On the Minimum Node and Edge Searching Spanning Tree Problems

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 EngineeringNational Dong Hwa University, Hualien 974, Taiwan

#4049

2

Outline Introduction The Hardness of MNSST and MESST Approximation Algorithms Conclusion

#4049

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

#4049

4

Introduction Examples for Node Searching Problem

3

2 2

3

2 2 2 2

#4049

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 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

#4049

6

Introduction Examples for Edge Searching Problem

3

2 2

2

2

#4049

7

Introduction The Minimum Node (Edge) Searching Spanning Tree

Problem

#4049

8

Introduction Node Searching Problem on Trees

Branch

u u

#4049

9

Introduction Edge Searching Problem on Trees

Branch

u u

#4049

10

Introduction Node (Edge) Searching Problem on Trees

Hub

uk

k

k

k+1

#4049

11

Introduction Node (Edge) Searching Problem on Trees

Avenue

u v

#4049

MNSST (MESST) IS NP-HARD

12

#4049

13

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.

s1 s2 s3

x1 x2 y1 y2 z1 z2

m = 3

n = 2

s1 s2 s3

x1 x2 y1 y2 z1 z2

#4049

14

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.

Main theorem:The 4-searchable node searching spanning tree problem is NP-hard.

#4049

15


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

#4049

16


44

33

The resulting graph is a bipartite graph.

3n

3n

m

n

7n

2×22+1

#4049

17


33

4 4

3 3

#4049

18


33

4 4

3 34

5

#4049

19


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.

Corollary:The 4-searchable node searching spanning tree problem on bipartite graphs is NP-hard.

#4049

20

4-Searchable Edge Searching Spanning Tree Problem

44

33

The resulting graph is a bipartite graph.

6n

3n

m + n

n

10n

2×31+1

#4049

21


44

33

For any tree T with minimum degree 3, ns(T) = es(T).

#4049

22


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.

Corollary:The 4-searchable edge searching spanning tree problem on bipartite graphs is NP-hard.

#4049

APPROXIMATION ALGORITHMS

23

#4049

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) = maxvV\{u} D(u, v).

Let u be the vertex s.t. L(u) = r = minvV 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.

#4049

25


4

3 3

3

2 2 22

2

2

2

Approximation solution

2

2

2

2

#4049

26


32

222

2

2 2 2

Optimal solution

#4049

27

Approximation Ratio by Hub Property

u

r - 1

#4049

28

Approximation Ratio by Hub Property

u

#4049

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) = maxvV\{u} D(u, v).

Let P be the path u~v s.t. L(u) = d = maxvV 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.

#4049

30


2

2

2

2

2

2 2

2

2

2

2

3

2

3

Approximation solution

#4049

31


Intuitively, the approximation ratio should be better than the previous one.

#4049

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.

#4049

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?

#4049

34

Call For PapersInternational Workshop on

Theories and Applications of Graphsin conjunction with ICSEC 2014

July 30, 2014, Khon Kaen, 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

http://itag2014.ntcb.edu.tw/

#4049

35

Thank you very much.

On the Minimum Node and Edge Searching Spanning Tree Problems

Documents