Complexity results for preference aggregation over (m)CP-nets: Pareto and majority voting

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 paradoxes, 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 has extensively been investigated, 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 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 date) did not have any explicit non-obvious lower bound.(a) A CP-net modeling dinner preferences.(b) The CP-net's extended preference graph. Figure 1: A CP-net and its preference graph.has a combinatorial structure [18,45,48]. By combinatorial structure, we mean that the set of candidates (or outcomes) is the Cartesian product of finite value domains for each of a set of features (also called variables, or issues, or attributes). The problem of aggregating agents' preferences over combinatorial domains (or multi-issue domains) is called a combinatorial vote [44,45].Interestingly, voting over combinatorial domains is rather common. For example, in 2012, on the day of the US presidential election, voters in California...

Complexity results for preference aggregation over (m)CP-nets: Max and rank voting

“…In this paper, we continue our thorough complexity investigation started in [45] by considering max and rank voting as defined by Rossi et al [53]. We expand our previous work and further explore the complexity of mCP-nets (and hence of global voting over CP-nets).…”

“…A precise complexity analysis of global voting was missing for a long time, as explicitly mentioned several times in the literature [35,38,39,41,42,58]. Only recently, a thorough complexity analysis of Pareto and majority global voting over CP-nets was carried out in [45].…”

“…Otherwise, voting paradoxes (i.e., the selection in the aggregation process of suboptimal outcomes) can happen. O-legality is a quite stringent constraint, which we can avoid to assume in this work, as it was also not assumed in [45], because we consider global voting. Our main goal in this paper is to carry out a precise complexity analysis of the dominance semantics in mCP-nets when the max and rank voting schemes [53] are considered.…”

A Knowledge Compilation Map for Conditional Preference Statements-based Languages

“…7, Corrolary 1] prove that , ≻ , ⊲⊳ and ∼ are PSPACE complete for 1-CP⋫∧ and for linearisable, locally complete formulas of 1-CP⋫. More precise hardness results for acyclic CP-nets are also proved by [28]. Proposition 12 completes the picture.…”

CP-Nets Structure Learning Based on mRMCR Principle

“…Compared with other selection, preference is given to choose more inclined items, which is popular among collaborative filtering, recommendation systems and product configurations. Currently, most applications of preferences are described based on the conditional preference network [3], such as voting based on CP-nets [4], [5], preference aggregation based on CP-nets [6], product recommendation based on CP-nets, etc. Preference processing combines acyclic graphs with probabilistic knowledge organically, and has obvious advantages in reasoning and dominating queries.…”