NADAS CON LA OBLIGACIÓN DE PERMITIR EL CONTROL DE LA ADMINISTRACIÓN, INFORMAR

The operator-counting LP heuristic for the net change constraints SEQ is the state equa- tion heuristic hSEQ. It achieves the second highest coverage after the landmark con- straints.

To measure the impact of the safety-based improvement of the state equation heuris- tic, we conducted an additional experiment where we extended SEQ with the corre- sponding upper bound net change constraints. This is implemented as changing con-

11.1. Individual Constraint Groups 10 20 30 40 10 20 30 40 ≥ 50 ≥ 50 hLP_SEQ(sI) h MIP SEQ (sI )

Figure 11.4: Initial heuristic values produced by operator-counting LP and MIP heuris- tics with lower bound net change constraints.

straints for safe variables into equations. As expected from Corollary 10.2, this has no effect on the number of expanded nodes. However, with the additional constraints we solve30 fewer tasks across 18 domains, which can be attributed to slower evaluations of the LP solver.

We do not have a theoretical result that the upper bound net change constraints are strictly weaker than the lower bound net change constraints. However, if we solely use all upper bound net change constraints, the LP heuristic returns very poor estimates. In 26 domains the search expands more states resulting in a coverage of only 661 tasks. So these constraints are indeed strictly weaker.

We also tried adding all positive and negative prevail constraints to the set of con- straints, but this never influences the heuristic value. Coverage is mostly unaffected by this and only shows differences of one or two tasks in some domains. Overall, coverage reduces by3 when using prevail constraints; both with LPs and MIPs.

Restricting operator-counting variables to the integers has a small positive effect on heuristic quality. Figure 11.4 shows initial heuristic values for the LP and the MIP heuristic. Switching to the MIP reduces the number of expansions by roughly an order of magnitude in the two PegSolitaire domains but did not lead to more solved tasks. In fact, the LP heuristic solves 2 tasks more than the MIP heuristic in each of the two domains. There are 7 more domains where the restriction to integers reduced the number of expansions but in all of them the number of expansions stays in the same order of magnitude. Only one task in the domain Woodworking is solved with a MIP and not solved with an LP.

We have also seen that hSEQ is closely related to general operator cost partitioning over atomic projections (hOCP

All1). If the abstractions contain no dead states the two heuris-

11. Experiments

Non-negative costs General costs Singleton goal patterns hOCP+_Int1 : 621 hOCP_Int1 : 622 All singleton patterns hOCP+

All1 : 560

hOCP All1: 601

hSEQ_{: 723}

Table 11.4: Coverage for different variants of optimal operator cost partitioning for atomic projections.

are the same iff no variable contains values that are unreachable or from which the goal value cannot be reached in the projection. This is the case for most planning tasks in our benchmark set. Though there are some exceptions: in the domain Airport planes navigate on runways, can take off and land. Planes that take off are assumed to leave for another airport, so they cannot land again. If a plane is initially on the ground and its goal is to move to another position on the ground, then taking off sets the position of the plane to “in the air”, and the goal value of this variable can no longer be reached. Another example is in the domain Parking that models cars maneuvering in a car park. Due to a modeling error it is possible for a car to park behind itself, which means it then blocks itself and can never move again. In both of these cases a variable domain contains a dead value and hSEQis not guaranteed to be the same as hOCP_All1. This situation also occurs in the domains ParcPrinter, TPP, and one task in the domain Trucks. In all cases hSEQand hOCP_All1 use the same number of expansions, so the dead values never influence the heuristic value on our benchmarks.

Table 11.4 compares the coverage of hSEQ_{to that of other methods to combine atomic}

projections. We have already seen in Part I that using projections to non-goal variables slows down non-negative cost partitioning without improving the heuristic value, and that this loss can be recovered to some degree by allowing negative costs. Just using general cost functions was not enough to improve coverage above that of hOCP+_Int1 or hOCP_Int1 . The state equation heuristic exceeds the coverage of both by over100 tasks. It has the heuristic accuracy of hOCP

All1 but it can be computed much faster. Due to the smaller

LP size hSEQ solves more tasks than hOCP_All1 in38 domains, an overall coverage increase of 122. This shows that the main reason for hSEQ_{’s superior performance is the more}

compact representation, though the general cost partitioning also plays an important role, as the comparison of hOCP+_All1 and hOCP_All1 shows.