The problem of managing and aggregating agent preferences has attracted extensive interest in the computer science community , because methods for representing and reasoning about preferences are very important in artificial intelligence (AI) applications, such as recommender systems , (group) product configuration [24, 11, 70], (group) planning [10, 66, 65, 69], (group) preference-based constraint satisfaction [9, 5, 14], and (group) preference-based query answering/information retrieval [57, 56, 23, 6].
In computer science, the study of preference aggregation has often been based on the solid ground of social choice theory, which is the branch of economics analyzing methods for collective decision making [2, 3]. Having a well-founded theory and practice on how to properly and efficiently manage and aggregate preferences of real software agents, and hence support the growth and use of these technologies, has been one of the main drivers for investigating social choice theory from a computational perspective. In social choice theory, the actual ways of representing agent preferences are rarely taken into consideration, also because the sets of candidates usually considered are relatively small in size. For this reason, most of the insights obtained in the computational social choice literature about the computational properties of preference aggregation functions (or voting procedures) have assumed that agent preferences over the set of candidates are extensively listed (see  and references therein). Although this is perfectly reasonable when we reason about, e.g., (political) elections among a not too numerous set of human candidates, this is not feasible when the voting domain (i.e., the set of candidates) has a combinatorial structure [45, 48, 18]. 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 had to vote also for eleven referenda . As another example, it may be the case that the inhabitants of a town have to make a joint decision about different related issues regarding their community, which could be whether and where to build new public facilities (such as a swimming pool or a library), or whether to levy new taxes. Note that these voting scenarios are often also called multiple elections or multiple referenda [15, 78, 75, 47, 48, 18]. Other examples are group product configurations and group planning [70, 48]. As for the latter, consider, e.g., a situation in which multiple autonomous agents have to agree upon a shared plan of actions to reach a goal that is preferred by the group as a whole, such as a group of autonomous robots coordinating during the exploration of a remote area/planet. Each robot has a specific task to accomplish, and the group as a whole coordinates to achieve a common goal. That is, 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, however, their individual preferences have to be blended in all together, so that the course of action of a robotic agent does not interfere with the tasks of the other agents, and the overall mission is successful. These examples show the great relevance of dealing with combinatorial votes, and hence the pressing necessity of finding ways to represent agent preferences over multi-issue domains and algorithms for aggregating them.
Combinatorial domains contain an exponential number of outcomes in the number of features, and hence compact representations for combinatorial preferences are needed [45, 48] (see also Section 6.1 for more background). The graphical model of CP-nets  is among the most studied of these representations, as proven by a vast literature on them. In CP-nets, the vertices of a graph represent features, and an edge from vertex to vertex models the influence of the value of feature on the choice of the value of feature . Intuitively, this model captures preferences like “if the rest of the dinner is the same, with a fish dish (’s value), I prefer a white wine (’s value)”, also called conditional ceteris paribus preferences; a more detailed example is given below.
Assume that we want to model one’s preferences for a dinner with a main dish and a wine. In the CP-net in Figure 0(a), an edge from vertex to vertex models that the value of feature influences the choice of the value of feature . More precisely, and are the possible values of feature , and they denote “eat” and “ish”, respectively, while and are the possible values of feature , and they denote “ed (wine)” and “hite (wine)”, respectively. The table associated with feature specifies that when a meat dish is chosen, then a red wine is preferred to a white one, and when a fish main is chosen, then a white wine is preferred to a red wine. The table associated with feature indicates that a meat dish is preferred to a fish one. These tables are called CP tables. A CP-net like this one can represent the above conditional ceteris paribus preference “given that the rest of the dinner does not change, with a meat dish (’s value), I prefer a red wine (’s value)”.
Every CP-net has an associated extended preference graph, whose vertices are all the possible outcomes of the domain, and whose edges connect outcomes differing on only one value. More precisely, there is a directed edge from an outcome to another, if the latter is preferred to the former according to the preferences encoded in the tables of the CP-net. Figure 0(b) shows the extended preference graph of the CP-net in Figure 0(a), having as vertices all the possible combinations for the dinner, and there is, e.g., an edge from to , because the combination meat and red wine is preferred to the combination meat and white wine. The preferences encoded in a CP-net are the transitive closure of its extended preference graph. Intuitively, an outcome is preferred to an outcome according to the preferences of a CP-net, if there is a directed path from to in the extended preference graph.
CP-nets are also used to model preferences of groups of individuals, obtaining a multi-agent model, called CP-nets , which is a set, or profile, of CP-nets, one for each agent. The preference semantics of CP-nets is defined via voting schemes: through its own individual CP-net, every agent votes whether an outcome is preferred to another. Various voting schemes were proposed for CP-nets [64, 52], and different voting schemes give rise to different dominance semantics for CP-nets. In this paper, we consider Pareto and majority voting as they were defined in . In the voting schemes proposed for CP-nets, the voting protocol adopted, i.e., the actual way in which votes are collected , is global voting [46, 48]. In this protocol, the results of the voting procedure are computed by having as input the CP-nets as a whole (see Section 6.2 for related works on different voting protocols over CP-nets).
Consider again the dinner scenario, and assume that there are three agents (Alice, Bob, and Chuck), expressing their preferences via CP-nets (see Figure 2). In Pareto voting, an outcome dominates an outcome , if all agents prefers to . In majority voting, an outcome dominates an outcome , if the majority of agents prefers to .
The outcome is not Pareto optimal, because there is an outcome (namely ), which is preferred to by all the agents. The outcome , instead, is Pareto optimal, because there is no outcome Pareto dominating . Hence, from a Pareto perspective, is better than . The outcome , however, is not majority optimal, because majority dominates (Alice and Chuck prefer to ). On the other hand, is majority optimal, because there is no outcome majority dominating . Hence, from a majority perspective, is better than . Moreover, again according to the majority voting scheme, is a very good outcome, because is also majority optimum, which means that majority dominates all other outcomes. On the contrary, in this example, there is no Pareto optimum outcome, i.e., there is no outcome Pareto dominating all other outcomes.
In the literature, a comparison between sequential voting (which is another voting protocol; see Section 6.2) and global voting over CP-nets was explicitly asked for and stated to be highly promising . However, global voting over CP-nets has not been as thoroughly investigated as sequential voting. In fact, unlike CP-nets, which were extensively analyzed, a precise complexity analysis of CP-nets has been missing for a long time, as explicitly mentioned several times in the literature [46, 49, 50, 51, 52, 67]—since the dominance semantics for CP-nets is global voting over CP-nets, in the following, we use them interchangeably. Furthermore, it was conjectured that the complexity of computing majority optimal and majority optimum outcomes in CP-nets is harder than NP and co-NP [49, 51].
The aim of this paper is to explore the complexity of CP-nets (and hence of global voting over CP-nets). In particular, we focus on acyclic binary polynomially connected CP-nets (see Section 2 for these notions) built with standard CP-nets, i.e., the constituent CP-nets of an CP-net rank all the features, and they are not partial CP-nets (which instead were allowed in the original definition of CP-nets ). Unlike what is often assumed in the literature, in this work, we do not restrict the profiles of CP-nets to be -legal (which means that there is a topological order common to all the CP-nets of the profile; see Section 6.3). We carry out a thorough complexity analysis for the (a) Pareto and (b) majority voting schemes, as defined in , of deciding (1) dominance, (2) optimal and (3) optimum outcomes, and (4) the existence of optimal and (5) optimum outcomes. Deciding the dominance for a voting scheme means deciding, given two outcomes, whether one dominates the other according to . Deciding whether an outcome is optimal or optimum for a voting scheme means deciding whether the outcome is not dominated or dominates all others, respectively, according to . Deciding the existence of optimal and optimum outcomes is the natural extension of the previous problems.
A summary of the complexity results obtained in this paper is provided in Figure 3. More precisely, deciding dominance and optimal outcomes is complete for NP and co-NP, respectively, for both Pareto and majority voting, while deciding the existence of optimal outcomes can be done in constant time for Pareto voting and is complete for for majority voting. Furthermore, deciding optimum outcomes and their existence is in LOGSPACE and P for Pareto voting, and complete for and between and for majority voting, respectively.
It thus turns out that Pareto voting is the easiest voting scheme to evaluate among the two analyzed here. More precisely, both Pareto and majority dominance are NP-complete, however, only the complexity of majority dominance carries over to deciding optimal and optimum outcomes and their existence, and causes a substantial increase of their complexity, e.g., deciding the existence of majority optimal and optimum outcomes is hard for and , respectively. This is due to the fact that majority voting is structurally more complex than Pareto voting. Intuitively, Pareto voting is based on unanimity, hence, to disprove Pareto dominance between two outcomes, it suffices to find one agent that does not agree with the dominance relationship. This particular structure of Pareto voting makes the other tasks not more difficult than the dominance test or even tractable. Our results hence prove the conjecture posed in [49, 51] about majority voting tasks over ()CP-nets being harder than NP and co-NP.
We show completeness results for most cases, and we provide tight lower bounds for problems that (up to date) did not have any explicit lower bound transcending the obvious hardness due to the dominance test over the underlying CP-nets. Many of our results are intractability results, where the problems are put at various levels of the polynomial hierarchy. However, although intractability is usually “bad” news, these results are quite interesting, as for most of these tasks, only EXPTIME upper bounds were known in the literature to date . Even more interestingly, some of these problems are actually tractable, as they are in P or even LOGSPACE, which is a huge leap from EXPTIME.
Our hardness results are given for binary acyclic polynomially connected ()CP-nets. This means that our hardness results extend to classes of ()CP-nets encompassing the CP-nets considered here, and in particular also to general CP-nets with partial CP-nets or multi-valued features. More generally, the hardness results proven here extend to any representation scheme as “expressive and succinct” as the class of CP-nets used in the proofs (see Section 2.5). Moreover, the membership results above P that we prove here extend to any “NP-representation” scheme (see Section 2.5). Our hardness results on the existence of optimal and optimum outcomes provide also lower bounds for the computational problems. Indeed, actually computing optimal or optimum outcomes cannot be easier than the bounds shown here, because otherwise it would be possible to decide their existence more efficiently.
Organization of the paper
The rest of this paper is organized as follows. Section 2 provides some preliminaries. In Section 3, we prove some basic complexity results for CP-nets. Sections 5 and 4 analyze the complexity of Pareto and majority voting, respectively: first, we analyze the complexity of dominance testing; then, we study the complexity of deciding whether an outcome is optimal and whether there exists an optimal outcome; and we conclude by dealing with the complexity of deciding whether an outcome is optimum and whether there exists an optimum outcome. In Section 6, we discuss related works. Section 7 summarizes the main results and gives an outlook on future research. For several results, we give only proof sketches in the body of the paper, while detailed proofs are provided in Appendix A.
In this section, we give some preliminaries, briefly recalling from the literature preference relations and aggregation, conditional preference nets (CP-nets), CP-nets for groups of agents (CP-nets), and the complexity classes that we will encounter in our complexity results. We also define a formal framework for preference representation schemes, because our membership results will be given for generic representations whose dominance test is feasible in NP.
2.1 Preference relations and aggregation
Before dwelling upon the details of CP-nets, which is the specific preference representation analyzed in this paper, we now give an introductory overview of the general concepts of preferences and their aggregation.
In this paper, a preference relation over a set of outcomes is a strict order over , i.e., is a binary relation over that is irreflexive (i.e., ), asymmetric (i.e., if , then ), and transitive (i.e., if and , then ). A preference ranking is a preference relation that is total (i.e., either or for any two different outcomes and ). Usually, given two outcomes and , their preference relationship stated in is denoted by , instead of , which means that, in , is strictly preferred to , or dominates . On the other hand, means that , and means that and , i.e., and are incomparable in . Observe that in a preference ranking, it cannot be the case that two outcomes are incomparable. Given a preference relation , an outcome is optimal in if there is no outcome such that . We say that is optimum in , if for all outcomes such that , it holds that . Clearly, if there is an optimum outcome in , then it is unique. For notational convenience, if the preference relation is clear from the context, we do not explicitly mention as a subscript in the notations above. In the following, if not stated otherwise, when we speak of preferences structures, we mean preference relations.
In preference aggregation, we deal with preferences of multiple agents. A preference profile is a set of preference relations. We assume that all the preferences of are defined over the same set of outcomes, i.e., the agents express their preferences over the same set of candidates. In this paper, we focus on voting procedures based on comparisons of pairs of outcomes (see, e.g.,  for a classification of different kinds of preference aggregation procedures). For this reason, we need to define the following sets of agents. For a profile , we denote by , , and , the sets of agents preferring to , preferring to , and for which and are incomparable, respectively.
The voting schemes considered in this paper are Pareto and majority. The definition of their dominance semantics over preference profiles, reported below, is a generalization of the respective definition over CP-nets given in .
An outcome Pareto dominates an outcome , denoted , if all agents prefer to , i.e., .
An outcome majority dominates an outcome , denoted , if the majority of the agents prefer to , i.e., .
For a preference profile and a voting scheme , if outcome does not dominate outcome , we denote this by . An outcome is optimal in , if for all , it holds that , while is optimum in , if for all , it holds that . Note that optimum outcomes, if they exist, are unique.
We now focus on CP-nets, which is the preference representation that we will more closely investigate in this work. As mentioned in the introduction, the set of outcomes of a preference relation is often defined as the Cartesian product of finite value domains for each of a set of features. Conditional preference nets (CP-nets)  are a formalism to encode conditional ceteris paribus preferences over such combinatorial domains. The distinctive element of 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. Below, we recall the syntax, semantics, and some properties of CP-nets; see Section 6.1 for more on conditional ceteris paribus preferences and preference representations in general.
Syntax of CP-nets
A CP-net is a triple , where is a directed graph whose vertices represent the features of a combinatorial domain, and and are a function and a family of functions, respectively. The function associates a (value) domain with every feature , while the functions are the CP tables for every feature , which are defined below. The value domain of a feature is the set of all values that may assume in the possible outcomes. In this paper, we assume features to be binary, i.e., the domain of each feature contains exactly two values, usually denoted and , and called the overlined and the non-overlined value (of ), respectively. For a set of features , denotes the Cartesian product of the domains of the features in . Thus, an outcome is an element of . Given a feature and an outcome , we denote by the value of in , while, given a set of features , is the projection of over . For two outcomes and , and a set of features , we denote by that for all ; we write , when this is not the case, i.e., when there is at least one feature such that . The CP tables encode preferences over feature values. Intuitively, the CP table of a feature specifies how the values of the parent features of influence the preferences over the values of . More formally, for a feature , we denote by the set of all features in from which there is an edge to . We call the set of the parents of (in ). We denote by the set of all the (strict) preference rankings over the elements of . Each function maps every element of to a (strict) preference ranking over the domain of . If , then is a single (strict) preference ranking over . Note that indifferences between feature values are not admitted in (classical) CP-nets. Each function is represented via a two-column table, in which, given a row, the element in the first column is the input value of the function , and the element in the second column is the associated (strict) preference ranking over . Since is total, in the table representing the function, there is a row for any combination of values of the parent features, i.e., for a feature , there are rows in the table of .
In the following, when we define CP tables, we often use a logical notation to identify for which specific values of the parent features, a particular row in the CP table has to be considered. Although this is the notation on which generalized propositional CP-nets  are based on, it is used here only for notational convenience. In this paper, we always assume that CP tables are explicitly represented in the input instances. In the CP tables, denotes being preferred to . We denote by the size of CP-net , i.e., the space in terms of bits required to represent the whole net (which includes features, edges, feature domains, and CP tables).
Semantics of CP-nets
The preference semantics of CP-nets can be defined in several different but equivalent ways . A first definition has a model-theoretic flavour [8, Definitions 2 and 3]. Intuitively, a preference ranking violates a CP-net , if there are two outcomes and that according to the CP tables of should be ranked , but they are not ranked in such a way in (i.e., , since is total). Formally, a preference ranking violates a CP-net , if there are two distinct outcomes and a feature such that (i.e., and differ only on the value of ), in the order , and . A preference ranking satisfies a CP-net , if does not violate . Given two outcomes and , a CP-net entails the preference , denoted , if for every preference ranking over that satisfies . The preference semantics of CP-nets can be equivalently defined via the concept of improving (or alternatively worsening) flip [8, Definition 4]: let be a feature, and let be an outcome. Intuitively, flipping the value of in from to a different one is an improving flip, if the new value of is preferred, given the values in of the parent features of . More formally, flipping from to a different value is an improving flip, if holds in . Given two outcomes and differing only on the value of a feature , there is an improving flip from to , denoted , if flipping the value of from to is an improving flip. In the following, we often omit the feature and simply write ; and when we say that we flip a feature, then we often mean that the flipping is improving. The (extended) preference graph of is the pair , where the nodes are all the possible outcomes of , and, given two outcomes , the directed edge from to belongs to if and only if .
It can be shown that, for a CP-net and two outcomes and , if and only if there is a sequence of improving flips from to [8, Theorems 7 and 8]. Therefore, for an agent whose preferences are encoded through a CP-net , we say that the agent prefers to , or that dominates (in ), denoted , if entails , or, equivalently, if there is an improving flipping sequence from to . If for two outcomes and , neither nor , then and are incomparable (in ), denoted (which is equivalent to the existence of preference rankings and that both satisfy such that and ).
Note here that, since there are no indifferences between features values in (classical) CP-nets, for any two outcomes and , either one dominates the other, or they are incomparable.
Consider the CP-net shown in Figure 4. For the outcomes and , it holds that , because . For the outcomes and , it holds that , because there is no path from to in . However, , because , and hence it is not the case that . Consider now the outcomes and . Then, by the improving flipping sequence .
Properties of CP-nets
A CP-net is binary, if all its features are binary. The indegree of a CP-net is the maximum number of edges entering in a node of the graph of . A CP-net is singly connected, if, for any two distinct features and , there is at most one path from to in . A class of CP-nets is polynomially connected, if there exists a polynomial such that, for any CP-net and for any two features and of , there are at most distinct paths from to in . A CP-net is acyclic, if is acyclic. It is well known that acyclic CP-nets always have a preference ranking satisfying , their extended preference graph is acyclic, the preferences encoded by are consistent (i.e., there is no outcome such that ), and there is a unique optimum outcome dominating all other outcomes (and, clearly, not dominated by any other), which can be computed in polynomial time .
It is known that dominance testing, i.e., deciding, for any two given outcomes and , whether , is feasible in NP over polynomially connected classes of binary acyclic CP-nets . However, it is an open problem whether dominance testing is feasible in NP over non-polynomially-connected classes of binary acyclic CP-nets. Also, the complexity of dominance testing for non-binary CP-nets is currently still open. Whereas dominance testing for the class of acyclic binary singly connected CP-nets whose indegree is at most six is NP-hard —we improve this result in Section 3, requiring only indegree three. Dominance testing is feasible in polynomial time on acyclic binary CP-nets whose graph is a tree or a polytree , and it is PSPACE-complete for cyclic CP-nets .
In the rest of this paper, we consider only binary acyclic (and often polynomially connected) classes of CP-nets. When the CP-net is clear from the context, we often omit the subscript “” from the notations introduced above.
In this section, we focus on CP-nets , which are a formalism to reason about conditional ceteris paribus preferences when a group of multiple agents is considered. Intuitively, an CP-net is a profile of (individual) CP-nets, one for each agent of the group. The original definition of CP-nets also allows for partial CP-nets. Here, we consider only CP-nets consisting of a collection of standard CP-nets. The difference is that we do not allow for non-ranked features in agents’ CP-nets, and hence there is no distinction between private, shared, and visible features (see  for definitions), i.e., all features are ranked in all the individual CP-nets of an CP-net.
As underlined in , the “” of an CP-net stands for multiple agents and also indicates that the preferences of agents are modeled, so a CP-net is an CP-net with . Formally, an CP-net consists of CP-nets , all of them defined over the same set of features, which, in turn, have the same domains. If is an CP-net, we denote by the set of all features of , and by the domain of feature in . Given this notation, , for all , and , for all features and all . Although the features of the individual CP-nets are the same, their graphical structures may be different, i.e., the edges between the features in the various individual CP-nets may vary. We underline here that, unlike in other papers in the literature, we do not impose that the individual CP-nets of the agents share a common topological order (i.e., we do not restrict the profiles of CP-nets to be -legal); see Section 6 for more on -legality.
An outcome for an CP-net is an assignment to all the features of the CP-nets, and given an CP-net , we denote by the set of all the outcomes in . The preference semantics of CP-nets is defined through global voting over CP-nets. In particular, via its own individual CP-net, each agent votes whether an outcome dominates another, and hence different ways of collecting votes (i.e., different voting schemes) give rise to different group dominance semantics for an CP-net. Let be an CP-net, and let and be two outcomes. With a notation similar to the one defined above, , , and are the sets of the agents of preferring to , preferring to , and for which and are incomparable, respectively.
In [64, 52], various voting schemes were proposed and analyzed to define multi-agent dominance semantics for CP-nets. In this paper, we focus on two of them, namely, Pareto and majority voting, whose dominance semantics definitions are the natural specializations to CP-nets of Pareto and majority dominance semantics defined above. Consider an CP-net , and let and be two outcomes. Then:
Pareto dominates , denoted , if all the agents of prefer to , i.e., .
majority dominates , denoted , if the majority of the agents of prefers to , i.e.,
For a voting scheme , optimal and optimum outcomes in CP-nets are defined in the natural way.
An CP-net is acyclic, binary, and singly connected, if all its CP-nets are acyclic, binary, and singly connected, respectively. A class of CP-nets is polynomially connected, if the set of CP-nets constituting the CP-nets in is a polynomially connected class of CP-nets. The indegree of an CP-net is the maximum indegree of its constituent individual CP-nets. Unless stated otherwise, we consider only polynomially connected classes of acyclic binary CP-nets. When the CP-net is clear from the context, we often omit the subscript “” from the above notations.
2.4 Computational complexity
We now give some notions from computational complexity theory, which will be required for the complexity analysis carried out in this paper. First, we briefly recall the complexity classes that we will encounter in this paper (along with some closely related ones), and then we recall the notion of polynomial-time reductions among decision problems, and some decision problems that are hard for some of these complexity classes. We assume that the reader has some elementary background in computational complexity theory, including the notions of Boolean formulas and quantified Boolean formulas, Turing machines, and hardness and completeness of a problem for a complexity class, as can be found, e.g., in[40, 60].
The class P is the set of all decision problems that can be solved by a deterministic Turing machine in polynomial time with respect to the input size, i.e., with respect to the length of the string that encodes the input instance. For a given input string , its size is usually denoted by . The class of decision problems that can be solved by nondeterministic Turing machines in polynomial time is denoted by NP. They enjoy a remarkable property: any “yes”-instance has a certificate for being a “yes”-instance, which has polynomial length and can be checked in deterministic polynomial time (in ). For example, deciding whether a Boolean formula over the Boolean variables is satisfiable, i.e., whether there exists some truth assignment to these variables making true, is a well-known problem in NP; in fact, any satisfying truth assignment for is clearly a certificate that is a “yes”-instance, i.e., that is satisfiable.
For a complexity class , we denote by co- the complementary class to , i.e., the class containing the complementary languages of those in . For example, the problem of deciding whether a Boolean formula is not satisfiable is in co-NP. The class P is contained in both NP and co-NP, i.e., .
By LOGSPACE, we denote the set of decision problems that can be solved by deterministic Turing machines in logarithmic space. For such machines, it is assumed that the input tape is read-only, and that these machine have a read/write tape, called work tape, for intermediate computations. The logarithmic space bound is given on the space available on the work tape. The class LOGSPACE is contained in P.
The class , defined originally in , is the class of problems that are a “conjunction” of two problems, one from NP and one from co-NP, i.e., . The class co- is the class of problems whose complements are in , equivalently, it can be defined as the class of problems that are a “disjunction” of two problems, one from NP and one from co-NP, i.e., .
The classes , , and , forming the polynomial hierarchy (PH) , are defined as follows: , and, for all , , , and . Here, (resp., ) is the set of decision problems solvable by nondeterministic (resp., deterministic) polynomial-time Turing machines with an oracle to recognize, at unit cost, a language in . Note that , , and . Sometimes a bound is imposed on the number of calls that are allowed to be issued to the oracle. For example, denotes the set of decision problems solvable by a deterministic polynomial-time Turing machine that is allowed to query a oracle at most logarithmically many times (in the size of the input). By definition, .222For the complexity class , an interesting characterization has recently been provided: is the class of languages involving the counting and comparison of the number of “yes”-instances in two sets containing instances of or languages . This is quite useful for reductions in voting settings where votes have to be counted and compared.
The classes and co- can be generalized to the classes and , respectively, for , that are the conjunction and the disjunction, respectively, of and ; in particular, . Note also that .
Reductions and hard problems
A decision problem is (Karp) reducible to a decision problem , denoted , if there is a computable function , called (Karp) reduction, such that, for every string , is defined, and is a “yes”-instance of if and only if is a “yes”-instance of . A decision problem is polynomially (Karp) reducible to a decision problem , denoted , if there is a polynomial-time (Karp) reduction from to . In this paper, we consider only Karp reductions.
To prove hardness for a complexity class, we show reductions from various problems known to be complete for the complexity classes that they belong to. We next define such problems, so that we can later refer to them by name.
Deciding the satisfiability of Boolean formulas, denoted Sat, is the prototypical NP-complete problem, which remains NP-hard even if only CNF formulas are considered [26, 41], i.e., Boolean formulas in conjunctive normal form with three literals per clause. The complementary problem Unsat of deciding whether a given Boolean formula is not satisfiable is co-NP-complete. It remains co-NP-hard even if only CNF formulas are considered, and it is the equivalent to the problem Taut of deciding whether a DNF formula is a tautology. A DNF formula is a Boolean formula in disjunctive normal form with three literals per term. CNFs and DNFs are actually linked (see [28, 29]).
The prototypical - and -complete problems are defined as follows: given a quantified Boolean formula (QBF) , where
is a sequence of alternating quantifiers , and
is a (non-quantified) Boolean formula over disjoint sets of Boolean variables,
decide whether is valid. The problem is -complete [71, 74], while is -complete [71, 74]. These problems remain hard for their respective classes even if is in CNF, when , and if is in DNF, when [71, 74]. We denote by QBF (resp., QBF)333Note the difference in the subscripts of the notations and QBF (resp., QBF). In the former notation, is the first quantifier of the sequence, and, for notational convenience, we place “” before “” in the subscript. On the other hand, in the latter notation, is the last quantifier of the sequence, and, for notational convenience, we place “” after “” in the subscript. the problem of deciding the validity of formulas , where is (resp., ), and is in CNF (resp.,
DNF). For odd, QBF (resp., QBF) is complete for (resp., ), while, for even , QBF (resp., QBF) is complete for (resp., ). Observe that (resp., ) is equivalent to Sat (resp., Taut).
Sometimes, it is preferable that in QBF formulas the non-quantified formula is CNF rather than DNF, or vice-versa. For example, to show -hardness it might be the case that we would prefer to start our reduction from formulas with being in CNF, rather than in DNF, as required by QBF. To achieve this, we can exploit De Morgan’s laws. Indeed, we have that is logically equivalent to , where . We thus extend the notation above. We denote by QBF (resp., QBF) the problem of deciding the validity of formulas , where is (resp., ), and is in DNF (resp., CNF). For odd , QBF (resp., QBF) is complete for (resp., ), while, for even , QBF (resp., QBF) is complete for (resp., ).
2.5 A framework for preference representation schemes
In this paper, most of the membership results that we will show hold for generic preference representation schemes. For this reason, we now introduce the general framework of representation schemes that we will refer to. Inspired by the concept of compact representations in [34, 36, 48, 18], we define preference representation schemes as suitable encodings for a class of preference relations, denoted . Formally, a preference representation scheme defines a computable representation function and a computable Boolean function such that, for any relation , is the encoding of according to , and evaluates to , if (i.e., if ), and to , otherwise. By , we denote the size of the representation of via .
Let and be two preference representation schemes. We say that is at least as expressive (and succinct) as , denoted , if there exists a function in FP (i.e., computable in deterministic polynomial time) that translates a preference relation represented in into an equivalent preference relation represented in , i.e., into a preference relation over the same outcomes and with the same preference relationships between them. More precisely, we require that and , for each pair of outcomes and . Observe that belonging to FP entails that there exists a constant (depending on ) such that , i.e., the size of is polynomially bounded in the size of .444Note that the above definitions are slightly different from the ones in : the counterpart of this paper’s function in  is not required to be computable, and the transformation function in  is only required to be polynomially bounded, but not polynomially computable.
A P- and an NP-representation is a preference representation scheme whose function is in P and NP, respectively. For example, the polynomially connected classes of acyclic binary CP-nets are NP-representation schemes.
When a compact representation is clear from the context, we often simply write instead of , and and instead of and , respectively. While doing so, we are identifying the preference relation with its actual representation.
3 Complexity of basic tasks on CP-nets
To precisely characterize the complexity of Pareto and majority voting tasks in Sections 5 and 4, we need to understand how complex is deciding, given a CP-net and two outcomes and , whether dominates or whether and are incomparable. Here, we prove that the former problem is NP-complete, while the latter is co-NP-complete. To achieve this, after giving some preliminary definitions on how to encode Boolean formulas into CP-nets, we show that deciding the satisfiability of Boolean formula can be reduced to the problem of deciding dominance between outcomes in CP-nets. This allows us to prove the NP-hardness of the dominance test in CP-nets, and the co-NP-hardness of deciding incomparability is shown as a byproduct of this property.
We first introduce a notation mapping Boolean assignments to outcomes of CP-nets; this notation will frequently be used later in the paper. In particular, to prove the hardness of voting tasks on CP-nets, we often provide reductions from problems regarding the satisfiability (or validity) of (quantified) Boolean formulas. For this reason, CP-nets will often have sets of features associated with sets of Boolean variables. For example, for a Boolean formula over the set of Boolean variables , we often define an CP-net that has as a subset of its set of features. Then, for a (partial or complete) assignment over , an outcome of encoding over the features set is such that, for the features in , if , then ; if , then ; and if is undefined, then . The values of the features in will be specified in each particular case.
We next define formula nets, which will be used in hardness proofs, and which are intuitively CP-nets aiming at having a particular preference relationship between two outcomes depending on the satisfiability of associated Boolean formulas in CNF. This will allow us to show that deciding dominance in CP-nets is NP-hard.555The NP-hardness of dominance in CP-nets was proven already in . Here, we show this result, because the construction proposed here, which allows us to prove a stricter result, is different from the one available in the literature, and it is required in multiple reductions in the rest of the paper.
Formally, let be a Boolean formula in CNF defined over the set of Boolean variables , and whose set of clauses is . We often omit the variable set from the notation of the Boolean formula, i.e., we write instead of , if this does not cause ambiguity. We denote by the -th literal of the -th clause. From , we build the CP-net in the following way (see Figure 6 for an example).
The features of are:
for each variable , there are features and (called variable features), and we denote by the set of variable features;
for each clause , there is a feature (called clause feature), and we denote by the set of clause features; and
for each literal , there is a feature (called literal feature), and we denote by the set of literal features.
All features are binary, with the usual notation for their values. When the formula is clear from the context, we often omit the subscript “” from the notation of the sets of features illustrated above. The edges of are: for each literal or , there are edges , , and . The CP tables of are:
for each variable , features and have the CP tables
and , respectively;
for each literal , if , then feature has the CP table
otherwise (i.e., ) has the CP table
for each clause , feature has the CP table
Note that is binary, acyclic, singly connected, its indegree is three, and the CP-net can be built in polynomial time in the size of .
Let be a Boolean formula in CNF defined over a set of Boolean variables, and let be an assignment on . Let be the outcome of encoding on the feature set , and assigning non-overlined values to all other features, and let be the outcome assigning overlined values to all and only variable and clause features. Then:
There is an extension of to satisfying if and only if ;
There is no extension of to satisfying if and only if .
The idea at the base of this proof is that the CP tables in are designed so that the features enact the role of variables, literals, and clauses of a CNF Boolean formula. Details of the proof are at page A.1. ∎
Let be a Boolean formula in CNF defined over a set of Boolean variables, and let and be two outcomes of assigning non-overlined values to all features and overlined values to all and only variable and clause features, respectively. Then:
is satisfiable if and only if ;
is unsatisfiable if and only if .
3.2 Complexity of dominance, incomparability, and optimality on CP-nets
As mentioned above, via formula nets, it is possible to show the NP-hardness and the co-NP-hardness of dominance and incomparability on CP-nets, respectively. We start by showing the NP-hardness of dominance on CP-nets. More formally, consider the following problem on CP-nets.
|Instance:||A CP-net , and two outcomes|