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«Discrete and Lexicographic Helly Theorems and their Relations to LP-Type Problems Thesis submitted for the degree of “Doctor of Philosophy” by ...»

-- [ Page 27 ] --

From the last two inequalities we obtain that c2 = c2II (B 2,0 ). Hence D2II (B 2,0 ) is a solution defining set for the type II 2-lpiercing problem on B 2 as well. We let

D2II (B 2 ) = D2II (B 2,0 ).

We note that none of the rectangles of D2II (B 2 ) is pierced by cBL. Due to Observation B.3.2, no point which lies not above and not to the right of cBL pierces any of the rectangles in D2II (B 2 ). In particularly, none of the rectangles in D2II (B 2 ) is pierced by c1, so all the rectangles in D2II (B 2 ) must be pierced by two points.

We will next define D1 (B 1 ) such that all the rectangles in D1 (B 1 ) must be pierced by one point. If bB ∈ Br 1 (this implies that the solution of the 1-piercing problem on B 1 \ Br ∪ {bB, bL } is not to the left of c1 ), we

–  –  –

We note that D1 (B 1 ) is a solution defining set for the 1-lpiercing problem on B 1. We include BR and BT in B3 that we define below. Thus we can not decrease the solution of the type BL 3-lpiercing problem on D1 (B 1 ) ∪ D2II (B 2 ) ∪ BR ∪ BT ∪ B∗ ∪ B ∗ (for any choice of D2II (B 2 )) by decomposing it into a 1-lpiercing problem on a proper subset of D1 (B 1 ) and a 2-lpiercing problem on the remaining elements. We let B3 = B R ∪ B T, and let D3T L (B) = D1 (B 1 ) ∪ D2II (B 2 ) ∪ B3 ∪ B∗ ∪ B ∗.

From the definitions above D3T L (B) is a solution defining set for the type BL 3-lpiercing problem on B.

Clearly |D3BL (B) \ (B∗ ∪ B ∗ )| ≤ (4 − 2) + (5 − 2) + (5 − 3) + (5 − 3) = 9, so |D3BL (B)| ≤ 15.

Until now, for every XX ∈ {T R, BR, T L, BL}, we have defined D3XX, a solution defining set for the type XX 3-lpiercing problem on B, if B is type XX 3-pierceable. It remains to handle the cases where B is not type XX 3-pierceable. In these cases the set of rectangle not pierced by cXX (i.e., B \ P (cXX )) is not 2-pierceable. By Corollary B.3.8 we take D2 (B \ P (cXX )), a solution defining set for the type XX 2-lpiercing problem on B \ P (cXX ) of at most 6 rectangles. In this way D2 (B \ P (cXX )) ∪ B∗ ∪ B ∗ is not type XX 3-pierceable. We let

–  –  –

Due to Observation B.4.1 D3 (B) is a solution defining set for the 3-lpiercing problem on B. We conclude the proof by noting that due to our cardinality calculations above we have |D3 (B)| ≤ |B∗ | + |B ∗ | + |D3T R (B) \ (B∗ ∪B ∗ )|+|D3BR (B)\(B∗ ∪B ∗ )|+|D3T L (B)\(B∗ ∪B ∗ )|+|D3BL(B)\(B∗ ∪B ∗ )| ≤ 4+2+5+6+7+10 = 34.

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