Mixed Integer Programming Model for Compaction and
8.1 Limitation of the Locality Heuristic
In our position-based compaction algorithm, the locality heuristic is used to restrict the relative motion of polygon
Q
with respect to polygonP
to a convex subset ofP
(Q
).On the one hand, such a restriction is essential in reducing the motion planning problem
1Flips and discrete rotations can be added to the MIP model, but we have not attempted to implement
embodied in compaction to a more manageable linear programming problem. On the other hand, it may exclude other solutions to a compaction problem that may result in a shorter length. A simple example is depicted in Figure 8.1. Figure 8.1 (a) shows the initial con- guration of a layout to be compacted. Figure 8.1(b) shows the result of applying our position-based compaction algorithm.
Obviously, for the layout in Figure 8.1(a), the best result that can be achieved by a compaction algorithm is a layout of length
l
in which polygonQ
and polygonP
are stacked vertically. The actual result achieve by our position-based compaction algorithm has length 2l
1.In this example, our position-based compaction algorithm failed to nd the optimal compaction result of length
l
because the convex subset selected by the locality heuristic preventsP
andQ
from moving to the relative position that yields the optimal solution. The starting edge selected by the locality heuristic is the edgeb
ofP
(Q
). The convex subset C ofP
(Q
) built fromb
is actually the half-plane delimited byb
and on the right-handside of
b
. The linear constraint built fromC constrainsQ
to stay on the right ofP
. Thus,the length 2
l
1 of the layout shown in Figure 8.1(b) is the best length achievable given the constraint.Clearly, if the edge
a
ofP
(Q
) (see Figure 8.1(c)) is selected as the starting edge forbuilding a convex subset C 1 of
P
(
Q
), then C1 is the half-plane delimited by and above
a
. The linear constraint built from C1 states that
Q
must lie on top ofP
. Hence, using C1 instead of
C to build the non-overlapping constraints would enable our position-based
compaction algorithm nd the optimal compaction of length
l
.An immediate idea to improve our position-based compaction algorithm is therefore to try several convex subsets of
P
(Q
) for each pair of polygonsP
andQ
and choose theones that collectively yield the best compaction result. However, the following theorem shows that making such a choice is NP-hard.
Theorem 8.1
Given a layout consisting ofN
polygonsP
1,P
2,:::;P
N. Assume that foreach pair of polygons
P
i andP
j (1;i;j
N
,i
6=j
), we can choose one from two di erentconvex subsets of
P
i(P
j) to build non-overlapping constraints forP
i andP
j in ourposition-based compaction algorithm. Then, nding a combination of the convex subsets that achieves the optimal solution for compaction is NP-hard.
Proof:
We again reduce the now familiar PARTITION problem to our problem. The reduction is shown in Figure 8.2. For an instancea
;a
;:::;a
N of PARTITION, weh l h a b P (−Q) c Q cP 2 l − 1 (a) (b) (c) Q c Q P c P l P c P P c P Q c Q Q 2h
Figure 8.1: (a) The initial layout. (b) The compaction result using the position-based compaction algorithm. (c) Illustration of the locality heuristic: the dotted lines illustrates the boundary of the convex subset identi ed by the locality heuristic.
P P P P P P 1 1 P P 0 2 3 4 5 6 7
Figure 8.2: Reduction of PARTITION to selecting the right combination of convex subsets.
P P P P P P 1 1 P P 0 2 3 4 5 6 7
Figure 8.3: Successful selection of the right combination of convex subsets solves PARTI- TION.
build rectangles
P
i of height 1 and lengtha
i, (i
= 1;:::;N
). We then add two additionalrectangles
P
0 andP
N+1 each of height 1 and lengthB
= 1 2P
Ni=1
a
i. Initially, these polygonsare placed in the con guration shown in Figure 8.2.
For rectangles
P
i andP
j (i
=6j
), the Minkowski sumP
i(P
j) is also a rectangle. Lett
,b
,l
andr
represent the top, bottom, left and right edges ofP
i(P
j) respectively. Letconvex sets
H
t,H
b,H
l andH
r be the half-planes inP
i(P
j) that are delimited by edget
,b
,l
andr
respectively. The non-overlapping constraint built usingH
t constrainsP
j tostay above
P
i. The other three cases can be deduced similarly.The choices of convex sets for pairs of polygons are as follows. First, for building non- overlapping constraints of
P
i (1i
n
) withP
the one that constrains
P
i to stay on top ofP
0 and the one that constrainsP
i to stay tothe right of
P
0. Second, for building non-overlapping constraints betweenP
i (1
i
n
)and
P
n+1, we o er the convex subset that constrainsP
i to stay to the leftP
n+1 and theone that constrains
P
i to stay belowP
n+1. Third, for building non-overlapping constraintsbetween
P
i andP
j (1i;j
n
,i < j
), we o er the convex subset that constrainsP
i tostay to the left
P
j and the one that constrainsP
i to stay belowP
j.Figure 8.3 shows the minimal length compaction of the layout in Figure 8.2. It is clear that PARTITION has a solution if and only if the layout in Figure 8.2 has a length 2
B
compaction. 2
In the next section, we will use a set of Boolean variables (each has value 0 or 1) to represent each of the convex subsets to be chosen. The problem of choosing the combination of convex subsets that yields the minimal length compaction is then reduced to a MIP problem. Theorem 8.1 illustrates that using the full power of MIP is justi ed since the problem is at least as hard as MIP (that is, NP-hard) in the worst case. More importantly, reducing the problem to MIP enables us to take the advantage of the powerful techniques developed recently for solving mixed integer programs, especially the techniques deployed by the MINTO package [Nem93] which we are currently using.
Remark 8.1
Note that in Figure 8.1(a), if the upper right vertex ofP
and the lower left vertex ofQ
are incident, then the pointc
Qc
P is incident on the vertex at the upper rightcorner of
P
(Q
). In this case, we are forced to make an arbitrary choice between usingedge