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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

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#4049 Outline Introduction The Hardness of MNSST and MESST Approximation Algorithms Conclusion 2

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#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

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

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#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

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

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

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#4049 Introduction Node Searching Problem on Trees Branch 8 uu

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

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

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

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#4049 MNSST (MESST) IS NP-HARD 12

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#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

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#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.

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

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#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

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

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

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#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.

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#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

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

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#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.

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#4049 APPROXIMATION ALGORITHMS 23

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#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

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

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

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

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#4049 Approximation Ratio by Hub Property 28 u

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#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

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

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

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#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

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#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

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#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

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#4049 Thank you very much. 35

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