Algorithms for Geometric Covering and Piercing Problems Robert Fraser PhD defence Nov. 23, 2012

Vignettes Discrete Unit Disk Cover Problem Various settings Strip-based decomposition of the plane Hardness of problem even in a strip Hausdorff Core Problem General simple polygons Polygons with a single reflex vertex MST with Neighborhoods Problem Parameterized algorithms Hardness of problem 1

The Discrete Unit Disk Cover (DUDC) Problem 2

Applications 3

Line-separated DUDC 4 d1d1 d2d2 d4d4 d3d3 d5d5 q1q1 q2q2 q4q4 q3q3 q5q5 p1p1 p2p2 p4p4 p3p3 p5p5 p6p6 p7p7 p9p9 p8p8

Strip-Separated DUDC 5 p1p1 p2p2 p4p4 p3p3 p5p5 q1q1 q2q2 q4q4 q3q3 q5q5 q6q6 q7q7 q9q9 q8q8 [Ambühl et al., 2006]

Assisted LSDUDC u1u1 u2u2 u4u4 u3u3 u5u5 q1q1 q2q2 q4q4 q3q3 q5q5 p1p1 p2p2 p4p4 p3p3 p5p5 p6p6 p7p7 p9p9 p8p8 l1l1 l2l2 6

DUDC algorithm: work from top to bottom, then bottom to top, then cover what remains. Uses assisted LSDUDC and WSDUDC algorithms. Strip-Based Decomposition 7

Within-Strip DUDC 8 }

If easy, implies a simple PTAS for DUDC based on shifting grids… MAX independent set of unit disk graph is easy in strips of fixed height! Within-Strip Discrete Unit Square Cover is easy in strips of fixed height! 9 p1p1 p2p2 p4p4 p3p3 p5p5 q1q1 q2q2 q4q4 q3q3 q5q5 q6q6 q7q7 q9q9 q8q8

Hardness of WSDUDC (1/2) Reduce from Vertex Cover on planar graphs of max degree 3. 10

Hardness of WSDUDC (2/2) Each edge has even number of added points: Convert to instance of WSDUDP: 11

12

Covering Gaps 13 p1p1 p2p2 p4p4 p3p3 p5p5 q1q1 q2q2 q4q4 q3q3 q5q5 q6q6 q7q7 q9q9 q8q8

3-Approximate Algorithm 14

Summary of DUDC Results 15

Trade-off in DUDC Algorithms 16

The Hausdorff Core Problem 17

Shrinking Disks 18

Why is this challenging? 19

Hausdorff Core Approximation Algorithm for Simple Polygons Approximation scheme based upon a discretization of the search space. Intuition: 1.Divide the disks into discrete segments. 2.Grow the disks slightly such that at least one segment of each expanded disk will be contained in the optimal solution. 3.Check for a solution among all intervals on all disks. 20

A Simpler Problem What if the polygon that we want to approximate has only one reflex vertex? 21

One Vertex Solution Find angle where max distance is minimized. 22

Summary of Hausdorff Core Results FPTAS for Hausdorff core on simple polygons: O((n 3 +n 2 ε -6 )log( ε -1 )) time. O(n 3 ) time exact algorithm for polygons containing a single reflex vertex. 23

The (Minimum Weight) MST with Neighborhoods Problem MSTN 24

The Maximum Weight MST with Neighborhoods Problem max-MSTN 25

Parameterized Algorithms 26

Parameterized max-MSTN Algorithm T opt TcTc T c 27

Parameterized max-MSTN Algorithm T opt TcTc T c Consider this edge 28

Hardness of MSTN Reduce from planar 3-SAT clause variable clause variable (with spinal path) Create instance of MSTN (resp. max-MSTN) so that: -Weight of the optimal solution may be precomputed for any instance; -Weight of solution corresponding to a non-satisfiable instance is greater than (resp. less than) the optimal solution by a significant amount. 29

Summary of MST Results

Nice Results and Open Problems Best known fast DUDC approximation algorithm. Hardness of WSDUDC. OPEN Is assisted LSDUDC NP-hard? OPEN Is there a nice PTAS for WSDUDC? First known solutions for the Hausdorff Core problem. OPEN Are there classes of polygons for which finding the Hausdorff Core is easy? Best known results for MSTN. First results for max-MSTN. Hardness of 2-GMST. OPEN Is there a PTAS for max-MSTN (even on unit disks)? OPEN Is there a combinatorial approximation algorithm for 2-GMST (particularly with an approximation factor of <4)? 31

Thanks! 32