Computational Intelligence and Neuroscience

Volume 2018, Article ID 2082875, 9 pages

https://doi.org/10.1155/2018/2082875

## Cutting Cycles of Conditional Preference Networks with Feedback Set Approach

^{1}Shandong University, Jinan, Shandong Province, China^{2}Yantai University, Yantai, Shandong Province, China

Correspondence should be addressed to Ke Li; nc.ude.uds@ekil

Received 5 February 2018; Revised 30 May 2018; Accepted 3 June 2018; Published 28 June 2018

Academic Editor: Silvia Conforto

Copyright © 2018 Zhaowei Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

As a tool of qualitative representation, conditional preference network (CP-net) has recently become a hot research topic in the field of artificial intelligence. The semantics of CP-nets does not restrict the generation of cycles, but the existence of the cycles would affect the property of CP-nets such as satisfaction and consistency. This paper attempts to use the feedback set problem theory including feedback vertex set (FVS) and feedback arc set (FAS) to cut cycles in CP-nets. Because of great time complexity of the problem in general, this paper defines a class of the parent vertices in a ring CP-nets firstly and then gives corresponding algorithm, respectively, based on FVS and FAS. Finally, the experiment shows that the running time and the expressive ability of the two methods are compared.

#### 1. Introduction

The evil of graph exists in cycles [1–3]. Several famous problems in computer science just like satisfiability, knapsack, and graph three-colorability problem are all related to cycles. With the awkward cycles, the above-mentioned questions are difficult to deal with.

Due to the importance of the problem, it has been extensively studied, although the problem was proven to be NP-complete for general graphs. Moreover, many graph problems are polynomially solvable if restricted to instances of acyclicity or even low cyclicity.

Generally, deleting cycles is considered as feedback set problem applied in many fields, such as circuit testing and deadlock resolution. Analyzing manufacturing processes and computational biology is used to delete cycles. Some different exact and approximate algorithms have been proposed incipiently based on Branch-Prune and linear programming. Measure-and-Conquer techniques and local search approaches have also been employed as usual method.

Feedback set [4, 5] includes feedback vertex set (FVS) and feedback arc set (FAS) or feedback edge set problems, which are classical NP problems. For different situations which can be undirected or directed graph, equal or unequal weighted graph, proper approaches have been proposed, but there is no uniform method in all cases.

A conditional preference network [1], abbreviated as a CP-net, is a simple and intuitive tool of graph model [6], which can represent preferences of agent, so do learning and aggregation and suits for describing qualitative multiattribute decision-making preference with dependencies. It can be converted into a weighted directed graph under usual conditions. Since it is a graph model, there always exist cycles. That will produce an effect on consistency of CP-nets where one single decision value does not appear more than one time in an arbitrary order sequence or satisfiability where there exists some preference dominant ranking for each decision value in the decision space. Moreover, the above-mentioned algorithms of FVS and FAS are not applicable because CP-nets are not general graph model [6, 7].

For cutting the cycles of CP-nets, two methods are presented based on feedback vertex set (FVS) and feedback arc set (FAS). The following are the main contributions of this paper.

(1) As a FAS problem, based on context of attribute priority, parent of vertex with cycles is defined with formalization firstly. Arcs (edges) of CP-nets are deleted by the relation; then an algorithm is presented [8, 9].

(2) As a FVS problem, based on the context of attribute relation, concept of relation reduction is given. Vertices of CP-nets are deleted by the through relation reduction; then another algorithm is proposed [10].

The rest of the paper is constructed as follows. In Section 2, some related works are presented. In Section 3, we present the main and basic definitions used throughout the paper. In Sections 4 and 5, FAS and FVS are proposed to deal with cycle of CP-nets, respectively. Section 6 presents the results of experiments. Finally, Section 7 summarizes the work and present studies in the future work.

#### 2. Related Work

In [11], Brafman and Domshlak tackle the complexity of determining whether one outcome is preferred to another outcome (dominance testing) which is known for tree-structured networks only; moreover, little is known about the consistency of cyclic CP-nets. In this paper they show how the complexity of dominance testing depends on the structure of the CP-net. In particular they provide a new polynomial time algorithm for polytrees. In addition, they show a class of cyclic CP-nets that is never consistent, while showing other classes on which consistency can be tested for efficiently.

The cyclic networks part of Domshlak in [1] proves that the consistency of cyclic CP-nets is not guaranteed and depends on the actual nature of the CPTs. This article holds that cyclic CP-nets usefulness requires further analysis. One can argue that it is possible to cluster the variables to preserve acyclicity. And it shows that further investigation of cyclic CP-nets, as well as a characterization of the different classes of utility functions that can be represented by cyclic and acyclic networks, remains of interest.

In [12], Liu et al. utilize treewidth which can decrease the solving complexity to solve some reasoning tasks on induced graphs, such as the dominance queries on the CP-nets in the future. And they present an efficient algorithm for computing the treewidth of induced graphs of CP-nets. It is revealed that by experiment the treewidth of induced graphs of CP-nets is much smaller with regard to the number of vertices.

#### 3. Preliminaries

In this section, some basic concepts of CP-nets are presented.

*Definition 1 (CP-net). *A conditional preference network (CP-net) is a graph model <*V*,* A*,* T*>, in which

(i) a set of variables makes up the** vertices** in the network,

(ii) a set of directed** arcs** connects pairs of vertices,

(iii) each vertex has a conditional preference** table** that qualifies the effects the parents have on the vertex.

The CP-net may be directed acyclic or directed cyclic graph [13]; i.e., it may exist with directed cycles, which is distinguished with classic Bayesian network [14, 15].

*Example 2 (auto configuration). *For a car configuration, we focus on two attributes which are* Cl* (Class) and* Co*(Color), where* Cl* has no parents and* Co*’s parent is* Cl*. Assume the following conditional preferences:

The table is sufficient to order all the outcomes completely:

The example can be described by the CP-net in Figure 1.