1. Multi-Agent explore and retrieve Chris Enedah Andreas Edlund

2. The problem ● Given: – A polygonal world in two dimensions. – Multiple disc robots. – Unknown number of artifacts at unknown positions. – A dropoff location. ● The problem: – Find and retrieve all artifacts and drop them off near the droppoff location. ● Applications: – Crashed airplane

3. Problem parts ● Roadmap placement ● Object retrieval

4. Roadmap placement ● Find the lowest number of guards in space that together can see every portion of space. ● Compute collision-free paths between these guards.

5. Sample space ● Uses rejection sampling.

6. Compute visible space ● For each node, compute the visible portions of all edges in the scene.

7. Compute visible space ● For each tripple (guard, cover edge, target edge). – First set the whole target edge to be one visible edge sgement. Guard Target edge Cover edge

8. Compute visible space ● Cast a “shadow” from the cover edge. Visible edge segments

9. Compute visible space ● Cut the visible edge segment using the shadow.

10. Result of the visibility computation ● Every node now has a set of visible edge segments. ● Each edge segment refers only to one guard. Edge Guard 1 Guard 2 Guard 3

11. Decomposition of the visible edge segments ● The edge is decomposed into smaller edge segments. Edge Guard 1 Guard 2 Guard 3 123456789 Guard1 = {1,4,5,9} Guard2={1,2,8,9} Guard3={5,6}

12. Set cover ● The whole Art-gallery problem is reduced to a set cover problem. ● The set cover problem: – Find the smallest number of sets in R such that all elements in X is accounted for. R={{1, 4, 6, 7, 9}, {3, 5, 6}, {2, 5, 6, 8, 9}, {1, 3, 6, 10} X={1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

13. Set cover ● The set cover problem is NP-hard so coming up with a perfect solution is infeasible. ● Use a greedy algorithm instead. R={{1, 4, 6, 7, 9}, {3, 5, 6}, {2, 5, 6, 8, 9}, {1, 3, 6, 10} X={1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

14. Set cover ● Select the largest set in R and remove it. ● Remove all elements in this set from X. ● Remove all elements in this set from all other sets in R. R={{3, 5}, {2, 5, 8,}, {3, 10} X={2, 3, 5, 8, 10} Selected: {1, 4, 6, 7, 9}

15. Set cover ● Same thing again. R={{3}, {3, 10} X={3, 10} Selected: {1, 4, 6, 7, 9} {2, 5, 8}

16. Set cover ● And again... R={} X={} Selected: {1, 4, 6, 7, 9} {2, 5, 8} {3, 10}

17. Placing the roadmap ● We want every pair of guards to connect to each other. – Place additional nodes at all the convex vertices of the scene. – Do A*-searches between every pair of guards.

18. Placing the roadmap

20. Problems ● The visibility computation is slow – O(mn 2 ) – m – number of samples – n – number of edges ● The rest “should” have the same time complexity as in the paper.