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Integrating Class Hierarchies
Yuzhong Qu
NJVR
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2,996 vocabularies
From 261 PLDs (many are from w3.org)
455,718 terms
396,023 classes, 59,868 properties, (many are in YAGO NS)
Instantiation found for
115,707 classes (29.2%), e.g. foaf:Person
25,963 properties (43.4%), e.g. dc:creator
1,874 vocabularies (62.6%)
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Select Vocabulary
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Class and property
Instantiated classes //and their ancestors
The amount of instantiation, e.g. k  10 (100?)
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Instantiated Class Hierarchy
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A
D
C
e1
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Homomorphism
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Let M ={S1, S2 ,…} be a partially ordered set (or poset),
and so does N= {C1, C2 ,…}
H:MN be a functional relation from M to N (partial?)
Si  Sj  H(Si)  H(Sj)
Note
Merging class hierarchies (taxonomies)
Abstractive summary of a given class hierarchy
|Range H|  K
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Distance
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C
S
H
|| H || max i {dist ( Si , H ( Si ))}
 max i {1  p( Si , H ( Si ))}
|| I || 0
|| H  F |||| H ||  || F ||
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Distance
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 max{dist (S , C ) | H (S )  C }
k
k
k
1
dist ( Si , H ( Si ))

m i
 p(S ) * dist (S , H (S ))
i
i
i
i
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Merge
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S
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Summary of instances (class hierarchy)
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Summary of instances (class hierarchy)
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
| Range(H) |  K
?
Si  H ( Si )

Minimize
 p(S ) * dist (S , H (S ))
i
i
i
i
OR
Maximize
 p(S ) p(S , H (S ))
i
i
i
i
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Instance category and taxonomy
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A
D
C
leaf node is weighted
e1
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Related Problem
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Huffman Coding
Minimum-cost flow problem
(Directed) Steiner Tree
Node-weighted Steiner Tree
(Weighted)Vertex Cover
(Weighted) Dominating Set
Maximum coverage problem (select no more than K sets)
Weighted version (elements are weighted)
Minimum Set Cover
Weighted version (sets are weighted)
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Huffman Coding (Minimum weighted path length)
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Huffman D A. A method for the construction of minimum-redundancy
codes. Proceedings of the IRE, 1952, 40(9): 1098-1101.
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Minimum-cost flow problem
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Given a directed graph with source s and sink t, where
each edge (u,v) has capacity c(u, v), flow f(u, v), and
cost a(u, v). You are required to send an amount of
flow d from s to t.
Minimize
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Minimum Steiner Tree
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Given a set V of points (vertices), interconnect them by a network
(graph) of shortest length, where the length is the sum of the
lengths of all edges.
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Minimum Steiner Tree
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Given an edge-weighted graph G = (V, E, w) and a subset S ⊆ V of
required vertices.
A Steiner tree is a tree in G that spans all vertices of S. The task is
to find a minimum-weight Steiner tree.
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Dominating Set problem
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A dominating set for a graph G = (V, E) is
a subset D of V such that every vertex not
in D is adjacent to at least one member of D.
The minimum dominating set is NP-hard
Its decision version is a classical NPcomplete decision problem
the problem is not fixed-parameter tractable in
the sense that no algorithm with running
time f(k)nO(1) for any function f exists unless
the W-hierarchy collapses to FPT=W[2].
if the input graph is planar, the problem
remains NP-hard, but a fixed-parameter
algorithm is known.
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Vertex Cover problem
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A vertex cover of a graph is a set of vertices such that each edge
of the graph is incident to at least one vertex of the set.
The minimum vertex cover is NP-hard
Its decision version, the vertex cover problem, was one of Karp's
21 NP-complete problems
“if G has a vertex cover of k vertices” is fixed-parameter tractable
O(kn + 1.2852k)
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Other Techniques
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Graph summarization
Graph edit distance
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Reference (Minimum cost flow)
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James B. Orlin. A polynomial time primal network simplex
algorithm for minimum cost flows. Mathematical Programming.
1997(78): 109–129.
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Reference (Steiner Tree)
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Klein P, Ravi R. A nearly best-possible approximation algorithm
for node-weighted Steiner trees. Journal of Algorithms, 1995,
19(1): 104-115.
Zelikovsky A. A series of approximation algorithms for the
acyclic directed Steiner tree problem. Algorithmica, 1997,
18(1): 99-110.
Charikar M, Chekuri C, Cheung T, et al. Approximation
algorithms for directed Steiner problems. Proceedings of the
ninth annual ACM-SIAM symposium on Discrete algorithms.
1998: 192-200.
Zosin L, Khuller S. On directed Steiner trees. Proceedings of
the thirteenth annual ACM-SIAM symposium on Discrete
algorithms. 2002: 59-63.
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Reference (Vertex Cover)
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Niedermeier R, Rossmanith P. On efficient fixed-parameter
algorithms for weighted vertex cover. Journal of Algorithms,
2003, 47(2): 63-77.
White L J, Gillenson M L. An efficient algorithm for minimum
k-covers in weighted graphs. Mathematical Programming,
1975, 8(1): 20-42.
Chen J, Kanj I A, Xia G. Improved parameterized upper bounds
for vertex cover. Mathematical Foundations of Computer
Science 2006. Springer Berlin Heidelberg, 2006: 238-249.
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Reference (Graph Summarization)
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Navlakha S, Rastogi R, Shrivastava N. Graph summarization
with bounded error. Proceedings of the 2008 ACM SIGMOD
international conference on Management of data. ACM, 2008:
419-432.
Tian Y, Hankins R A, Patel J M. Efficient aggregation for graph
summarization. Proceedings of the 2008 ACM SIGMOD
international conference on Management of data. ACM, 2008:
567-580.
Zhang N, Tian Y, Patel J M. Discovery-driven graph
summarization. Data Engineering (ICDE), 2010 IEEE 26th
International Conference on. IEEE, 2010: 880-891.
Gao X, Xiao B, Tao D, et al. A survey of graph edit distance.
Pattern Analysis and applications, 2010, 13(1): 113-129.
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Reference (Document Summarization)
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Celikyilmaz A, Hakkani-Tur D. A hybrid hierarchical model
for multi-document summarization. ACL 2010: 815-824.
Shen C, Li T. Multi-document summarization via the
minimum dominating set. COLING 2010: 984-992
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Acknowledgement
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Q&A
Discussion
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