Modeling Qualitative Preferences Using the CP-net Model

Modeling Qualitative Preferences Using the CP-net Model

The ability to make decisions is a corner-stone of many AI applications: • Decision-support expert systems • Autonomous agents • Configuration software • … To make good decision, we must be able to assess and compare different alternatives.

Assessing Alternatives • We compare alternatives based on their: • Likelihood • Desirability Our Focus: Assessing/specifying outcome desirability

Specifying Preferences Uncertainty Involved? yes no Utility function desirable Enough to rank potential outcomes Need to weigh the contributions of different outcome Need to recognize the best feasible alternative

Utility functions • Qantify outcome desirability • Capture the difference in desirability between outcomes • Necessary when: • Uncertainty is involved, or • Drawbacks: • Complicated preference elicitation. • Generally hard optimization. Serious practical concern Serious comput. concern

When a utility function cannot be and/or need not be obtained, one should resort to other, more qualitative forms of preference representation.

What We Want from a Qualitative Preference Model • Simple elicitation process based on intuitive and natural statements about preferences • No need for an expert decision analyst • Makes automatic online elicitation feasible • As expressive as possible, subject to above • Supports an efficient optimization process

Overview • CP-net model for qualitative preferences. • Preference Optimization. • Complexity analysis of outcome comparisons. • Consistency testing. • Various Enhancements. • Potential applications. • Future research directions and open problems.

Ceteris Paribus (cp) Statemenents Ceteris Paribus (Lat.) – all else being equal “ I prefer to have wine with my meal, all else being equal” That is: given two identical meals, one with wine and one without, I prefer the former.

Conditional CP Statements “ I prefer red wine to white wine with my meal, ceteris paribus, given that meat is served” That is: given two identical meals in which meat is served, I prefer red wine to white wine. Tells us nothing about two identical meals in which meat is NOT served.

Domain variables . • Outcome space . • Preferences over the outcome space. Outcomes and Preferences • Example on Product (Computer System) Configuration: • Domain variables are the system’s properties: • Processor Speed, Processor Manufacturer, Screen Size, etc. • Dom(Screen Size) = {15in, 17in, 19in, 21in} • Possible preferential statements of a customer: • I prefer 1000 MHz on 800 MHz • I prefer 19inscreen on 17inscreen if video card is Sony’s

Preferential Independence I prefer 1000 MHz to 800 MHz(all else being equal) If my preferences over the values of a variable v does not depends on the values of some other variables, then v is preferentially independent of all other variables. I prefer 19in screen to 17in screen if video card is Sony’s (all else being equal) If my preferences over the values of a variable v depends on the values of some other variables v1, …,vk , then v is conditionallypreferentially independent of all other variables V-{v1, …,vk }, given an assignment on v1, …,vk .

Preferential Independence A subset of variables is preferentially independent of its complement if and only if, for all assignments holds Let be a partition of into three disjoint non-empty sets. and are conditionallypreferentially independent given if and only if, for all holds

CP-nets (Boutilier, Brafman, Hoos, Poole, UAI ‘99) An intuitive, qualitative, graphical model of preferences, that captures statements of conditional preferential independence. • DAG in which each node represents a domain variable. • The immediate parents P(v) of a variable v in the network are those variables that affect user’s preference over the values of v. • P(screen size) = { video card manuf. } • P(operating system) = { processor speed, screen size }

A B C D E F Example of a CP-net

A B C worst best Consistency Any CP-net defines a partial order over the outcome space.

A B C A B D E C D E F F Preferential Optimization Finding the preferentially optimal outcome is straightforward!

Domain variables . • Outcome space . • Preferences over the outcome space. • Constraints on the domain variables. Adding Constraints Constraints in Product (Computer System) Configuration: 17in screens are currently unavailable. 15in screens of NEC are incompatible with the Sony’s graphical card. Windows XP requires at least 800MHz processor speed.

 >    Constraint-based Preferential Optimization (Boutilier et al. 97’) Branch & Bound algorithm for determining the set of feasible, preferentially non-dominated outcomes was suggested. • Starting with an empty set of solutions, the algorithm continuously extends it by adding new non-dominated solutions. • The current set of the solutions serves a lower bound for the forthcoming candidates.

Choose any variable v with no parents in G. X Y Z Let v1 >… > vk be the preference ordering of D(v) given the assignment on P(v) in K. K(x,y) : a1 > … > ak A Constraint-based Preferential Optimization E A B F C G Initialize the set of local results R = . D H

fori = 1tokdo v = vi X Y Z ifCiis inconsistent or exist j<i s.t. Cj Ci continue with next iteration else Let K’ be the partial assignment induced by v = vi and Ci . Reduce G to G’ by removing the variables assigned by K’. E B F Let G’1,…, G’m be the strongly connected components of G’ C G forj= 1tomdo Sj = Search (G’j, KK’, Ci) D H ai Strengthen the set of constraints C by v = vi to obtain Ci ai E B F C G D H K(x,y) : a1 > …> ai> …> ak

ai E B F X Y Z C G D H E B F if Sj  for all j m C G foreach o  K’ S1 …  Smdo D if for each o’  R holds Ko’  Ko H ai fori = 1tokdo v = vi Strengthen the set of constraints C by v = vi to obtain Ci ifCiis inconsistent or exist j<i s.t. Cj Ci continue with next iteration else Let K’ be the partial assignment induced by v = vi and Ci . Reduce G to G’ by removing the variables assigned by K’. Let G’1,…, G’m be the strongly connected components of G’ forj= 1tomdo Sj = Search (G’j, KK’, Ci) Add o to R return R

Constraint-based Preferential Optimization

So what is the price of comparison (dominance testing) between two outcomes?

Flipping Sequence - Example

Where, for , • outcome differs from the outcome in the value of exactly one variable • , given the values of in (and ) Dominance Testing In (Boutilier et al.) dominance testing was treated as a search for a flipping sequence from the (purported) less preferred outcome to the (purported) more preferred outcome through a sequence of more preferred outcomes:

CP-net graph Complexity Remarks Tree Linear? Backtrack free algorithm in BBHP ‘99 Dominance Testing for CP-nets with Binary Variables.

Complexity of Dominance Testing First result (Boutilier et al.): Dominance testing for binary, tree CP-nets is backtrack free.

Example: A C B D E

The algorithm for trees is not good for polytrees A B C

CP-net graph Complexity Remarks Tree Lower bound Dominance Testing for CP-nets with Binary Variables.

CP-net graph Complexity Remarks Tree Lower bound Polytree k - maximal indegree Dominance Testing for CP-nets with Binary Variables.

Start of Analysis Denote by the maximal number of times that a variable may be required to flip its value on a irreducible flipping sequence from to . Lemma 1: Given a dominance testing problem where is a singly connected, binary CP-net, for each variable we have thatolds:

Framework for Polytrees • Using the upper bound established by Lemma 1, we provide a polynomial time procedure that determines the maximal number of feasible, possibly required value flips for a given variable. • We provide a polynomial time algorithm that determines whether or not exist a flipping sequence from to . This algorithm is based on top-down execution of the previously defined procedure on the variables of the CP-net.

CP-net graph Complexity Remarks Tree Lower bound Polytree k - maximal indegree Singly-connected DAG NP-complete Reduction from 3SAT Dominance Testing for CP-nets with Binary Variables.

CP-net graph Complexity Remarks Tree Lower bound Polytree - maximal indegree Singly-connected DAG NP-complete Reduction from 3SAT -connected DAG NP-complete Minimal flipping sequences are polynomially bounded Dominance Testing for CP-nets with Binary Variables.

CP-net graph Complexity Remarks Tree Lower bound Polytree - maximal indegree Singly-connected DAG NP-complete Reduction from 3SAT -connected DAG NP-complete Minimal flipping sequences are polynomially bounded DAG ? EXPTIME or in NP? Dominance Testing for CP-nets with Binary Variables.

    Recall that ... • Starting with an empty set of solutions, the B&B algorithm continuously extends it by adding new non-dominated solutions. • The current set of the solutions serves a lower bound for the forthcoming candidates. • New candidate is compared to all solutions generated until now.

The Big Picture • Generally, COP is much harder than CSP. • If any non-dominated solution is enough, then the CP-net based COP is not harder than the corresponding CSP. • If some non-dominated solutions are required, and the CP-net is reasonably restricted, then the CP-net based COP is still not harder than the corresponding CSP. • In general, if some (even just 2) non-dominated solutions are required, then the CP-net based COP may be much harder.

A B C worst best acyclic Recall that … Any CP-net defines a partial order over the outcomes.

A B Consistency of Cyclic CP-nets Cyclic CP-nets are not necessarily consistent …

Strongly Connected Components Context variables of a SCC Contexts of a SCC = assignments to its context variables Localizing the Analysis

Modeling Qualitative Preferences Using the CP-net Model