Parameterized Complexity Results
for Probabilistic Network
Structure Learning
Stefan Szeider
Vienna University of Technology, Austria
Workshop on Applications of
Parameterized Algorithms and Complexity
APAC 2012
Warwick, UK, July 8, 2012
Joint work with:
Serge Gaspers
UNSW, Sydney
Papers:
•
Mikko Koivisto
U Helsinki
Mathieu Liedloff
U d’Orleans
Sebastian Ordyniak
Masaryk U Brno
Algorithms and Complexity Results for Exact Bayesian Structure Learning. Ordyniak and
Szeider. Conference on Uncertainty in Artificial Intelligence (UAI 2010).
•
An Improved Dynamic Progamming Algorithm for Exact Bayesian Network Structure
Learning. Ordyniak and Szeider. NIPS Workshop on Discrete Optimization in Machine
Learning (DISCML 2011).
•
On Finding Optimal Polytrees. Gaspers, Koivisto, Liedloff, Ordyniak, and Szeider.
Conference on Artificial Intelligence (AAAI 2012).
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2
Stefan Szeider
Probabilistic Networks
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•
Bayesian Networks (BNs) were
introduced by Judea Perl in 1985
(2011 Turing Award Winner)
•
A BN is a DAG D=(V,A) plus tables
associated with the nodes of the
network
•
In addition to Bayesian networks, also
other probabilistic networks have been
considered: Markov Random Fields,
Factor Graphs, etc.
3
Judea Pearl
Stefan Szeider
Example:
A Bayesian
Network
Rain
Sprinkler
T
F
F
0.4
0.6
T
0.01
0.99
Rain
Sprinkler
Sprinkler
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Rain
T
F
0.8
0.2
Rain
Grass Wet
T
F
F
F
0.8
0.2
F
T
0.01
0.99
T
F
0.9
0.1
T
T
0.99
0.01
4
Grass
Stefan Szeider
Applications
•
•
•
diagnosis
•
•
•
•
information retrieval
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computational biology
document
classification
image processing
decision support
etc.
A BN showing the main pathophysiological
relationships in the diagnostic reasoning
focused on a suspected pulmonary embolism
event
5
Stefan Szeider
Computational Problems
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6
Stefan Szeider
Computational Problems
• BN Reasoning: Given a BN, compute the
probability of a variable taking a specific value
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6
Stefan Szeider
Computational Problems
• BN Reasoning: Given a BN, compute the
probability of a variable taking a specific value
• BN Learning: Given a set of sample data, find
a BN that fits the data best.
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Stefan Szeider
Computational Problems
• BN Reasoning: Given a BN, compute the
probability of a variable taking a specific value
• BN Learning: Given a set of sample data, find
a BN that fits the data best.
• BN Structure Learning: Given sample
data, find the best DAG
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Stefan Szeider
Computational Problems
• BN Reasoning: Given a BN, compute the
probability of a variable taking a specific value
• BN Learning: Given a set of sample data, find
a BN that fits the data best.
• BN Structure Learning: Given sample
data, find the best DAG
• BN Parameter Learning: Given sample
data and a DAG, find the best
probability tables
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Stefan Szeider
Computational Problems
• BN Reasoning: Given a BN, compute the
probability of a variable taking a specific value
• BN Learning: Given a set of sample data, find
a BN that fits the data best.
• BN Structure Learning: Given sample
data, find the best DAG
• BN Parameter Learning: Given sample
data and a DAG, find the best
probability tables
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Stefan Szeider
BN Learning and Local Scores
BN
Sample Data
n1 n2
0 1
1 1
* 1
0 *
1 0
....
....
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n3
0
0
0
1
1
n4
1
1
0
1
1
n5
1
1
*
*
0
n6
0
0
1
1
1
n2
n1
....
....
....
....
....
....
n4
n3
n6
n5
7
...
Stefan Szeider
BN Learning and Local Scores
Local Score Function
f(n,P)=score of node n with P ⊆V
as its in-neighbors (parents)
BN
Sample Data
n1 n2
0 1
1 1
* 1
0 *
1 0
....
....
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n3
0
0
0
1
1
n4
1
1
0
1
1
n5
1
1
*
*
0
n6
0
0
1
1
1
n2
n1
....
....
....
....
....
....
n4
n3
n6
n5
7
...
Stefan Szeider
Our Combinatorial Model
• BN Structure Learning:
Input: a set N of nodes and a local sore
function f (by explicit listing of nonzero
tuples)
Task: find a DAG D=(N,A) such that
the sum of f(n,PD(n)) over all nodes n
is maximum.
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Stefan Szeider
Exact Bayesian Network Structure Learning
Example
Problem: Given a local score function f over a set of nodes N find a
Exact Bayesian Network Structure Learning
An optimal BN
structure
Problem: Given a local score function f over a set of nodes N find a
Idea: Restrict
Bayesian
network
(BN)
on N score
whose
score is maximum
to f . gr
Bayesian
network
(BN) on
N whose
is maximum
with respectwith
to f .respect
undirected
A local score function
Pf (n, P)f (n, P)
n nP
a {d}
a {d} 1
1
a {b,ac,{b,
d} c, 0.5
d} 0.5
b {a,
b f }{a, f }1
1
c {e}
1
!
7
c {e}
1
d
;
1
d
; 1
1
e {b}
e {b} 1
1
f {d}
f d}{d} 1
1
g {c,
g {c, d}
A local score function f
a
a
A good su
an optima
IdA super-s
efficiently
the tiling a
I
b
7!
e
1
c
b
f
e
d
c
g
f
An optimal BN for f
A local
scoreStructure
functionLearning
f
Exact Bayesian
Network
is NP-hard.An optimal BN for
g
f
Heuristic Algorithms: find first an undirected graph, then greedily orient
Exact Bayesian Network Structure Learning is NP-hard.
the edges,
then doProgramming
local searchover
foraimproving
the orientation
Dynamic
Tree Decomposition
Idea: Ignore r
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Dynamic Programming over a Tree Decomposition
a, b, c, d
9
Stefan Szeider
NP-Hardness
Theorem: BN Structure Learning is NP-hard.
[Chickering 1996]
•
Proof: By reduction from FAS
(NP-h for max deg 4, [Karp 1972])
u
u
u’
f(x,{u})=1
v
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u’
x
x’
v
10
f(x’,{u’})=1
f(v,{x})=1
f(v,{x’})=1
f(v,{x,x’})=2
Stefan Szeider
Problem:An
Given
a local
optimal
DAGscore
for f function f (
on N whose score is maximum with resp
Exact Probabilistic Network Learning is NP-hard.
A local score function f
Very Few Polynomial Cases
1
2
v P fv(P)
3 {1}
1
Known Results
4 {1} 0.1
5 1{1} 0.5
2
5 {1, 2} 1
6 {4} 0.9
6 {3, 4} 1
5
3 7 {5} 40.9
7 {4, 5} 1
Theorem: An optimal branching
3
4
can be found in polynomial
time. (Chu and Liu 1965, Edmonds 1967)
6
7
A polytree
7!
3
5
A local score function f
6
7
Exact Probabilistic
Network
Learning is
A branching
A
Known Res
Restricting the structure of the resulting DAG has lead to the following
R
results:
h
1
2
i
I An optimal branching can found in polynomial time (Chu and Liu,
a
1965;Edmonds, 1967).
I Finding an optimal polytree is NP-hard (Dasgupta, 1999).
Theorem: Finding an optimal
polytree is NP-hard (Dasgupta 1999)
3
4
5
Our Result
Can parameters make Main
theResult: For every constant k an optimal k-branching can be found
6
7
in polynomial time.
problem tractable?
A polytree
1
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2
Restricting the structure of the resulting
Stefan Szeider
results:
Parameterized Complexity
• Identify a parameter k of the
problem
• Aim: solve the problem in time
f(k)nc
fixed-parameter tractable (FPT)
•
• W-classes:
XP: solvable in time nf(k)
FPT ⊆W[1] ⊆W[2]⊆···⊆ XP
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Downey and Fellows:
Parameterized Complexity
Springer 1999
Stefan Szeider
Parameters for BN
Structure Learning?
1. Parameterized by the treewidth (of the
super structure)
2. Parameterized Local Search
3. Parameterized by distance from being a
Branching
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Stefan Szeider
1.
BN Structure learning
parameterized by the treewidth
(of the super structure)
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Stefan Szeider
“poster” — 2011/12/9 — 21:33 — page 1 — #1
An Improved Dynam
Bayesian
3rd Workshop on Discrete Optimization in Machine Learning (DISCML’11), Sierra Nevada, Spain, De
Improved Dynamic
Programming Algorithm for E
TheAnSuper
Structure
Bayesian Network Structure Learning
“poster” — 2011/12/9 — 21:33 — page 1 — #1
rkshop on Discrete Optimization in Machine Learning (DISCML’11), Sierra Nevada, Spain, Dec 17, 2011
Se
Improved Dynamic Programming Algorithm for Exact
Bayesian Network Structure Learning
•
Sebastian Ordyniak and Stefan Szeider1
Super structure Sf of a local score function f is the undirected
graph Sf=(N,E) containing all edges that can participate in an
Bayesian
Network Structure Learning
optimal DAG [Perrier, Imoto, Exact
Miyano
2008]
Sebastian Ordyniak and Stefan Szeider1
Exact Bayesian Network Structure Learning
Super-structure
Problem:
Given
a
local
score
function
f
over
a
set
of nodes N find a
ture Learning
Super-structure
on NRestrict
whosethe
score
is to
maximum
with
respect
to f .
Problem: Given a local score function f over a setBayesian
of nodes Nnetwork
find a (BN)Idea:
search
BNs whose
skeleton
is contained
a set of nodes N find a
Idea: Restrict the search to BNs whose skeleton is contained inside an
Bayesian network (BN) on N whose score is maximum with respect to f .
undirected
maximum with respect to f .
undirected graph (the super-structure).
n
P
f (n, P)
a {d}
1
a {b, c, d} I 0.5
A good super-structure contains
b {a, f }
1 optimal solution.
an
c
d
b
c {e} I A1super-structure
7!can beb
efficiently
obtained, e.g., using
d
;
1
the
e {b}
1 tiling algorithm.
gf
f
{d}
1
e
1
An optimal BN for g
f {c, d}
a
s NP-hard.
A local score function f
n a P
f (n, P)
a
a
a {d}
1
a {b, c, d} 0.5
b {a, f }
1I A good super-structure contains
c
solution.b
d1 an optimal7!
c cc {e}
d
b
d
;
1I A super-structure can be
e {b}
1 efficiently obtained, e.g., using
f {d}
1 the tiling algorithm.
ef
e
g gf{c, d} g 1
f
An optimal BN for f
local score
function
super structure
Network Structure Learning is NP-hard.
Exact Bayesian
e Decomposition
A local
Dynamic Lower Bound Approach
Idea: Ignore records that do not represent optimal BNs.
score function f
BN
a
d
c
g
e
An optimal BN for f
f
Exact Bayesian Network Structure Learning
is NP-hard.
Dynamic
Lower Bound Approach
b, c, da TreeXDecomposition
X
Dynamic Programminga,over
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graph (the super-structure).
Idea: Ignore records that do not represent optimal BNs.
Dynamic
Programming over a Tree Decomposition
15
Stefan Szeider
Hardness
Theorem: BN Structure Learning is W[1]-hard when
parameterized by the treewidth of the super
structure.
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Stefan Szeider
Hardness
Theorem: BN Structure Learning is W[1]-hard when
parameterized by the treewidth of the super
structure.
Proof: by reduction from multi-colored clique (MCC).
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Stefan Szeider
Hardness
Theorem: BN Structure Learning is W[1]-hard when
parameterized by the treewidth of the super
structure.
Proof: by reduction from multi-colored clique (MCC).
v11
v12
v13
v11
v12
a13
v33
v21
v32
v22
v31
v23
7!
v13
a12
v33
v21
v32
v22
a23
v31
v23
MCC instance, k=3
Figure 7: An example for the graph G and the super-structure Sf according to the construction used in the proof of Theorem 3.
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Stefan Szeider
Hardness
Theorem: BN Structure Learning is W[1]-hard when
parameterized by the treewidth of the super
structure.
Proof: by reduction from multi-colored clique (MCC).
v11
v11 v12
v12 v13 v13
v11 v11
v12 v12
a13 a13
v33
7!
v21 v21
v33
v32
v32
v31
v22 v22
v31
v23 v23
MCC instance, k=3
7!v33
v13 v13
a12 a12
v33
v21 v21
v32 v32
v22 v22
a23 a23
v31 v31
v23 v23
Super Structure
An example
forgraph
the graph
G and
the super-structure
Sf according
to the
FigureFigure
7: An7:example
for the
G and
the super-structure
Sf according
to the
con-construction used in the proof of Theorem 3.
struction used in the proof of Theorem 3.
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12
v12
v13
v13
v11
v11
v
v121612
v13
v13
Stefan Szeider
Hardness
v11
v12
v13
v11
v12
v13
Theorem: BN Structure Learning is W[1]-hard when
parameterized by the treewidth of the super
7!
structure.
a13
v33
v21
v32
v33
v22
v31
a12
v21
v32
v23
v22
a23
v31
v23
Proof: by reduction from
(MCC).
Figure multi-colored
7: An example for the graph clique
G and the super-structure
S according to the conf
struction used in the proof of Theorem 3.
v11
v33
v11 v12
v12 v13 v13
7!
v21 v21
v33
v32
v32
v31
v22 v22
v31
v23 v23
7!v33
v12 v v13
v11 v12 v12
v11 v11
13 v13
v11
a13 a13
a13
a12 a12
vv3333
v21 v21v21
v32 vv3232
v22
v22 v22
a23 a23
v31 vv3131
v23
v23 v23
7!
v12
v13
a12
v33
v21
v32
v22
a23
v31
v23
G
D(E 0 )
An example
forgraph
the graph
G and
the super-structure
Sf according
to the
FigureFigure
7: An7:example
for the
G and
the super-structure
Sf according
to the
con-con0
struction used in the proof of Theorem 3.
struction used in the proof of Theorem 3.Figure 8: The example graph G from Figure 7 together with an edge set E , given as the
bold edges in the illustration of G, and the resulting dag D(E 0 ).
MCC instance, k=3
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Super Structure
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v11
v
v121612
v13
v13
BN corresponding to
clique
Stefan Szeider
XP-tractability
Theorem: BN Structure Learning is in XP when
parameterized by the treewidth of the super
structure.
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Stefan Szeider
XP-tractability
Theorem: BN Structure Learning is in XP when
parameterized by the treewidth of the super
structure.
Proof: dynamic programming along a nice tree decomposition
For each node U of the tree decomp. we compute a table T(U).
T(U) gives the the highest possible score any partial solution (below U)
can achieve
• for a given chosen parents of the nodes in the bag, and
• for a given reachability between nodes in the bag by directed paths.
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Stefan Szeider
XP-tractability
Theorem: BN Structure Learning is in XP when
parameterized by the treewidth of the super
structure.
Proof: dynamic programming along a nice tree decomposition
For each node U of the tree decomp. we compute a table T(U).
T(U) gives the the highest possible score any partial solution (below U)
can achieve
• for a given chosen parents of the nodes in the bag, and
• for a given reachability between nodes in the bag by directed paths.
Corollary: If the super-structure has bounded
degree, then we get fixed-parameter tractability
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Stefan Szeider
Experiments
• Alarm Network: 37 nodes
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Stefan Szeider
ms = 1
for all P 2 Pf (n) do
if P \ F = ; and ms  f (n, P ) then
ms = f (n, P )
return ms
for each n 2 Ut do
ub = ub + MaxScore(n, ;)
return ub
Experiments
hh
hh
• Bottleneck: huge tables:
• SharpLib for dynamic programming
• Dynamic Lower Bound approach (LB)
• Ongoing improvements
Figure 2: The pseudo-code for the functions MaxScore (Left) and UpperBound (Right).
compute the score according to the Akaike Information Criterion (AIC) [1], which is the default
scoring criterion provided by BNLEARN. It is however possible and straightforward to use any other
type of local score function as an input to the algorithm. The local score function together with the
super-structure is then fed into a C++ implementation of the algorithm from Section 4. For our implementation we use SHARPLIB [21], a library for decomposition-based algorithms. To compute the
tree decomposition of the super-structure the algorithm uses the Greedy-Fill-In heuristic [2]. Using
upper bound and lower bound heuristics from LIBTREEWIDTH [15] we found that the Greedy-Fill-In
heuristic is able to compute an optimal tree decomposition for the considered super-structures using
negligible time. For all our experiments we use a single Opteron 6176SE core with 128 GB of RAM.
Sf
SSKEL
STIL (0.01)
STIL (0.05)
STIL (0.1)
Characteristics of Sf
|V (Sf )| |E(Sf )|
(Sf ) tw(Sf )
37
46
6
3
37
62
7
5
37
63
7
5
37
65
7
5
running time (s)
no LB
LB
14
7
20785
6393
46560 16712
44554 16928
memory usage (MB)
no LB
LB
337
180
15679
5346
38525
18786
38520
18918
Table 1: The characteristics of the considered super-structures and the running time and memory usage needed
without (no LB) and with (LB) the dynamic lower bound approach.
We consider two kinds of super-structures as an input to the algorithm:
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• The skeleton SSKEL of the Alarm network. 19
Stefan Szeider
2.
BN Structure Learning
parameterized local search
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Stefan Szeider
Local Search
• Once we have a candidate BN, we can try
to find a better one by local editing.
• We consider three local editing operations:
ADD - add an arc
•
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DEL - delete an arc
REV - reversing an arc
We parameterize by the number k of edits
we can make in one step.
21
Stefan Szeider
k-Local Search
• Parameterizing LS problems by the number
of changes in one step has been suggested
by Fellows 2003
• Since then a series of paper on that
parameterized local search have been
published, considering, among others:
TSP, Max-SAT, Max-Cut,Vertex Cover,Weighted
SAT, Cycle Traversals, Stable Matchings, etc.
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Stefan Szeider
Results
operations complexity
Theorem:Except for {ADD}
and {DEL} all problems are
W[1]-hard.
Hardness even holds for instances
of bounded degree.
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ADD
PTIME
DEL
PTIME
REV
W[1]-h
ADD,DEL
W[1]-h
ADD,REV
W[1]-h
DEL,REV
W[1]-h
ADD,DEL,REV
W[1]-h
Stefan Szeider
Results
operations complexity
Theorem:Except for {ADD}
and {DEL} all problems are
W[1]-hard.
Hardness even holds for instances
of bounded degree.
ADD
PTIME
DEL
PTIME
REV
W[1]-h
ADD,DEL
W[1]-h
ADD,REV
W[1]-h
DEL,REV
W[1]-h
ADD,DEL,REV
W[1]-h
Does it help to bound the degree?
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Stefan Szeider
W[1]-hardness for bounded
degree
• All known parameterized LS problems
are FPT for instances of bounded degree.
• Almost all W[1]-hard problems are FPT
for graphs of bounded degree.
• Red/Blue Nonblocker isn’t an exception.
• We have now a new proof by reduction
from Independent Set (we don’t need the
reduced graph to have bounded degree).
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Stefan Szeider
Reduction from IS for {REV}
BN local search
Independent Set
v
w
u
v
w’
u’
v’
u
w
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Stefan Szeider
Reduction from IS for {REV}
BN local search
Independent Set
v
w
u
v
w’
u’
v’
u
w
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Stefan Szeider
Reduction from IS for {REV}
BN local search
Independent Set
v
w
u
v
w’
u’
v’
u
w
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Stefan Szeider
Reduction from IS for {REV}
BN local search
Independent Set
v
w
u
v
w’
u’
v’
u
w
degree reduction
u
u
w1
w1
w2
u
w3
w2
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w3
Stefan Szeider
3.
BN Structure Learning
parameterized by distance from being a
branching
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Stefan Szeider
Beyond Branchings
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Stefan Szeider
A local score function f
(
An optimal DAG for f
Exact Probabilistic Network Learning is NP-hard.
Beyond Branchings
Known Results
1
2
1
Theorem: An optimal branching can
3
5
be found in polynomial
time.4
2
3
3
5
4
[Chu and Liu 1965, Edmonds 1967]
6
7
6
A polytree
7
A branching
Restricting the structure of the resulting DAG has lead to the following
results:
I
I
An optimal branching can found in polynomial time (Chu and Liu,
1965;Edmonds, 1967).
Finding an optimal polytree is NP-hard (Dasgupta, 1999).
Our Result
Main Result: For every constant k an optimal k-branching can be found
in polynomial time.
1
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2
Stefan Szeider
A
R
h
i
a
Problem:An
Given
a local
optimal
DAGscore
for f function f (
on N whose score is maximum with resp
Exact Probabilistic Network Learning is NP-hard.
A local score function f
Beyond Branchings
1
2
v P fv(P)
3 {1}
1
Known Results
4 {1} 0.1
5 1{1} 0.5
2
5 {1, 2} 1
6 {4} 0.9
6 {3, 4} 1
5
3 7 {5} 40.9
7 {4, 5} 1
Theorem: An optimal branching can
3
be found in polynomial
time.4
[Chu and Liu 1965, Edmonds 1967]
6
7!
3
5
A local score function f
7
6
7
Exact Probabilistic
Network
Learning is
A branching
A
A polytree
Known Res
Restricting the structure of the resulting DAG has lead to the following
R
results:
h
1
2
i
I An optimal branching can found in polynomial time (Chu and Liu,
a
1965;Edmonds, 1967).
I Finding an optimal polytree is NP-hard (Dasgupta, 1999).
Theorem: Finding an optimal
polytree is NP-hard [Dasgupta 1999]
3
4
5
Our Result
Main Result: For every constant k an optimal k-branching can be found
6
7
in polynomial time.
1
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27
2
A polytree
Restricting the structure of the resulting
Stefan Szeider
results:
Problem:An
Given
a local
optimal
DAGscore
for f function f (
on N whose score is maximum with resp
Exact Probabilistic Network Learning is NP-hard.
A local score function f
Beyond Branchings
1
2
v P fv(P)
3 {1}
1
Known Results
4 {1} 0.1
5 1{1} 0.5
2
5 {1, 2} 1
6 {4} 0.9
6 {3, 4} 1
5
3 7 {5} 40.9
7 {4, 5} 1
Theorem: An optimal branching can
3
be found in polynomial
time.4
[Chu and Liu 1965, Edmonds 1967]
6
7!
3
5
A local score function f
7
6
7
Exact Probabilistic
Network
Learning is
A branching
A
k-branching = polytree suchA that
polytree after
Known Res
Restricting
the structure
of thehas
resulting DAG has lead to the following
R
deleting certain k arcs,
every
node
results:
h
1
2
at most one parent. An optimal branching can found in polynomial time (Chu and Liu, i
I
1965;Edmonds, 1967).
Finding an optimal polytree is NP-hard (Dasgupta, 1999).
Theorem: Finding an optimal
polytree is NP-hard [Dasgupta 1999]
I
3
4
a
5
Our Result
Main Result: For every constant k an optimal k-branching can be found
6
7
in polynomial time.
1
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2
A polytree
Restricting the structure of the resulting
Stefan Szeider
results:
XP-Result
Theorem: finding an optimal k-branching is
polynomial for every constant k.
(i.e., the problem is in XP for parameter k.)
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Stefan Szeider
XP-Result
Theorem: finding an optimal k-branching is
polynomial for every constant k.
(i.e., the problem is in XP for parameter k.)
Approach:
(1) guess the deleted arcs
(2) solve the induced problem for branchings
turns out to be quite tricky!
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Stefan Szeider
A k-branching (k=4)
+ +
= D k-branching
= X deletion set
= X‘ co-set
= R rest
+
= R∪X’=D-X
polytree with in-deg
≤1
|X|≤k and |X’|≤k. Hence for constant k we can try all X and X‘
in polynomial time. For each choice of X∪X’ we compute the
best set R such that R∪X’ is a polytree with in-deg ≤1.
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Stefan Szeider
Matroid Approach
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•
•
•
Let X, X’ be two disjoint sets of arcs.
•
The acyclicity matroid where a set R is
independent iff R∪X∪X’ has no undirected cycles.
•
Lemma: D is a k-branching iff D-X is independent in
both matroids.
We define define two matroids over N×N.
The in-degree matroid where a set R is
independent iff R ∩ (X∪X’) = ∅ and R has
indegree ≤1.
30
Stefan Szeider
Matroid Intersection
• Hence, for each guessed set X we need to
find an optimal set D that is independent in
both matroids.
• Optimality can be expressed by considering
weighted matroids: w(u,v)= f(v,{u})−f(v,∅).
• Solution can be found by means of
a poly time algorithm for weighted
matroid intersection
(Brezovec, Cornuejols, Glover 1986).
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Stefan Szeider
FPT?
• We don’t know whether the k-branching
problem is FPT.
• We have FPT algorithms for special cases
whenever the guess-phase is FPT.
• For instance: the arcs in X∪X’ form an in-tree
and each node has a bounded number of
potential parent sets.
• BN structure learning remains NP-hard
under this restriction.
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Stefan Szeider
A slight generalization is
W[1]-hard
Define a k-node branching as a polytree that
becomes a branching by deletion of k nodes.
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Stefan Szeider
A slight generalization is
W[1]-hard
Define a k-node branching as a polytree that
becomes a branching by deletion of k nodes.
Theorem: Finding an optimal
k-node branching is W[1]-hard.
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Stefan Szeider
A slight generalization is
W[1]-hard
Define a k-node branching as a polytree that
becomes a branching by deletion of k nodes.
Theorem: Finding an optimal
k-node branching is W[1]-hard.
Proof: reduction from multicolored clique.
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Conclusion
• BN Network Structure Learning
• important and notoriously difficult
• parameterized complexity approach:
• treewidth of super structure
• parameterized local search
• k-branchings and k-node branchings
• Other parameters?
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Stefan Szeider