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
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Outline Introduction The Hardness of MNSST and MESST Approximation Algorithms Conclusion
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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
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Introduction Examples for Node Searching Problem
3
2 2
3
2 2 2 2
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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
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Introduction Examples for Edge Searching Problem
3
2 2
2
2
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Introduction The Minimum Node (Edge) Searching Spanning Tree
Problem
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Introduction Node Searching Problem on Trees
Branch
u u
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Introduction Edge Searching Problem on Trees
Branch
u u
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Introduction Node (Edge) Searching Problem on Trees
Hub
uk
k
k
k+1
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Introduction Node (Edge) Searching Problem on Trees
Avenue
u v
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MNSST (MESST) IS NP-HARD
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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
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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.
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4-Searchable Node Searching Spanning Tree Problem
Proof.3-Dimension Matching Problem 4-Searchable Node Searching Spanning Tree Problem
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4-Searchable Node Searching Spanning Tree Problem
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The resulting graph is a bipartite graph.
3n
3n
m
n
7n
2×22+1
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4-Searchable Node Searching Spanning Tree Problem
33
4 4
3 3
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4-Searchable Node Searching Spanning Tree Problem
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4 4
3 3
4
5
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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.
Corollary:The 4-searchable node searching spanning tree problem on bipartite graphs is NP-hard.
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4-Searchable Edge Searching Spanning Tree Problem
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The resulting graph is a bipartite graph.
6n
3n
m + n
n
10n
2×31+1
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4-Searchable Edge Searching Spanning Tree Problem
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For any tree T with minimum degree 3, ns(T) = es(T).
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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.
Corollary:The 4-searchable edge searching spanning tree problem on bipartite graphs is NP-hard.
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APPROXIMATION ALGORITHMS
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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.
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Approximation Algorithm by Hub Property
4
3 3
3
2 2 22
2
2
2
Approximation solution
2
2
2
2
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Approximation Algorithm by Hub Property
32
222
2
2 2 2
Optimal solution
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Approximation Ratio by Hub Property
u
r - 1
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Approximation Ratio by Hub Property
u
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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.
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Approximation Algorithm by Avenue Property
2
2
2
2
2
2 2
2
2
2
2
3
2
3
Approximation solution
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Approximation Algorithm by Avenue Property
Intuitively, the approximation ratio should be better than the previous one.
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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.
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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?
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Call For Papers
International Workshop onTheories and Applications of Graphs
in conjunction with ICSEC 2014July 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
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Thank you very much.
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