of features V, Z ⊆ F, a conditional ceteris paribus preference statement sounds like: “Given the specific instantiation γ of values for the features in Z, outcomes varying over V, and all else being equal, are ranked according to the following preference relation restricted over V ”
Summary
The problem of managing and aggregating agent preferences has attracted extensive interest in the computer science community (see, e.g., the comprehensive survey by Brandt et al [21]), because methods for representing and reasoning about preferences are very important in artificial intelligence (AI) applications, such as recommender systems [70], (group) product configuration [15,30,77], (group) planning [14,72,73,76], (group) preference-based constraint satisfaction [9,12,18], and (group) preference-based query answering/information retrieval [10,28,63,64]. The robots have their own specific preferences over a vast amount of variables/features emerging from the contingency of the situation to complete their individual tasks, their individual preferences have to be blended together These examples show the great relevance of dealing with combinatorial votes, and the pressing necessity of finding ways to represent agent preferences over multi-issue domains and algorithms for aggregating them. We carry out a thorough complexity analysis for the (a) Pareto and (b) majority voting schemes, as defined by Rossi et al [71], of deciding (1) dominance, (2) optimal and (3) optimum outcomes, and (4) the existence of optimal and (5) optimum outcomes. We give only proof sketches in the body of the paper, while detailed proofs are provided in Appendix A
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