1. Efficient Closed Pattern Mining in Strongly Accessible Set Systems
Mario Boley, Tamás Horváth, Axel Poigné, Stefan Wrobel
Fraunhofer IAIS, Sankt Augustin & University of Bonn Germany

2. Closed Frequent Patterns
data mining definition: frequent patterns that cannot be further enlarged without changing their support
example: closed frequent itemsets
compact representation of frequent itemsets
number of frequent itemsets can be exponentially larger than that of closed frequent itemsets A B C D E 1

3. The Closed Set Mining (CSM) Problem
in many cases, the closed frequent pattern mining problem is an instance of the Closed Set Mining problem:
Given
a finite ground set E,
a membership oracle MF: 2E  {0,1} defining a family F  2E with   F,
and a closure operator : F  F,
list the family (F) of closed sets, i.e., (F) = { (X) : XF } .
example: closed frequent itemsets
E: set of items
F: family of frequent itemsets
oracle MF decides whether an itemset is frequent
for every XF, (X) is the intersection of the transactions containing X

4. Results on Mining Closed Sets
several positive complexity results in different communities, e.g.,
Formal Concept Analysis (Wille, ’82, Ganter & Wille, ’99)
e.g., polynomial delay algorithm (Ganter & Reuter, ‘91)
assumption: F is the power set of E
Closed Frequent Itemset Mining (Pasquier, Bastide, Taouil, & Lakhal, ’99)
e.g., incremental polynomial time algorithm (Boros, Gurvich, Khachiyan, & Makino, ’03)
assumption: F is an independence system (closed under taking subsets)
question for this talk: What about closed frequent pattern mining problems with set systems not even closed under intersection?

5. Example: Track Mining
given a database of GPS-based recordings of spatio-temporal movements (tracks), list the set of closed frequent connected subgraphs of movements of people or cars in a street network
closed frequent connected subgraphs: ‘homogeneous’ connected subnetworks
model:
street network: undirected graph G = (V,E)
tracks: subsets of E
embedding operator: subset relation
easy to decide
underlying set system: F = { X  E : X is frequent and connected }
F is not closed under intersection

6. Example frequency threshold = 1 F = { {a,b,c,d,e,f,g,h,k}, {a,i,j,k} }
intersection is not connected a i b j c g h k d f e

7. Generators and Inductive Generators
C: -closed element, i.e., C  (F)
generator of C:
X  F such that C = (X)
inductive generator of C:
C’ {e}  F such that
- C’ is -closed,
- C = (C’  {e}) for some e  E \ C’
example:
() = , (a) = (ac) = ac, (ab) = (abd) = (abcd) = abcd
abcd has a generator (e.g., ab), but no inductive generator
ac has an inductive generator (i.e., a)

8. The Closed Set Mining (CSM) Problem
Lemma: The CSM problem can be solved with polynomial delay if
the membership oracle and the closure operator can be computed in polynomial time and
for every -closed set except (), there exists an inductive generator.
proof sketch:
traverse the digraph of -closed sets in depth-first manner
(C’,C) is an edge iff there is an e  E \ C’ such that C’ {e} is an inductive generator of C
-closed sets are stored in prefix trees

9. Main Result for Strongly Accessible Set Systems
set system (E,F) is strongly accessible if F and for every X,Y  F satisfying X  Y, there exists an eY \ X such that X  {e}  F .
there is a sequence X=X0, X1,…,Xk=Y s.t. |Xi \ Xi-1| = 1 for i = 1,…,k
Thm: For any finite strongly accessible set system (E,F)
(i) given by a polynomial membership oracle and
(ii) for any polynomially computable closure operator  : F  F,
(F) can be listed with polynomial delay.
proof sketch:
show that every -closed set has an inductive generator
apply the previous lemma

10. Appl. 1: Closed Frequent Itemset Mining
Thm: The closed frequent itemset mining problem can be solved with polynomial delay.
proof sketch:
family of frequent itemsets is an independence system
strongly accessible
frequency can be decided in polynomial time
set system is given by a polynomial membership oracle
closure operator: (X) = intersection of the transactions containing X
can be computed in polynomial time

11. Appl. 2: Closed Frequent Connected Subgraph Mining
Given an undirected graph G = (V,E), a transaction database D of subgraphs of G, and an integer frequency threshold t > 0,
list the family of closed frequent connected subgraphs of D.
Thm: The above problem can be solved with polynomial delay.
proof sketch:
F: set of frequent connected subgraphs of G
not closed under intersection
F is strongly accessible
membership is decidable in polynomial time
closure of a frequent connected subgraph X: largest connected supergraph of X in the intersection of the transactions containing X.
it is indeed a closure operator and can be computed in polynomial time

12. Closed Frequent Connected Subgraph Mining
Example: frequency threshold = 2 …

13. Appl. 3: Closed Frequent Subpath Mining
data mining definition: a path P is closed frequent if it is frequent and has strictly larger support than any path P’ containing P
there is no closure operator corresponding to this definition
example:
D = { abc }
frequency threshold = 1
F = { ,a, b, c, ab, ac, bc }
closed 1-frequent paths: C = { ab, ac, bc }
suppose there is a closure operator  s.t. (F) = C
because of extensivity: (a) must be ab or ac, say ab
(a) = ab is not a subset of (ac) = ac contradicting monotonicity b a c

14. Appl. 3: Closed Frequent Subpath Mining
alternative definition: let P be a path in G
compute the intersection GP of the transactions containing P
return the intersection of the maximal paths in GP that contain P
example:
D = { abc }
frequency threshold = 1
F = { ,a, b, c, ab, ac, bc }
closed 1-frequent paths: C’ = { a, b, c, ab, ac, bc }
Thm: The set of closed frequent path w.r.t. the alternative definition can be listed with polynomial delay. b a c

15. An Open Problem accessible set systems:
for all X  F \ {} there is an e  X such that X \ {e}  F
there is a sequence  =X0, X1,…, Xk=X s.t. |Xi \ Xi-1| = 1 for i = 1,…,k
abcd has no inductive generator
Question: Can the positive result on strongly accessible set systems be generalized to accessible set systems?