Minimum Vertex Cover in Rectangle Graphs

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

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

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

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

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

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

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

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

9. k-outerplanargraphs 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

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*23k) time). Dynamic programming on the 3k-tree width decomposition

11. Tree decomposition • A tree decomposition: • Tree with a vertex set called bagassociated 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. g a h b c f e d a a g c f a g h b c f d c e

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. g a h b c f e d a a g c f a g h b c f d c e

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. g a h b c f e d a a g c f a g h b c f d c e Treewidth of graph G:

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

15. Approximation ratio

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