Download presentation

Presentation is loading. Please wait.

Published byShannon Humphreys Modified over 3 years ago

1
演 算 法 實 驗 室演 算 法 實 驗 室 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
#4049 Outline Introduction The Hardness of MNSST and MESST Approximation Algorithms Conclusion 2

3
#4049 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 3

4
#4049 Introduction Examples for Node Searching Problem 4 3 2 2 3 2 2 2 2

5
#4049 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 slided by 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 5

6
#4049 Introduction Examples for Edge Searching Problem 6 3 2 2 2 2

7
#4049 Introduction The Minimum Node (Edge) Searching Spanning Tree Problem 7

8
#4049 Introduction Node Searching Problem on Trees Branch 8 uu

9
#4049 Introduction Edge Searching Problem on Trees Branch 9 u u

10
#4049 Introduction Node (Edge) Searching Problem on Trees Hub 10 u k k k k+1

11
#4049 Introduction Node (Edge) Searching Problem on Trees Avenue 11 uv

12
#4049 MNSST (MESST) IS NP-HARD 12

13
#4049 3-Dimension Matching Problem 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. 13 s1s1 s2s2 s3s3 x1x1 x2x2 y1y1 y2y2 z1z1 z2z2 m = 3 n = 2 s1s1 s2s2 s3s3 x1x1 x2x2 y1y1 y2y2 z1z1 z2z2

14
#4049 4-Searchable Node Searching Spanning Tree Problem Given a simple connected undirected graph G=(V, E), determine if it has a spanning tree whose node-search number is 4. 14 Main theorem: The 4-searchable node searching spanning tree problem is NP-hard.

15
#4049 4-Searchable Node Searching Spanning Tree Problem Proof. 3-Dimension Matching Problem 4-Searchable Node Searching Spanning Tree Problem 15

16
#4049 4-Searchable Node Searching Spanning Tree Problem 16 44 3 3 The resulting graph is a bipartite graph. 3n3n 3n3n m n 7n7n 2×22+1

17
#4049 4-Searchable Node Searching Spanning Tree Problem 17 3 3 44 33

18
#4049 4-Searchable Node Searching Spanning Tree Problem 18 3 3 44 33 4 5

19
#4049 4-Searchable Node Searching Spanning Tree Problem 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. 19 Corollary: The 4-searchable node searching spanning tree problem on bipartite graphs is NP-hard.

20
#4049 4-Searchable Edge Searching Spanning Tree Problem 20 44 3 3 The resulting graph is a bipartite graph. 6n6n 3n3n m + n n 10n 2×31+1

21
#4049 4-Searchable Edge Searching Spanning Tree Problem 21 44 3 3 For any tree T with minimum degree 3, ns(T) = es(T).

22
#4049 4-Searchable Edge Searching Spanning Tree Problem 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. 22 Corollary: The 4-searchable edge searching spanning tree problem on bipartite graphs is NP-hard.

23
#4049 APPROXIMATION ALGORITHMS 23

24
#4049 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) = max v V\{u} D(u, v). Let u be the vertex s.t. L(u) = r = min v V L(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. 24

25
#4049 Approximation Algorithm by Hub Property 25 4 3 3 3 2 22 2 2 2 2 Approximation solution 2 2 2 2

26
#4049 Approximation Algorithm by Hub Property 26 3 2 2 22 2 2 2 2 Optimal solution

27
#4049 Approximation Ratio by Hub Property 27 u r - 1

28
#4049 Approximation Ratio by Hub Property 28 u

29
#4049 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) = max v V\{u} D(u, v). Let P be the path u~v s.t. L(u) = d = max v V L(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. 29

30
#4049 Approximation Algorithm by Avenue Property 30 2 2 2 2 2 2 2 2 2 2 2 3 2 3 Approximation solution

31
#4049 Approximation Algorithm by Avenue Property 31 Intuitively, the approximation ratio should be better than the previous one.

32
#4049 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. 32

33
#4049 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? 33

34
#4049 Call For Papers International Workshop on Theories and Applications of Graphs in conjunction with ICSEC 2014 July 30, 2014, Khon Kaen, Thailand Website: http://itag2014.ntcb.edu.twhttp://itag2014.ntcb.edu.tw Important Dates: Submission: May 1, 2014 Notification: June 1, 2014 Final version: June 15, 2014 Registration: July 1, 2014 34

35
#4049 Thank you very much. 35

Similar presentations

OK

1 Approximability Results for Induced Matchings in Graphs David Manlove University of Glasgow Joint work with Billy Duckworth Michele Zito Macquarie University.

1 Approximability Results for Induced Matchings in Graphs David Manlove University of Glasgow Joint work with Billy Duckworth Michele Zito Macquarie University.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on statistics in maths cheating Ppt on ministry of corporate affairs new delhi Ppt on classical dances of india Ppt on principles of object-oriented programming php Ppt on 21st century skills common Ppt on 2d transformation in computer graphics by baker Ppt on product specification template Ppt on question tags sentence Ppt on mobile tv seminar Poster template free download ppt on pollution