On the Minimum Node and Edge Searching Spanning Tree Problems

1. On the Minimum Node and Edge Searching Spanning Tree Problems Sheng-Lung Peng Department of Computer Science and Information Engineering National Dong Hwa University, Hualien 974, Taiwan

2. Outline Introduction The Hardness of MNSST and MESST Approximation Algorithms Conclusion

3. Introduction • Node Searching Problem • Placing a searcher on a vertex • Removing a searcher from a vertex • A contaminated edge is clear if both of its end-vertices contain searchers • The objective is to clear the graph by using the minimum number of searchers, denoted as ns(G) for a graph G • Equivalent to the gate matrix layout, interval thickness, pathwidth, vertex separation, and narrowness problems

4. Introduction 3 3 2 2 2 2 2 2 Examples for Node Searching Problem

5. Introduction • Edge Searching Problem • Placing a searcher on a vertex • Removing a searcher from a vertex • Moving a searcher from a vertex along an edge • A contaminated edge is clear if it is slidedby a searcher • The objective is to clear the graph by using the minimum number of searchers, denoted as es(G) for a graph G • ns(G) – 1  es(G)  ns(G) + 1 for any graph G

6. Introduction 3 2 2 2 2 Examples for Edge Searching Problem

7. Introduction The Minimum Node (Edge) Searching Spanning Tree Problem

8. Introduction u u • Node Searching Problem on Trees • Branch

9. Introduction u u • Edge Searching Problem on Trees • Branch

10. Introduction k+1 k u k k • Node (Edge) Searching Problem on Trees • Hub

11. Introduction u v • Node (Edge) Searching Problem on Trees • Avenue

12. MNSST (MESST) is NP-hard

13. 3-Dimension Matching Problem m = 3 s1 s2 s3 s1 s2 s3 x1 x2 y1 y2 z1 z2 x1 x2 y1 y2 z1 z2 n = 2 Given mutually disjoint sets X, Y, and Z, |X| = |Y| = |Z| = n, and a set S = {(x, y, z) | x  X, y  Y, z  Z}, |S| = m, determine if there is a matching M with |M| = n, where M is called a matching if M  S and no elements in M agree in any coordinate.

14. 4-Searchable Node Searching Spanning Tree Problem Main theorem: The 4-searchable node searching spanning tree problem is NP-hard. Given a simple connected undirected graph G=(V, E), determine if it has a spanning tree whose node-search number is 4.

15. 4-Searchable Node Searching Spanning Tree Problem Proof.3-Dimension Matching Problem  4-Searchable Node Searching Spanning Tree Problem

16. 4-Searchable Node Searching Spanning Tree Problem 4 4 2×22+1 3 3 7n n m 3n 3n The resulting graph is a bipartite graph.

17. 4-Searchable Node Searching Spanning Tree Problem 4 4 3 3 3 3

18. 4-Searchable Node Searching Spanning Tree Problem 5 4 4 3 3 4 3 3

19. 4-Searchable Node Searching Spanning Tree Problem Corollary: The 4-searchable node searching spanning tree problem on bipartite graphs is NP-hard. Given a simple connected undirected graph G=(V, E), the problem of determining if it has a spanning tree whose node-search number is 4 is NP-hard.

20. 4-Searchable Edge Searching Spanning Tree Problem 4 4 2×31+1 3 3 10n n m + n 3n 6n The resulting graph is a bipartite graph.

21. 4-Searchable Edge Searching Spanning Tree Problem 4 4 3 3 For any tree T with minimum degree 3, ns(T) = es(T).

22. 4-Searchable Edge Searching Spanning Tree Problem Corollary: The 4-searchable edge searching spanning tree problem on bipartite graphs is NP-hard. Given a simple connected undirected graph G=(V, E), the problem of determining if it has a spanning tree whose edge-search number is 4 is NP-hard.

23. Approximation Algorithms

24. Approximation Algorithm by Hub Property Given a graph G = (V, E), for each u  V, compute the shortest distance D(u, v) for every other vertex v. Let L(u) = maxvV\{u}D(u, v). Let u be the vertex s.t.L(u) = r = minvVL(v). Note that r is the radius of G and u is the center of G. Compute a spanning tree T by BFS (breadth first search) starting from vertex u. Compute ns(T) (es(T)) using an optimal algorithm.

25. Approximation Algorithm by Hub Property 2 2 2 3 2 2 4 2 3 3 2 2 2 2 2 Approximation solution

26. Approximation Algorithm by Hub Property 2 2 2 2 2 3 2 2 2 Optimal solution

27. Approximation Ratio by Hub Property u r - 1

28. Approximation Ratio by Hub Property u

29. Approximation Algorithm by Avenue Property Given a graph G = (V, E), for each u  V, compute the shortest distance D(u, v) for every other vertex v. Let L(u) = maxvV\{u}D(u, v). Let P be the path u~vs.t.L(u) = d = maxvVL(v) and P passes a center of G. Note that d is the diameter of G. Compute a spanning tree T by BFS (breadth first search) starting from the path P. Compute ns(T) (es(T)) using an optimal algorithm.

30. Approximation Algorithm by Avenue Property 2 2 2 2 3 2 2 3 2 2 2 2 2 2 Approximation solution

31. Approximation Algorithm by Avenue Property Intuitively, the approximation ratio should be better than the previous one.

32. Conclusion We prove that the minimum node (edge) searching spanning tree problem is NP-hard even on bipartite graphs. We propose two approximation algorithms for the minimum node (edge) searching spanning tree problem.

33. Future Work The lower bound for an n-vertex tree is too low in the analysis of Algorithm 1 (by hub property). Can it be improved? What is the tight approximation ratio of Algorithm 2 (by avenue property)? What is the time complexity for the problems on some special classes of graphs (e.g., chordal graphs)? (It is easy for AT-free graphs.) Are the graphs with 2 (or 3)-searchable spanning trees easy to be recognized?

34. Call For Papers International Workshop onTheories and Applications of Graphsin conjunction with ICSEC 2014July 30, 2014, KhonKaen, Thailand • Website: http://itag2014.ntcb.edu.tw • Important Dates: • Submission: May 1, 2014 • Notification: June 1, 2014 • Final version: June 15, 2014 • Registration: July 1, 2014

35. Thank you very much.