Experiments with weaker relaxations

The impact of different relaxations on the search is shown in Figure 5.7. We used 5 sets of problems:

• The verification problems from Ghosh et al. (Section 5.4.1) [66]

• The PSR domain (middle-size set from IPC 2004)

• Multi-agent planning problems from Kominis and Geffner [95] (Sec- tion 5.5)

• Instances of the trapping game (blocker) (Section 5.4.2), played on graphs with 4 to 47 nodes and

• Random instances of Min-Cut domain, with graphs of size 12–20 and 3–4 roadblocks (Section 5.3.1).

Each planner was run with up to 1h CPU time and 3GB memory per problem. Our implementation is built on Fast Downward planner [78] and uses Clasp v2.1.3 as an ASP solver.

If there are no assignments offalse to secondary variables in the bodies of the axioms, the naive version of hmax coincides with 3-value semantics hmax

(here, this happens only in the verification problems). For all the other prob- lems, the naive relaxation is as uninformed as blind search. The ASP-based relaxation is more accurate than the 3-value semantics relaxation only when some secondary facts can be derived from the disjunction in a relaxed state. In our test set, this occurs only in the PSR and Min-Cut domains. However, the weaker relaxation is much faster to compute. (Results for the ASP-based

hmax _{heuristic are omitted from the other domains, in which it expands ex-}

actly the same number of nodes but takes far more time to do so.) Although all the heuristics except the naive reduce the amount of search, blind search is still often the fastest; its main limitation is memory. This has also been demonstrated in STRIPS planning, e.g. in results from the IPCs 9_{. The}

3-valuedhmax heuristic is faster than blind search on the hardest of Kominis and Geffener’s problems and results with axiom-aware PDB heuristic on the Min-Cut domain show that even the ASP-based relaxation can be sufficiently informative to compensate for the overhead of computing it. We did not try the PDB heuristic on other problems since we do not know which are good abstractions, if indeed any exist.

9

Chapter 6

Preferred operators in partial

expansion A

?

To compute optimal plans, we use A? search guided by an admissible heuris- tic. We have, however, also developed a new search algorithm which can result in significant savings in terms of memory and time when the heuristic is expensive to compute, yet accurate and state have many successors. This situation is characteristic of many planning problems, including the PSR problem (Section 3.3.2). In the PSR problem, each state has at most as many possible successors as there are switches in the network – the number of switches can be in the order of thousands (e.g. transmission network for New South Wales), and in the network used in our experiments there were 45. While not all of those successor states are necessarily valid, they still need to be generated before we can determine that they are invalid. The algorithm that will be presented in this chapter can save us from generating some of those states as well.

Our algorithm, which we call _PrefPEA?_{, combines preferred operators}

with partial expansion A?_{. While preferred operators have been used in}

non-optimal planning, they have not been combined with optimal search algorithms such as A?_{. In our case, the preferred operators are obtained as a}

side effect of computing h+_{. While h}+ _{is expensive to compute, the optimal}

plan for a relaxed problem can provide us with more information than just heuristic cost estimate. We only partially expand the nodes, therefore and avoid generating all of the successors and reduce the number of heuristic computations (as well as computation necessary for determining whether the states are valid).

Techniques that we build on will be described in Section 6.1. In Sec- tion 6.2 we will explain how we combined those two ideas to reduce the num- ber of heuristic evaluations in cases where the heuristic is informative yet

expensive to compute. Section 6.3 will present the experiments and results.

6.1

Background

Here, we will give an overview of search techniques that we build on, namely identifying preferred oruseful actions, and partial expansion A? _(PEA?_).

For the description of the A? _{see the original paper by Hart et al. [111].}

We assume that the reader is familiar with the algorithm and the terms used. Here, we will just give a few definitions of the terms used in this chapter, all which are borrowed from Hart et al.’s paper. For a given node n, cheapest known cost of reachingn from the start state is denotedg(n). Heuristic cost estimate of nis denotedh(n) and it estimates the minimum cost of reaching the cheapest goal state from the given state. If the heuristic is known to never overestimate the cost, it is an admissible heuristic. As stated earlier, admissible heuristics are used in optimal planning because certain optimal search algorithms, like A?_{, guarantee that the solution returned is optimal,}

provided that the heuristic is admissible. Evaluation function, orf(n), is the sum g(n) and h(n) values. The cost of the cheapest path to a goal state is denoted f?_{. A}? _{can be described as best-first search on the f-value.}

A? _{is a graph search algorithm, but in this work we are interested in}

presenting and solving the planning problems as state-space search. There- fore, our graph is a state-space of the planning problem, and our succes- sor operators are actions. We will use the term node to refer to a tuple

n=hs, g(s), h(s)i. We defineg(n) =g(s), h(n) = h(s) and f(n) =f(s).