Efficient Heuristic Approach to Dominance Testing in CP-nets

Transcription

1 Efficient Heuritic Approach to Dominance Teting in CP-net Minyi Li Swinburne Univerity of Technology Quoc Bao Vo Swinburne Univerity of Technology Ryzard Kowalczyk Swinburne Univerity of Technology ABSTRACT CP-net (Conditional Preference Network) i one of the extenively tudied language for repreenting and reaoning with preference. The fundamental operation of dominance teting in CP-net, i.e. determining whether an outcome i preferred to another, i very important in many real-world application. Current technique for olving general dominance querie i to earch for improving flipping equence from one outcome to another a a proof of the dominance relation in all ranking atifying the given CP-net. However, it i generally a hard problem even for binary-valued, acyclic CPnet and tractable earch algorithm exit only for pecific problem clae. Hence, there i a need for efficient algorithm and technique for dominance teting in more general problem etting. In thi paper, we propoe a heuritic approach, called DT*, to dominance teting in arbitrary acyclic multi-valued CP-net. Our propoed approach guide the earch proce efficiently and allow ignificant reduction of earch effort without impacting oundne or completene of the earch proce. We preent reult of experiment that demontrate the computational efficiency and feaibility of our approach to dominance teting. Categorie and Subject Decriptor I.2.11 [Ditributed Artificial Intelligence]: Multiagent Sytem General Term Algorithm, Deign Keyword CP-net; Dominance Teting; Heuritic 1. INTRODUCTION The problem of eliciting, repreenting and reaoning with qualitative preference over multi-attribute domain arie in many field uch a planning, deign, and collective deciion making [5, 6, 7, 8]. A the number of alternative outcome of uch domain i exponentially large in the number of attribute, it i unpractical to expre preference explicitly by giving out the ordering over the alternative outcome pace. Therefore, the AI reearch community ha developed language for repreenting preference in uch domain in a uccinct way, exploiting tructural propertie uch a conditional preferential independence. The formalim of CP-net Cite a: Efficient Heuritic Approach to Dominance Teting in CP-net, Minyi Li, Quoc Bao Vo and Ryzard Kowalczyk, Proc. of th Int. Conf. on Autonomou Agent and Multiagent Sytem (AAMAS 2011), Tumer, Yolum, Sonenberg and Stone (ed.), May, 2 6, 2011, Taipei, Taiwan, pp Copyright c 2011, International Foundation for Autonomou Agent and Multiagent Sytem ( All right reerved. (Conditional Preference Network) [3] i among the mot popular one, ince it preerve a good readability that i imilar to the way uer expre their preference in natural language. CP-net provide a compact repreentation of preference ordering in term of natural preference tatement under a ceteri paribu (all ele being equal) interpretation. Ceteri paribu emantic induce a graph, known a induced preference graph [2, 3]; and an outcome α i aid to dominate another outcome β if there exit a directed path, alo called a equence of improving flip, coniting of ucceively improving outcome in the graph from β to α [1, 3]. Unfortunately, reaoning about the preference ordering (dominance relation) expreed by a CP-net i far from eay [3, 5]. With the exception of pecial cae uch a CP-net with tree or polytree tructured conditional dependencie, dominance teting ha been hown to be PSPACE-complete even with binary domain and acyclic dependence [5]. Some general pruning rule have been tudied in [3] to reduce the earch effort. But they might not be able to guide the earch efficiently when the number of variable i large or the tructure of the CP-net i complex. Another work propoed by Santhanam et al. [9] explore an approach to dominance teting with acyclic CP-net via Model Checking. However, their approach mainly applie to binary-valued conditional preference tatement. The complexity and feaibility of their approach to dominance teting in multi-valued CP-net i till an open quetion. Hence, there i a need for efficient algorithm and technique for dominance teting in more general problem etting. To thi end, we addre the problem of dominance teting by propoing an efficient heuritic algorithm, called DT*, to guide the earch proce for improving flipping equence from the wore outcome to the better outcome of the given query 1. The propoed approach can be applied to arbitrary acyclic multi-valued CP-net. It ue a numerical approximation of the given CP-net and conider the hamming ditance between the currently conidered outcome and the target outcome of the given query, i.e., the number of variable that the two outcome differ from each other. We how that our propoed approach efficiently guide the earch proce for improving flipping equence. It allow ignificant reduction of earch effort without impacting oundne or completene of the earch proce. Moreover, when there are no flipping equence poible, it return the quick failure for the dominance query without having to earch all poible branche. We experimentally evaluate the propoed algorithm in different tructure etting, including tree-tructured CP-net, directed-path ingly connected CP-net and arbitrary acyclic CP-net, and with different domain ize from binary to multi-valued. The experimental reult preented in thi 1 The propoed heuritic will be decribed in the context of improving flipping equence, but it can be applied to worening earch according to the ame principle 353

2 paper demontrate that the propoed approach i computationally efficient. It allow dominance querie for CP-net that are quite large and complex to be anwered in reaonable time. The remainder of thi paper i organized a follow. Section 2 retate the neceary background on CP-net and dicue ome exiting pruning technique for the earch proce in dominance teting. Second 3 introduce the propoed approach in technical detail and Section 4 preent the experimental reult. Finally, Section 5 dicue the concluding remark and outline ome direction for future reearch. 2. PRELIMINARIES 2.1 CP-net Let V = {X 1,..., X n} be a et of n variable, for each X V, D (X) i the value domain of X. A variable X i binary if D (X) = {x, x}. If X = { X i1,..., X ip } V, with i1 < < i p then D (X) denote D (X i1 ) D ( X ip ) and x denote an aignment of variable value to X (x D (X)). If X = V, x i a complete aignment; otherwie x i called a partial aignment. For any aignment x D (X), we denote by x [X] the value x D (X) (X X) aigned to variable X by that aignment; and x [W] denote the aignment of variable value w D (W) aigned to the et of variable W X by that aignment. If x and y are aignment to dijoint et X and Y, repectively (X Y = ), we denote the combination of x and y by xy. Let X, Y, and Z be nonempty et that partition V and a preference relation over D (V), X i (conditionally) preferentially independent of Y given Z iff for all x, x D (X), y, y D (Y), z D (Z): xyz x yz iff xy z x y z A CP-net N [3] over V i an annotated directed graph G over X 1,..., X n, in which node tand for the problem variable. Each node X i annotated with a conditional preference table (CPT), denoted by CP T (X), which aociate a total order X u with each intantiation u of X parent P a (X), i.e. u D (P a (X)). For intance, let V = {X 1, X 2, X 3}, all three being binary, and aume that the preference of an agent can be defined by a CP-net whoe tructural part i the directed acyclic graph G = {(X 1, X 2), (X 1, X 3), (X 2, X 3)}; thi mean that the agent preference over the value of X 1 i unconditional, preference over the value of X 2 (rep. X 3) i fully determined given the value of X 1 (rep. the value of X 1 and X 2). The preference tatement contained in the conditional preference table are written with the uual notation, that i, x 1 x 2 : x 3 x 3 mean that when X 1 = x 1 and X 2 = x 2 then X 3 = x 3 i preferred to X 3 = x 3. Figure 1 illutrate an example of CP-net. 2.2 Dominance Teting One of the mot fundamental querie in any preference repreentation formalim i whether ome outcome α dominate (i.e., i trictly preferred to) ome other outcome β, called Dominance Teting. A dicued in [3, 9], uch dominance querie in CP-net are required whenever we wih to generate more than one nondominated olution to a et of hard contrain. In thi paper, we aume the tructure of the CP-net i acyclic, i.e. doe not contain any dependency cycle. In uch cae, two outcome α and β can tand in one of three poible relation with repect to N : either N = α β (α i trictly preferred to β); or N = β α (β i trictly preferred to α); or N = α β (α and β are incomparable: N = α β and N = β α). The third cae mean that the given CP-net N doe not contain enough Figure 1: An example CP-net N information to prove that either outcome i preferred to the other. Given an acyclic CP-net, comparion between two outcome that differ in the value of a ingle variable are eay: we only need to check the CPT of that variable and determine which outcome aign it to a more preferred value. The better (improved) outcome can be conidered a a product of a ingle improving flip in the value of a variable X from the wore outcome. For any pair of outcome that differ on more than one variable, an outcome α i aid to dominate another outcome β with repect to an acyclic CPnet N (N = α β) if there exit a equence of improving flip from β to α. Otherwie, N = α β. The following definition of improving flipping equence i introduced in [3]. DEFINITION 1 (IMPROVING FLIPPING SEQUENCE). A equence of outcome β = γ 1, γ 2,..., γ m 1, γ m = α uch that β = γ 1 γ 2 γ m 1 γ m = α i an improving flipping equence with repect to an acyclic CP-net N if and only if, 1 i m, outcome γ i i different from the outcome γ i+1 in the value of exactly one variable X, and γ i+1[x] γ i[x] given the parent context u of X aigned by γ i and γ i+1. For intance, conider the preference tatement over two binary variable X 1 and X 2, x 1 x 1, x 2 x 2, the equence x 1 x 2, x 1x 2, x 1x 2 i an improving flipping equence from the outcome x 1 x 2 to the bet outcome x 1x Some General Search Technique Given an acyclic CP-net N, a query N = α β can be treated a a earch for an improving flipping equence from the le preferred outcome β to the more preferred outcome α. The earch proce can be implemented a an improving earch tree rooted at β, T (β). The children of every node 2 γ in T (β) are thoe outcome that can be reached by a ingle improving flip from γ. Conequently, every rooted path in T (β) correpond to ome improving flipping equence from the outcome β with repect to N. Taking different direction in T (β) lead to different improving equence; however, taking a different direction during the tree traveral may alo lead to a dead end, i.e., reach the optimal outcome of N without viiting the target outcome α of the query. Recent work have tudied the computational complexity of teting dominance relation in CP-net, e.g. [3, 5]. The reult how that dominance teting in general CP-net i PSPACE-complete and it remain PSPACE-complete even though the CP-net i acyclic [5]. Since the hardne of dominance teting, everal earch technique for dominance querie have been tudied in [3] in order to reduce the earch effort. Suffix Fixing. Let X i1 > > X in be an arbitrary topological ordering conitent with the CP-net N, an rth (r 1) 2 A node in the improving earch tree i alo an outcome. 354

3 uffix of an outcome α i the ubet of the outcome value α [Xi r] α [ X ir+1 ]... α [Xin ]. The rth uffix of outcome α and β match iff r j n, α [ X ij ] = β [ Xij ]. For a query N = α β, uffix fixing rule out the exploration of any poible flipping equence that detroy of the uffix of the currently conidered outcome that matche the target outcome α. It prune the ubtree that improve the value of a variable within the matching uffix. For intance, conider the CP-net N in Figure 1 and the query N = x 1 x 2x 3x 4x 5 x 1x 2x 3x 4 x 5. Let α = x 1 x 2x 3x 4x 5 and β = x 1x 2x 3x 4 x 5, if pruned uing uffix fixing and conider the variable ordering X 1 > X 5 > X 2 > X 3 > X 4, the 2 nd uffix x 3x 4 of α and β matche. Thu, the value of X 3 and X 4 will never be improved in the earch tree T (β), although given the aignment β[x 1] = x 1 (rep. β[x 2] = x 2, β[x 3] = x 3), x 3 x 3 (rep. x 4 x 4). A hown in [3], any complete earch algorithm for the improving earch tree remain complete if pruning uing uffix fixing i ued. Leat-variable flipping. For every node γ in the improving earch tree, leat-variable flipping rule retrict flip to the variable that are leat-improvable. Formally, a variable X i leatimprovable in an outcome γ with repect to N if there i ome value x D (X) uch that x u γ [X] (where u = γ[p a(x)] i the parent context aigned by γ), and no decendent of X in γ ha thi property. For a query N = α β, leat-variable flipping rule retrict attention to thoe variable that are not part of any matching uffix with the target outcome α and require that the only neighbour of a node γ can be expanded in the earch tree T (β) are thoe in which ome leat improvable variable with repect to γ i improved. However, leat-variable flipping rule i only complete for a retricted cla of CP-net [3], i.e. tree-tructured CP-net and binary-valued, directed-path ingly connected CP-net. For multiply-connected network, and network with multivalued variable, it doe not guarantee completene. That mean, leat-variable flipping may fail to find any improving equence from β to α although there doe exit at leat one. In uch cae, it doe not provide a correct anwer to the given query. For intance, conider the CP-net N in Figure 1 and the query N = x 1x 2x 3x 4x 5 x 1 x 2 x 3 x 4 x 5. 3 Starting with the root node β = x 1 x 2 x 3 x 4 x 5, the only leat improvable variable that can be flipped i X 2. Unfortunately, flipping X 2 to value x 2 lead to outcome x 1x 2 x 3 x 4 x 5, from which the target outcome α = x 1x 2x 3x 4x 5 i unreachable. All branche in the improving earch tree grow toward the optimal outcome x 1 x 2x 3x 4x 5 without going through the target outcome α of the query. Figure 2 how the complete improving earch tree T (β) uing leat-variable flipping. However, there in fact exit a equence of improving flip from β to α: x 1 x 2 x 3 x 4 x 5, x 1 x 2 x 3 x 4 x 5, x 1 x 2x 3 x 4 x 5, x 1 x 2x 3x 4 x 5, x 1x 2x 3x 4 x 5, x 1x 2x 3x 4x 5. When the number of variable i large or the tructure of the CPnet i complex, uffix fixing may not be able to guide the earch efficiently while leat-variable flipping rule doe not guarantee completene for general acyclic CP-net. To thi end, we will preent another efficient heuritic approach to dominance teting. The propoed approach ignificantly prune the earch tree without impacting oundne or completene of the earch proce. 3 Thi example ha alo been dicued in Example 7 in [3] Figure 2: Improving earch tree for query N = x 1x 2x 3x 4x 5 x 1 x 2 x 3 x 4 x 5 uing Leat-variable flipping rule 3. HEURISTIC FOR DOMINANCE TESTING In thi ection, we preent our propoed heuritic approach, called DT*, to dominance teting in arbitrary acyclic CP-net. In broad term, we firt define a penalty function baed on a numerical approximation propoed by Domhlak et al. [4] that approximate acyclic CP-net uing weighted oft contraint. Then, an evaluation function i defined baed on the hamming ditance between the currently conidered outcome and the target outcome and their penaltie a a heuritic to guide the earch proce. 3.1 Penalty function For a variable X, let D (X) be the domain ize of X and thu there are D (X) degree of penaltie of X, denoted by d 1,..., d D(X). Without lo of generality, we aume the degree of penaltie of a variable X range between 0 and D (X) 1; that i, d 1 = 0,..., d D(X) = D (X) 1. For intance, conider the agent CP-net in Figure 1, ince all variable are binary, there are only two degree of penaltie, i.e., d 1 = 0 and d 2 = 1 for each variable. For a variable X, conider a preference ordering over the value of X given an intantiation of X parent, let the rank of the mot preferred value of X be 0 and the rank of the leat preferred valued of X be D (X) 1, given an outcome γ, the degree of penalty of a variable X in γ i then the rank of the value γ[x] in the preference ordering over X given the parent context u = γ[p a(x)]. We denote by d γ X (dγ X { d 1,..., d D(X) } ) the degree of penalty of X with repect to γ. For intance, conider a variable X uch that D(X) = {x, x, x }. Aume that, under a parent context u = γ[p a(x)] aigned by an outcome γ, x x x. If γ[x] = x, then d γ X = d1 = 0; if γ[x] = x, then d γ X = d2 = 1; if γ[x] = x, then d γ X = d3 = 2. CP-net impoe a rich tructure to allow variable to have different degree of importance: variable higher-up in the tructure of the network are conidered to be more important than the lower level variable [1, 2, 3]. Thu, it i more important to obtain a preferred value for a variable than any of it decendent. We now analye the importance weight of a variable in a CP-net. Given an acyclic CP-net N and conider an improving flip from an outcome γ to another outcome γ that flip the value of a ingle variable X, changing the value of X may alo affect the preference tatu of X children. Thu, the reulting change from γ to γ include: (i) the degree of penalty of X decreae from d γ X to dγ X (dγ X > dγ X ); 355

4 Algorithm 1: agweightcp(n ) Input: N, an acyclic CP-net 1 Order variable of N in a revere topological ordering; 2 foreach X N do 3 if Ch (X) = then 4 w X 1; 5 ele w X wy ( D (Y ) 1); 7 end 8 end and (ii) the degree of penalty of X children change, which in the wort cae, reult in the degree of penalty of each chilren Y increaing from d γ Y = d1 to dγ Y = d D(Y ) 1. Conequently, in order to preerve the preference ordering induced by the given CP-net, the importance weight of a variable in that CP-net mut be larger than the um of the maximum penaltie of it children. We now provide the formal definition of the variable importance weight in an acyclic CP-net. DEFINITION 2 (IMPORTANCE WEIGHT). Given an acyclic CPnet N over a et of variable V. For each variable X V, let Ch (X) denote the et of children of X in N, the importance weight of variable X, denoted by w X, i recurively defined by: w X = 1 + w Y ( D (Y ) 1) (1) Algorithm 1 provide a imple implementation to compute importance weight of variable. It take linear time in the ize of the network. Following a revere topological ordering, it firt aign the importance weight to the variable that have no decendent (line 3 4) and then iteratively aign the importance weight to the upper level variable according to Equation (1). Note that there are everal way to aign importance weight to the variable and the way we ue here i different from [4]. In thi paper, we conider the tight lower bound of the importance weight aignment, i.e. the um of maximum penaltie of the variable children wy ( D (Y ) 1). EXAMPLE. Conider an agent CP-net over a et of 5 variable V = {X 1,..., X 5} in Figure 1. In thi example, ince all variable are binary, i.e. X V, D (X) = 2.We can aign the importance weight to each variable in a revere topological ordering of variable: w X4 = 1; w X2 = 1 + w X4 (2 1) = 2; w X3 = 1 + w X4 (2 1) = 2; w X1 = 1 + (w X2 (2 1) + w X3 (2 1)) = 5; w X5 = 1 + w X2 (2 1) = 3. The importance weight of each variable in thi CP-net i attached on top of the variable repectively in Figure 3. Given an acyclic CP-net N and an outcome γ, the penalty of a variable X in γ i the degree of penalty of X in γ multiplied by the importance weight of X. The penalty of γ i then defined by the um of penaltie of the domain variable. We define the following penalty function for an acyclic CP-net baed on the work by Domhlak et al. [4]. DEFINITION 3 (PENALTY FUNCTION). Given an acyclic CPnet N over a et of variable V and an outcome γ. The penalty function pen, mapping from an outcome γ O to [0, + ], i defined a follow: γ O, pen (γ) = w X d γ X (2) X V Figure 3: Variable importance weight in N EXAMPLE (CONT.) Conider our running example in Figure 1 and the outcome γ = x 1 x 2 x 3 x 4 x 5. A the agent unconditionally prefer X 1 = x 1 to X 1 = x 1 (rep. X 5 = x 5 to X 5 = x 5), d γ X 1 = 1 (rep. d γ X 5 = 1). On the other hand, x 2 x 2 (rep. x 3 x 3, x 4 x 4) given the parent context X 1 = x 1 and X 5 = x 5 (rep. X 1 = x 1, X 2 = x 2 and X 3 = x 3) and thu d γ X 2 = 1 (rep. d γ X 3 = 0, d γ X 4 = 0). Conequently, the penalty of outcome x 1 x 2 x 3 x 4 x 5 i: pen (γ) = w X1 1 + w X2 1 + w X3 0 + w X4 0 + w X5 1 = =. In order to compute the penalty of an outcome, we imply need to weep through the network from top to bottom (i.e., from ancetor to decendant), and to check the degree of penalty of the currently conidered variable given it parent context. And finally we compute the penalty of the outcome baed on Equation (2). Conequently, the penalty computation for a particular outcome take polynomial time in the ize of the network. We now prove that our algorithm for aigning penaltie over alternative outcome preerve the trict preference ordering induced by the original CP-net. THEOREM 1. Given an acyclic CP-net N, we have: α, β O, if N = α β then pen (β) > pen (α) PROOF. N = α β if and only if there exit a equence of improving flip from β to α, denoted by Seq (β, α) = γ 1 (= β), γ 2,..., γ m 1, γ m (= α), with repect to the conditional preference table in N. Each improving flip from γ i to γ i+1 in Seq (β, α) that improve the value of a ingle variable X, pen (γ i) pen (γ i+1) = w X (d γ i X ) dγ i+1 X +σ, where σ w (Y ) ( D (Y ) 1) and ( d γ i X ) dγ i+1 X 1. Thu, pen (γi) pen (γ i+1) w X w (Y ) ( D (Y ) 1) = wx (wx 1) = 1 > 0. Conequently, with each improving flip from γ i to γ i+1, pen (γ i) > pen (γ i+1). Following from the tranitivity: pen (γ 1 (= β)) > pen (γ 2) > > pen (γ m 1) > pen (γ m) (= α) and thu pen (β) > pen (α). COROLLARY 1. Given an acyclic CP-net N, α, β O, if pen (β) > pen (α) then N = α β or N = α β if pen (β) = pen (α) then N = α β LEMMA 1. Given an acyclic CP-net N over a et of variable V,let α, β be any pair of outcome that N = α β; IS the et of all poible improving flipping equence from β to α with repect to the CPT in N ; HD(β, α) the hamming ditance between β 356

5 and α (Note that both in binary-valued and multi-valued CP-net, the hamming ditance i defined by the number of variable that the two outcome differ from each other.); Seq(β, α) IS an improving flipping equence from β to α; Seq(β, α) i the length of Seq(β, α) and thu the number of improving flip from β in thi equence i Seq(β, α) 1, then, HD(β, α) Seq(β, α) 1 pen(β) pen(α) PROOF. A each improving flip flip the value of a ingle variable, if N = α β, there mut be at leat HD(β, α) flip that flip the value of each variable X that β and α differ, from β[x] to α[x]. Thu, Seq(β, α) IS, Seq(β, α) 1 HD(β, α). On the other hand, any improving flip from γ i to γ i+1 in Seq(β, α) that flip the value of a ingle variable X, pen(γ i) pen(γ i+1) 1 (ee the proof of Theorem 1). Thu, pen(γ i) pen(γ i+1) 1. Aume γ i an outcome in Seq(β, α) that improved from β by t flip, pen(β) pen(γ) t and pen(β) t pen(γ). Thu, pen(β) pen(α) t pen(γ) pen(α). If t > pen(β) pen(α), then pen(β) pen(α) t < 0 and pen(γ) pen(α) < 0. According to Corollary 1, N = γ α or N = γ α (N = α γ), contradicting the fact that γ i in the improving equence Seq(β, α). Hence, the number of improving flip from β in Seq(β, α) can not be greater than pen(β) pen(α), Seq(α, β) 1 pen(β) pen(α). 3.2 The propoed DT* algorithm The penalty function mentioned above provide an order-preerving numerical approximation for a given CP-net. We alo how the upper bound and lower bound of the number of improving flip from a wore outcome to a preferred outcome in Lemma 1. In thi ection, thee reult are ued a a heuritic in the earch proce for improving flipping equence. The propoed algorithm ha a number of deirable propertie: it often return the quick failure for the dominance query if no flipping equence i poible; it often quickly how that back-tracking i needed when there i no poible flipping equence to the target outcome following the currently conidered path; and, it efficiently guide the earch direction without compromiing oundne or completene of the earch proce. Given an acyclic CP-net N and a pair of outcome α and β, for the query N = α β, we build the earch tree T (β) and earch for an improving flipping equence to the target outcome α a dicued in [3]. We introduce the evaluation function f for the heuritic earch trategy a follow: DEFINITION 4 (EVALUATION FUNCTION). Given an acyclic CP-net N and the query N = α β (α, β O). The evaluation function f, mapping from a node (i.e., an outcome) γ in the improving earch tree T (β) to [0, + ], i defined by: f(γ) = pen (γ) HD(γ, α) pen (α) (3) Our propoed heuritic algorithm DT* (ee Algorithm 2) i adapted from the A* heuritic earch algorithm with f (γ) being the evaluation function. It maintain a priority queue of node to be expanded, known a the fringe. On the one hand, the lower f value for a node γ, the higher it priority i. On the other hand, we only conider the outcome that the f value i non-negative. That mean, Algorithm 2: DT*(N = α β) Input: a dominance query (an acyclic CP-net N ; a pair of outcome α and β; and determining whether N = α β) Output: T rue: N = α β; F ale: N = α β 1 if f(β) < 0 then 2 return F ale; 3 ele 4 fringe INSERT(MAKE-NODE(β), fringe); 5 while fringe do 6 γ REMOVE-FIRST(fringe); 7 if GOAL-TEST(γ = α) then 8 return T rue; 9 ele foreach X N do 11 if IMPROVABLE(γ, X) &&X ANY-MATCHING-SUFFIX(γ, α) then 12 γ SINGLE-FLIP(γ, X); 13 if NOT-REPEATED(γ ) && f(γ ) 0 then 14 INSERT-ASC(MAKE-NODE(γ ), fringe) end 16 end 17 end 18 end 19 end 20 return F ale 21 end an outcome γ will be added into the fringe only if f(γ) > 0. In eence, an outcome with a negative f value mean that there i no poible improving flipping equence from that outcome to the target outcome α (ee Lemma 2). Before adding the original node β into the fringe, the f value of β will be computed and the algorithm will return F ale if f(β) < 0 (line 1 2). In thi cae, the query fail (N = α β) even before building the root node of the improving earch tree. Otherwie, β will be added into the fringe a the root node of the improving earch tree T (β) (line 4). At each iteration of DT*, the firt node γ, i.e. the node with the lowet f value, i removed from the fringe and being expanded (line 4). The children of a node in T (β) are thoe outcome that can be reached by a ingle improving flip from that node. Our propoed algorithm applie uffix fixing rule, retricting attention to thoe variable in γ that are not part of any matching uffix with the target outcome α (line 11). Moreover, it require that a child γ of a node be added into the fringe if and only if: (i) γ ha not been travered before; and (ii) f(γ ) 0 (line 13). For the current node γ under conideration, we add each child γ of γ that meet the above requirement into the fringe in acendant order of the f value of the node in the fringe (line 14). DT* continue until: the currently conidered node for expanion equal to the target outcome α, then it end and return T rue (N = α β) (line 7 8); or the fringe i empty, it return F ale (N = α β) (line 20). In order to prove the completene of our propoed heuritic algorithm, we firt proof the follow lemma. LEMMA 2. Given an acyclic CP-net N and a query N = α β (α, β O), γ O, if f(γ ) < 0, then γ would not be part of any poible improving flipping equence from β to α. 357

6 Figure 4: Improving earch tree PROOF. During the execution of DT* algorithm, for any outcome γ (including β), f(γ ) = pen(γ ) HD(γ, α) pen(α). Aume that there exit an improving flipping equence Seq(γ, α) = γ 1(= γ ), γ 2,..., γ m 1, γ m(= α) from γ to the target outcome α. Baed on Lemma 1, we know that there mut be at leat HD(γ, α) flip improved from γ. For any improving flip from γ i to γ i+1, pen(γ i) pen(γ i+1) 1. Conequently, for any outcome γ that improved from γ by HD(γ, α) flip, pen(γ ) pen(γ ) HD(γ, α) and thu pen(γ ) HD(γ, α) pen(γ ). Hence, pen(γ ) HD(γ, α) pen(α) pen(γ ) pen(α). Becaue f(γ ) < 0, pen(γ ) pen(α) pen(γ ) HD(γ, α) pen(α) < 0. Conequently, pen(γ ) pen(α) < 0 and N = α γ. γ will not be part of any poible improving flipping equence to α, contradicting the fact that there exit an improving flipping equence Seq(γ, α) from γ to the target outcome α. We now prove the completene of our propoed heuritic algorithm. THEOREM 2. DT* i complete for any arbitrary acyclic CPnet. PROOF. DT* travere the tree earching all neighbour; it follow lowet evaluated value path and keep a orted priority queue of alternate path egment along the way. If at any point the path being followed ha a higher evaluated value than other encountered path egment, the higher evaluated value path i kept in the fringe and the proce i continued at the lower value ub-path. Thi continue until the currently conidered node for expanion i the target outcome or the fringe i empty. During the execution of DT* algorithm, there are three kind of node will be pruned: (i) the outcome that have been explored previouly; (ii) the outcome that improve the value of the variable that i part of ome matching uffix with the target outcome; and (iii) the outcome with negative f value. Obviouly, checking repeated node doe not affect the completene of the algorithm. Alo, a hown in [3], any complete earch algorithm for the improving earch tree remain complete if pruning uing uffix fixing rule i ued. Furthermore, we have already proved in Lemma 2 that an outcome γ with f(γ ) < 0 will not be part of any poible improving equence from β to α, o pruning the third kind of outcome alo doe not affect the completene of the algorithm. Conequently, DT* i complete for any acyclic CP-net. EXAMPLE (CONT.) We now demontrate the execution of DT* algorithm with the CP-net in our running example (Figure 1) and conider the query N = x 1x 2x 3x 4x 5 x 1 x 2 x 3 x 4 x 5. Let α = x 1x 2x 3x 4x 5 and β = x 1 x 2 x 3 x 4 x 5, we firt conider the f value of the le preferred outcome β of the query. A f(β) = pen(β) HD(β, α) pen(α) = 5 3 = 2 > 0, we build the earch tree T (β) with β being the root node and add β into the fringe. In the 1 th iteration of DT*, γ = x 1 x 2 x 3 x 4 x 5 i removed from the fringe to be expanded. There are three improvable variable from γ : X 1, X 2 and X 5. Hence, there are three children node: x 1 x 2 x 3 x 4 x 5, x 1x 2 x 3 x 4 x 5 and x 1 x 2 x 3 x 4x 5. The f value of thee three children node are computed accordingly. f(x 1 x 2 x 3 x 4 x 5) = 0, f( x 1x 2 x 3 x 4 x 5) = 1 and f( x 1 x 2 x 3 x 4x 5) = 0. A none of the f value of thee three children node i negative, all of them are added into the fringe according to the acendant order of the f value. In the 2 nd iteration, the firt outcome γ = x 1 x 2 x 3 x 4 x 5 with the lowet f value i removed from the fringe (Aume that the node with the ame f value will be travered in the order from left to right). There are three poible children node of γ : x 1x 2 x 3 x 4 x 5, x 1 x 2x 3 x 4 x 5 and x 1 x 2 x 3 x 4x 5. A f(x 1x 2 x 3 x 4 x 5) = = 1 < 0; f(x 1 x 2x 3 x 4 x 5) = = 0; and f(x 1 x 2 x 3 x 4x 5) = = 4 < 0. There i only one outcome x 1 x 2x 3 x 4 x 5 will be added into the fringe. In the 3 rd iteration, we continue with the outcome γ = x 1 x 2x 3 x 4 x 5. There are three poible outcome can be reached by a ingle flip from γ : x 1x 2x 3 x 4 x 5, x 1 x 2x 3x 4 x 5 and x 1 x 2x 3 x 4x 5. We compute the f value of thee three outcome: f(x 1x 2x 3 x 4 x 5) = = 2 < 0; f(x 1 x 2x 3x 4 x 5) = = 0; and f(x 1 x 2x 3 x 4x 5) = = 4 < 0. Only one outcome x 1 x 2x 3x 4 x 5 can be added into the fringe. Similarly, in the 4 th iteration, we explore the outcome γ =x 1 x 2x 3x 4 x 5 and add only one outcome x 1x 2x 3x 4 x 5 into the fringe. In the 5 th iteration, we explore the outcome γ = x 1x 2x 3x 4 x 5. In eence, there are two variable can be improved from γ : X 4 and X 5. However, a X 4 i in the 3 rd matching uffix with the target outcome α (uing the topological order X 1 > X 5 > X 2 > X 3 > X 4), we only conider flipping the value of X 5. And thi tep produce the target outcome α, which will be explored in the lat iteration and the algorithm return T rue to thi query. Note that a we have dicued in Section 2.3, an algorithm baed on Leat-variable flipping rule i incomplete in thi cae. 4. EXPERIMENT We now decribe the reult of experiment that how the feaibility of our approach to dominance teting with repect to (i) the average number of viited node during the earch proce; (ii) the number of variable var and the domain ize dz that can be efficiently handled in practice; and (iii) the tructure of CP-net. We compare the performance of the propoed DT* algorithm with (i) a tandard depth-firt earch algorithm that applie uffix fixing during the earch, called DF; and (ii) an algorithm uing Leat-variable flipping rule, called LVF. We generate random preference network by varying the number of variable, the tructure of the network and the preference of the variable. For directed-path ingly connected CP-net and arbitrary acyclic CP-net, we retrict the maximum in-degree of each node in the generated CP-net to. For multivalued CP-net, we retrict the maximum domain ize dz to 5. We conduct the following ix et of experiment. At each et of experiment, we generate 00 CP-net randomly and uing each reulting preference network, we evaluate 5 dominance querie by picking ditinct pair of outcome at random. 358

7 Node DF LVF DT Node Variable DF LVF DT Figure 5: Avg. number of viited node with binary-value treetructured CP-net Figure 7: Avg. number of viited node with binary-valued, directed-path ingly connected CP-net Node (a) Avg. number of viited node Node Variable (a) Avg. number of viited node Percentage Percentage (b) LVF Incompletene (b) LVF Incompletene Variable DF LVF DT DF LVF DT Figure 6: Multi-valued tree-tructured CP-net Figure 8: Multi-valued polytree CP-net Set 1: binary-valued tree-tructured CP-net. We vary the number of variable var from 2 to 30 and only generate treetructured dependence. From Figure 5 we can oberve that on average, the number of viited node by both DT* and LVF algorithm are much le than DF algorithm. Note that for binary-valued tree-tructured CP-net, LVF (Leat-variable flipping rule) i guaranteed to be complete and backtrackfree. 4 However, on average, DT* i more efficient than the LVF algorithm for dominance teting in tree-tructured CPnet. The average execution time of DT* approach with 30 variable i le than 0.03 econd. It offer more than three order of magnitude improvement in performance over the DF algorithm. Set 2: multi-valued tree-tructured CP-net. We vary var from 2 to. The reult of multi-valued tree-tructured CP-net (ee Figure 6(a)) i imilar to the et of experiment with binary-valued tree-tructured CP-net. However, LVF algorithm doe not guarantee completene in multi-valued CPnet. Figure 6(b) how the percentage of cae in which the LVF algorithm i incomplete, i.e., it give an incorrect 4 The author can alo refer to [3] Page 161, TreeDT algorithm for binary-valued, tree-tructured CP-net anwer to the query. In general, the percentage of incompletene of LVF algorithm i increaing a the number of variable increae. When there are variable, in more than 28% cae that LFV algorithm fail to find the improving flipping equence for the given query although there doe exit at leat one. On the other hand, according to the experiment data, DT* complete the earch proce in about 12 econd on average in the cae of variable. Set 3: binary-valued, directed-path ingly connected CP-net. In thi et of experiment, the number of variable var i from 2 to 25. Note that LVF algorithm guarantee completene in binary-valued, directed-path ingly connected CP-net while it may require back-tracking during the earch. The average number of viited node in thi et of experiment i hown in Figure 7. Both LVF and DT* algorithm are much more efficient than the DF algorithm. When there are 25 variable, the average execution time of DT* i about 5.7 econd, which i more than two order of magnitude le than the DF algorithm. Set 4: multi-valued, directed-path ingly connected CP-net. We vary var from 2 to 12. Figure 8(a) how that the average number of viited node of both LVF and DT* algorithm are much le than DF algorithm. Although the reult how that 359

8 Node Node (a) Avg. number of viited node (a) Avg. number of viited node Percentage Percentage (b) LVF Incompletene (b) LVF Incompletene DF LVF DT DF LVF DT Figure 9: Binary-value arbitrary acyclic CP-net when the number of variable i large, the LVF algorithm may viit le node than DT* algorithm, the percentage of incompletene of LVF i on the other hand, increaing a the number of variable increae (ee Figure 8(b)). When there are 12 variable, thi percentage i more than 25%. According to the experimental data, with 12 variable and each variable with the maximum domain ize of 5, the average execution time of DT* approach i till le than 50 econd. Set 5: binary-valued arbitrary acyclic CP-net. We vary var from 2 to 20. Similar to the reult preented in Set 4, when the number of variable i large (more than ) the average number of viited node of DT* algorithm i more than that of LVF algorithm (ee Figure 9(a)). However, for binaryvalued CP-net in general, LVF doe not guarantee completene and the percentage of cae that the LVF algorithm return incorrect anwer i increaing a the number of variable increae (Figure 9(b)). When there are 20 variable, thi percentage i more than 20%. While on average, DT* algorithm return a correct anwer to the given query in about 20 econd. Set 6: multi-valued arbitrary acyclic CP-net. In the lat et of experiment, we vary var from 2 to. The reult with arbitrary acyclic CP-net in multi-valued etting i imilar to that in binary-valued etting (ee Figure (a) and Figure (b)). When there are variable, the percentage of incomplete cae the LVF algorithm i more than 20%; on the other hand, DT* guarantee to return a correct anwer in about 9 econd on average. In ummary, our experiment how that on average, our propoed DT* algorithm i much more efficient than the DF algorithm. It i a relatively efficient a LVF algorithm while guaranteeing oundne and completene of the earch proce. From the experiment, we can alo conclude that our propoed DT* algorithm allow dominance querie for CP-net that are quite large and complex to be anwered in reaonable time. Figure : Multi-valued arbitrary acyclic CP-net 5. CONCLUSION AND FUTURE WORK In thi paper, we have tudied the problem of dominance teting in CP-net. We have propoed a heuritic algorithm DT* for dominance teting with arbitrary acyclic CP-net. The propoed approach ignificantly reduce the earch effort without impacting oundne and completene. We have alo experimentally hown that the propoed algorithm i computationally efficient. Nonethele, the preent work i only applicable for acyclic CPnet. The invetigation of technique to deal with cyclic preference need to be further explored. 6. ACKNOWLEDGEMENTS Thi work i partially upported by the ARC Dicovery Grant DP REFERENCES [1] C. Boutilier, R. I. Brafman, H. H. Hoo, and D. Poole. Reaoning with conditional ceteri paribu preference tatement. In UAI, page Morgan Kaufmann Publiher, [2] C. Boutilier, R. I. Brafman, H. H. Hoo, and D. Poole. Preference-baed contrained optimization with CP-net. Computational Intelligence, 20:137 7, [3] C. Boutilier, R. I. Brafman, H. H. Hoo, and D. Poole. CP-net: A tool for repreenting and reaoning with conditional ceteri paribu preference tatement. Journal of Artificial Intelligence Reearch, 21: , [4] C. Domhlak, S. D. Pretwich, F. Roi, K. B. Venable, and T. Walh. Hard and oft contraint for reaoning about qualitative conditional preference. J. Heuritic, 12(4-5): , [5] J. Goldmith, J. Lang, M. Truzczynki, and N. Wilon. The computational complexity of dominance and conitency in CP-net. J. Artif. Int. Re., 33(1): , [6] C. Lafage and J. Lang. Logical repreentation of preference for group deciion making. In KR, page , [7] J. Lang. Graphical repreentation of ordinal preference: Language and application. In ICCS, page 3 9, 20. [8] F. Roi, K. B. Venable, and T. Walh. mcp net: Repreenting and reaoning with preference of multiple agent. In AAAI. [9] G. R. Santhanam, S. Bau, and V. Honavar. Dominance teting via model checking. In AAAI,