Download Abstracting Planning Problems with Preferences and Soft Goals

Abstracting Planning Problems with Preferences and Soft Goals
Lin Zhu and Robert Givan
Electrical and Computer Engineering, Purdue University, West Lafayette IN 47907 USA
{lzhu, givan}@purdue.edu
Abstract
In this paper we describe the planner PATTERN P LAN that participates in the Fifth International Planning Competition. We
put emphasis on solving the over-subscription planning problem, which characterized the major difference between this
planning competition and the previous ones. Our solution is
to abstract the over-subscription problem into a generalized
orienteering problem and use the solution of this problem
to guide the search of solving the original over-subscription
problem.
Introduction
The goal in traditional planning research is usually to generate a plan that satisfies a fixed set of subgoals, sometimes with the additional objective of minimizing costs.
This also has been the theme in previous planning competitions(McDermott 2000; Bacchus 2001; Long & Fox 2003;
Hoffmann & Edelkamp 2005). In many real-world planning problems, however, it is necessary to explicitly consider the trade-off between the rewards of achieving goals
against their costs. Very recently this problem was considered in the setting of over-subscription planning (Smith
2004). In this setting, there are a large number of possible goals with differing rewards. Given a limit of time and
resources, the planning system must choose to accomplish
a subset of those goals that maximizes the rewards. In the
Fifth International Planning Competition (Gerevini & Long
2005), this problem is considered more generally by setting
the objective as optimizing a linear combination of rewards
and costs. We also call these problems over-subscription
planning problems.
Given an over-subscription planning problem, there is a
cost and a reward associated with each valid plan. For
any over-subscription problem, there is a underlying relation between rewards and costs. The over-subscription
problem is implicitly a linear optimization problem on this
relation. Because exactly generating this relation is difficult, previous approaches employed various approximation methods (Smith 2004; van den Briel et al. 2004;
Nigenda & Kambhampati 2005; Benton, Do, & Kambhampati 2005).
We also solve the over-subscription problem approximately by generating an abstraction problem. By design
the reward-cost relation on the abstraction problem approximates that on the original problem. A solution with good
quality on the abstraction problem thus hopefully leads to
a solution with good quality on the original problem. With
such a guide, we employ a forward-chaining method to solve
the original over-subscription problem.
In the following section, we will describe the abstraction
method and the planning method in detail.
Abstraction
We take an approach that is similar in spirit with a previous
method that constructs orienteering problems (Keller 1989;
Blum et al. 2003) for over-subscription planning (Smith
2004). Our approach differs in two ways. First, we recognize that a generalization of the orienteering problem is
necessary. Second, we introduce a novel fully automatic
method that translates the over-subscription problem into a
generalized orienteering problem.
We start by translating the PDDL planning problems into
the SAS+ formalism (Bäckström & Nebel 1995), based on
methods previously published (Helmert 2004; Edelkamp &
Helmert 1999). In SAS+ variables can have non-binary finite domains. More compact representations are achieved
with SAS+ than with STRIPS or PDDL.
Starting with an empty abstract problem, we then incrementally introduce variables to the abstract problem until a
pre-defined threshold (e.g., a certain number of variables) is
reached. Only those variables that are both a precondition
and an effect of some action are included. For each such action, the cost of that action is associated with the variable.
The variables are added to the abstract problem in descending order of their minimal associated costs. This abstract
state space represents the part of the original problem that is
most sensitive to costs. Any action or any set of goals can
be projected onto this abstract state space by ignoring other
variables.
In the next step all the goals with rewards are embedded
into this abstract state space. A goal can be embedded into
an abstract state if and only if the abstract state satisfies the
goal after the goal is projected into the abstract state space.
Note that some abstract state might contain no goal. Also
note it is possible that the same goal is embedded into multiple abstract states. Because of this we have to introduce a
generalization of the orienteering problem.
Now we are ready to define the generalized orienteering
problem. This problem consists of the abstract state space,
the distance between any pair of states, the goals that are
associated with states, and a reward for each goal. Starting from the projection state of the initial original state, the
objective is to optimize a linear combination of the rewards
and costs by moving on the abstract state space. A reward
for each goal can be collected only once, even if the same
goal is available in multiple abstract states.
We solve the generalized orienteering problem by adapting heuristic local search methods that are used to solve the
original orienteering problems (Smith 2004).
The solution for the generalized orienteering problem
defines an ordering on the goals of the original oversubscription problem. We then use a black-box costoptimizing planner to solve enlarging subsets of the goals
in sequence. The subset goal initially contains the first goal
solved in the generalized orienteering problem. At each
stage the planner tries to achieve all the goals in the subset
goal. A new goal is added to the subset goal if the planner succeeds. The black-box planner is stopped whenever a
subset goal is not achievable within a time limit. We call the
plan that successfully achieved the second last subset goal a
preliminary plan. The final plan is a prefix of the preliminary plan that achieves the linear optimization of costs and
rewards.
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