Point of Interest Coverage

1. Point of Interest Coverage Chris Ellis COT6410 10/6/08

2. Points of Interest (POI) • Surveillance of locations or objects is useful for study or intelligence gathering • POI can be located in hostile/dangerous environments • Autonomous tracking agents can be used

3. Tracking Agents • Remote or AI controlled • Mobile (land or air) • Equipped with cameras or sensor devices • Safer than deploying humans to hostile environments (http://www.european-security.com/imgbiblio/warrior_01s.jpg) (www.iirobotics.com/catalog/images/rovio-main.jpg)

4. Concerns • What if there are more POI than trackers? • How do we optimally place trackers to cover the POI?               

5. Formalizing the Problem • Represent POI as vertices in a weighted, fully connected graph • Assign Euclidean distance between two POI as weight of edge between corresponding vertices • Choose a fixed integer R as radius of vision from Tracker • Restrict Tracker placement to existing vertices

6. Problem Declaration • Optimization Problem • Given: a graph G = (V,E) and integer R • Problem: What is the minimum number of trackers with vision range R required to cover all vertices v  V? • A vertex is “covered” if a tracker is placed on it, or there is an edge with weight < R connected to another vertex with a tracker located on it

7. Problem Declaration • Decision Problem (POIC) • Given: a graph G and integers R and K • Problem: Can all vertices v  V be covered by K trackers with vision range R? • POIC  NP • Oracle produces a set of vertices S  V  |S| = K • For each vertex v  V if v  S or v is incident to some vertex in S with edge weight less than R, mark that vertex as covered • If any vertex v  V is not covered, return “No”, otherwise return “Yes”

8. Problem Reduction • Reduce to Vertex Cover • Convert an arbitrary instance of VC to POIC • Given an instance of VC • a graph G = (V,E) and integer K • Construct an instance of POIC • a graph G’ = (V’,E’) and integer K’, R

9. Problem Reduction • Let V’ = V • Let K’ = K • Let R = 2 • For each edge e = (vi,vj) in E construct an edge e’ = (vi,vj) with weight 1 • For each edge in the compliment of G construct an edge with weight eleventy-billion

10. Caveat • “True” instances of POIC are based on real-world locations • Vertices and edges conform to the constraints on points in 2D or 3D Euclidean space • My definition of POIC does not

11. Proof of Correctness • One to one mapping from v  V to v’  V’ • vi covers vk (VC sense) iff vi’ covers vk’ (POIC sense) • vi is covered by vk (VC sense) iff vi’ is covered by vk’ (POIC sense) • VC of G exists iff POIC of G’ exists