1. Minimum Vertex Cover in Rectangle Graphs R. Bar-Yehuda, D. Hermelin, and D. Rawitz 1

2. Vertex Cover in Rectangle Graphs R’  R s.t. R - R’ is pairwise non-intersecting: 2

3. Vertex Cover in Rectangle Graphs R’  R s.t. R - R’ is pairwise non-intersecting: 3

4. Vertex Cover in Rectangle Graphs R’  R s.t. R - R’ is pairwise non-intersecting: 4

5. Applications Critical section scheduling rectangles are jobs x-axis = time y-axis = shared resource dump min number of jobs to get a feasible schedule Automated map labeling rectangles are labels dump min number of labels to get a pairwise non-intersecting labeling 5

6. Previous Work NP-hard [Fowler et al. 1981] PTAS for equal height rectangles [Agarwal et al. 1998] PTAS for squares [Chan 2003] PTAS for bounded aspect-ratio rectangles [Erlebach et al. 2005] EPTAS for unit-squares [Marx 2005] f(1/  )  n O(1) instead of n f(1/  ). EPTAS for squares [van Leeuwen 2006] PTAS for non-crossing rectangles [Chan and Har-Peled 2009] no pair of rectangles intersect as follows: 6

7. Our results 1.EPTAS for non-crossing rectangles Improving [Chan and Har-Peled 2009] from a PTAS to an EPTAS. Extends to pseudo-disks. 1.(1.5 +  )-approx. for general rectangles First approximation for general rectangles which is better than 2. Extends to the weighted case. 7

8. Non-Crossing Rectangles Arrangement graph A R of R: V(A R ) := intersection points (joints) E(A R ) := rectangle edges Our EPTAS is based on Baker's scheme and uses the planarity of the arrangement graph

9. k-outerplanar graphs Label vertices of a plane graph by level. All vertices on exterior face level 1. All vertices on exterior face when vertices of levels 1 … i are removed are on level i+1. Graph is k-outerplanar when at most k levels. Theorem: k-outerplanar graphs have treewidth at most 3k – 1. 3-outerplanar 9

10. Vertex cover on k-outerplanar graphs For fixed k, finding a minimum Vertex Cover in a k-outerplanar graph can be solved in linear time (O(n*2 3k ) time). – Dynamic programming on the 3k-tree width decomposition 10

11. Tree decomposition A tree decomposition: – Tree with a vertex set called bag associated to every node. – For all edges {v,w}: there is a set containing both v and w. – For every v: the nodes that contain v form a connected subtree. b c d e f aa b c c c d e h hf f a g g a g 11

12. Tree decomposition A tree decomposition: – Tree with a vertex set called bag associated to every node. – For all edges {v,w}: there is a set containing both v and w. – For every v: the nodes that contain v form a connected subtree. b c d e f aa b c c c d e h hf f a g g a g 12

13. Tree decomposition A tree decomposition: – Tree with a vertex set called bag associated to every node. – For all edges {v,w}: there is a set containing both v and w. – For every v: the nodes that contain v form a connected subtree. b c d e f aa b c c c d e h hf f a g g a g 13 Treewidth of graph G:

14. Using modified Baker’s scheme For each i in {1,2, …, k} do – V i = all vertices in levels i, i+k, i+2k, i+3k, … – V-V i is (k-1)-outerplanar. – Thus V-V i has 3(k-1)-tree width – R i =R(V i )={r  R : r contains a point in V i } – Thus R-R i =R-R(V i ) has 6(k-1)-tree width Let t=arcmin i {|R i |} Construct decomposition tree of R-R t From the V-V t tree Using DP on the tree find minimum VC of R-R t 14

15. Approximation ratio 15

16. Open Problems 1.EPTAS for weighted non-crossing rectangles? 2.PTAS for general rectangles? Hardness? 3.Constant approx. for independent set? 16

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