A closer look at GeometricGreedy

Based on the results of the previous subsection, it is tempting to “forget about trees” and analyze GeometricGreedy directly. We have seen that GeometricGreedy is a natural sweepline algorithm for solving the online satisfied superset problem. In fact, as it does the minimum work necessary at every time, one might at first think that it is optimal. The minimal example in Figure2.12shows that this is not the case. Informally, GeometricGreedy is not optimal, because, as the example shows, it may be worth to do more work locally than what is strictly necessary, as this may reduce the amount of work we need to do in the future.

Figure 2.12: Smallest example

where GeometricGreedy is not op- timal. X = (1,3,2). (left) Geomet- ricGreedy output. (right) A smallest

satisfied superset.

We have already seen in §2.5.3that the performance of Greedy- Future (and by extension OnlineGreedy) is very good, matching several upper bounds. This makes OnlineGreedy, due to its simplic- ity in geometric view, perhaps the most promising contender for dynamic optimality (even if, in tree-view, OnlineGreedy is far from simple). In §4we analyze the behaviour of OnlineGreedy (in fact, GeometricGreedy) on a variety of input sequences, and in §5we give additional interpretations of GeometricGreedy.

It appears quite plausible that GeometricGredy computes a constant approximation of the optimum (this must be the case, if GreedyFuture is constant-competitive, as conjectured, see e.g.

2.7. The geometric view 49

size for X , and let costGG(X ) denote the cost of GeometricGreedy for X . Due to Theorem2.24 and Theorem2.28, these quantities exactly match the offline BST optimum, respectively the cost of GreedyFuture for X , according to the second offline BST model.

The worst known example [31] for the approximation ratio of GeometricGreedy is a construction where the ratio is arbitrarily close to4₃, see Figure2.13. The following conjecture is therefore natural, although much stronger than the conjectured dynamic optimality of GreedyFuture.

Problem 28. Is costGG(X ) ≤4₃·OPT(X ) for every X ?

Figure 2.13: Example where the approximation ratio of Geomet- ricGreedy is 4₃. X = (2,3,1,3,1,...).

(left) GeometricGreedy output.

(right) Smallest satisfied superset.

In fact, an even stronger conjecture is suggested by Demaine et al. [31], and earlier by Munro [76], who speculate that the cost of GreedyFuture matchesOPT(X ) with an additive error of at most

O(m), or even just m. This conjecture is sensitive to constant factor

modifications of the cost, therefore we need to be careful in spec- ifying exactly which cost model we use. As far as we understand, it was formulated in what we call the second offline model, there- fore, if true, it would also hold for GeometricGreedy. The example in Figure2.14shows that such a strong form of the conjecture is false. The figure presents an input sequence X ∈ S30, i.e. a per-

mutation of length m = 30, such that costGG(X ) ≥OPT(X ) + m + 1. The exact sequence shown in the figure is X = (18,9,17,16,25,19,4, 10, 26, 24, 8, 13, 5, 28, 23, 20, 7, 12, 1, 29, 27, 3, 22, 11, 14, 2, 30, 21, 6, 15).

Observe that the example does not settle Problem28, since the approximation ratio is not known to be more than4₃in this example. The example in Figure2.14was obtained with computer search. It is not immediately clear how to extend it into an infinite family of examples, although visually it does seem to have a certain structure. It seems plausible that there are permutations of arbitrary size with a compact description, on which the error of GeometricGreedy is larger than m (computer experiments suggest that such examples are easier to come by as m grows, until the computation itself becomes too costly). Constructing structured examples (whether permutations or not) on which Greedy performs suboptimally remains an interesting direction. Whether such constructions will give new insight on the behavior of Greedy remains to be seen.

A last remark on the example in Figure2.14is that the cost of GeometricGreedy is com- pared here not with the exact optimum, but with the cost of GeometricGreedy ran sideways (which is an upper bound onOPT). It is easy to see that running GeometricGreedy (or any valid satisfied superset algorithm) on reversed, mirrored, or 90-degrees rotated variants of the input still produces a valid solution. The fact that time and (key-)space in the BST model can be interchanged is one of the more surprising and non-trivial insights of the geometric view. The exact relation between the costs for these variants is, however, poorly understood.

Problem 29. Is the cost of GeometricGreedy asymptotically the same on X and Xrot, where

Xrotis X rotated by 90 degrees?

Based on the above discussion, the possibility that the following conjecture holds is not ruled out (again, this is a statement stronger than the dynamic optimality of GreedyFuture).

Problem 30. Is costGG(X ) ≤OPT(X ) + O(m) for every X ∈ [n]m?

Regardless of GeometricGreedy, one may try to attack the geometric satisfied superset problem directly, using various tools from the field of geometric approximations. One possi- ble direction would be to formulate the problem as a linear program. Surprisingly little has

Figure 2.14: Counterexample to additive error conjecture for Greedy. Per- mutation sequence X of length 30. (left) GeometricGreedy output with

costGG(X ) = 140. (right) a different solution showingOPT(X ) ≤ 109.

come so far from such attempts. One aspect of the satisfied superset problem that makes it unusual is its non-monotonicity: adding more points to a satisfied point set may render it unsatisfied. The hardness of the problem is also poorly understood (see Problem3). De- maine et al. [31] prove the NP-hardness of the problem in the general case where the input may contain multiple points in the same column, as well as in the same row. The following intriguing question is thus open.

Problem 31. Is minimum satisfied superset NP-hard for point sets with one point in every row?