Slide 1
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The Nemhauser-Trotter Reduction and Lifted
Message Passing for the Weighted CSP
Hong Xu T. K. Satish Kumar Sven Koenig
[email protected], [email protected], [email protected]
June 8, 2017
University of Southern California
the 14th International Conference on Integration of Artificial Intelligence and Operations Research Techniques in
Constraint Programming (CPAIOR 2017)
Padova, Italy
Slide 2
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Agenda
The Weighted Constraint Satisfaction Problem (WCSP)
The Constraint Composite Graph (CCG)
Computational Techniques Facilitated by the CCG
The Nemhauser-Trotter (NT) Reduction
Min-Sum Message Passing (MSMP)
Conclusion
1
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Executive Summary
Using the Constraint Composite Graph (CCG) of a WCSP,
• The Nemhauser-Trotter (NT) Reduction, a polynomial-time procedure, can
solve about 1/8 of the benchmark instances without search.
• The Min-Sum Message Passing (MSMP) algorithm, widely used in the
probabilistic reasoning community, produces significantly better solutions on
the CCG than on the WCSP’s original form. This further bridges the
probabilistic reasoning and CP communities.
2
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Agenda
The Weighted Constraint Satisfaction Problem (WCSP)
The Constraint Composite Graph (CCG)
Computational Techniques Facilitated by the CCG
The Nemhauser-Trotter (NT) Reduction
Min-Sum Message Passing (MSMP)
Conclusion
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The Weighted Constraint Satisfaction Problem: Motivation
Many real-world problems can be solved using the WCSP:
• RNA motif localization (Zytnicki et al. 2008)
• Communication through noisy channels using Error Correcting Codes in
Information Theory (Yedidia et al. 2003)
• Medical and mechanical diagnostics (Milho et al. 2000; Muscettola et al.
1998)
• Energy minimization in Computer Vision (Kolmogorov 2005)
• · · ·
3
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Weighted Constraint Satisfaction Problem (WCSP)
• N variables x = {X1
, X2
, . . . , XN
}.
• Each variable Xi
has a discrete-valued domain Di
.
• M weighted constraints {Es1
, Es2
, . . . , EsM
}.
• Each constraint Es
specifies the weight for each combination of assignments
of values to a subset s of the variables.
• Find an optimal assignment of values to these variables so as to minimize
the total weight: E(x) = M
i=1
Esi
(xsi
).
• Known to be NP-hard.
4
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WCSP Example on Boolean Variables
X
1
X
2
X
3
X
2
1
0
1
0
X
3
1.0
0.6 1.3
1.1
X
1
1
0
1
0
X
3
0.7
0.4 0.9
0.8
X
1
1
0
1
0
X
2
0.7
0.5 0.6
0.3
X
1
1
0
0.2
0.7
X
3
1
0
1.0
0.1
X
2
1
0
0.8
0.3
E(X1
, X2
, X3
) = E1
(X1
) + E2
(X2
) + E3
(X3
)+
E12
(X1
, X2
) + E13
(X1
, X3
) + E23
(X2
, X3
) 5
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WCSP Example: Evaluate the Assignment X1
= 0, X2
= 0, X3
= 1
X
1
X
2
X
3
X
2
1
0
1
0
X
3
1.0
0.6 1.3
1.1
X
1
1
0
1
0
X
3
0.7
0.4 0.9
0.8
X
1
1
0
1
0
X
2
0.7
0.5 0.6
0.3
X
1
1
0
0.2
0.7
X
3
1
0
1.0
0.1
X
2
1
0
0.8
0.3
E(X1
= 0, X2
= 0, X3
= 1) = 0.7 + 0.3 + 1.0 + 0.5 + 1.3 + 0.9 = 4.7
(This is not an optimal solution.) 6
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WCSP Example: Evaluate the Assignment X1
= 1, X2
= 0, X3
= 0
X
1
X
2
X
3
X
2
1
0
1
0
X
3
1.0
0.6 1.3
1.1
X
1
1
0
1
0
X
3
0.7
0.4 0.9
0.8
X
1
1
0
1
0
X
2
0.7
0.5 0.6
0.3
X
1
1
0
0.2
0.7
X
3
1
0
1.0
0.1
X
2
1
0
0.8
0.3
E(X1
= 1, X2
= 0, X3
= 0) = 0.2 + 0.3 + 0.1 + 0.7 + 0.6 + 0.7 = 2.6
This is an optimal solution. Using brute force, it requires exponential time to find. 7
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Agenda
The Weighted Constraint Satisfaction Problem (WCSP)
The Constraint Composite Graph (CCG)
Computational Techniques Facilitated by the CCG
The Nemhauser-Trotter (NT) Reduction
Min-Sum Message Passing (MSMP)
Conclusion
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Two Forms of Structure in WCSP
X
1
X
2
X
3
X
4
X
1
1
0
1
0
X
2
0.7
0.5 0.6
0.3
Numerical Structure
Graphical Structure • Graphical: Which variables
are in which constraints?
• Numerical: How does each
constraint relate the
variables in it?
How can we exploit both forms
of structure computationally?
8
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Minimum Weighted Vertex Cover (MWVC)
1
2
2 0
1
1
(a)
1
2
2 0
1
1
(b)
1
2
2 0
1
1
(c)
1
2
2 0
1
1
(d)
Each vertex is associated with a non-negative weight. Sum of the weights on the
vertices in the VC is minimized. 9
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Projection of Minimum Weighted Vertex Cover
onto an Independent Set
X
1
+
X
3
X
2
X
5
X
6
X
4
X
7
∞
1 1
1 1
2
1
X
1
X
2
X
3
X
4
X
5
X
6
X
7
1
1
1
1
2
3
1 = necessarily present
in the vertex cover
0 = necessarily absent
from the vertex cover
X
1
1
0
1
0
X
4
5
4 7
6
1
(Kumar 2008, Fig. 2) 10
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Projection of MWVC onto an Independent Set
Assuming Boolean variables in WCSPs
• Observation: The projection of MWVC onto an independent set looks
similar to a weighted constraint.
• Question 1: Can we build the lifted graphical representation for any given
weighted constraint? This is answered by (Kumar 2008).
• Question 2: What is the benefit of doing so?
11
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Lifted Representations: Example
X
1
X
2
X
3
X
2
1
0
1
0
X
3
1.0
0.6 1.3
1.1
X
1
1
0
1
0
X
3
0.7
0.4 0.9
0.8
X
1
1
0
1
0
X
2
0.7
0.5 0.6
0.3
X
1
1
0
0.2
0.7
X
3
1
0
1.0
0.1
X
2
1
0
0.8
0.3
E(X1
, X2
, X3
) = E1
(X1
) + E2
(X2
) + E3
(X3
)+
E12
(X1
, X2
) + E13
(X1
, X3
) + E23
(X2
, X3
) 12
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Lifted Representations: Example
X
2
1
0
1
0
X
3
1.0
0.6 1.3
1.1
X
1
1
0
1
0
X
3
0.7
0.4 0.9
0.8
X
1
1
0
1
0
0.7
0.5 0.6
0.3
X
1
1
0
0.2
0.7
X
3
1
0
1.0
0.1
X
2
1
0
0.8
0.3
X
1
A
4
0.2
0.7
X
2
A
5
0.8
0.3
X
3
A
6
1.0
0.1
X
1
A
1
0.2
0.5
X
2
0.1
X
2
A
2
0.4
0.6
X
3
0.7
X
1
A
3
0.3
0.4
X
3
0.5
X
2
13
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Constraint Composite Graph (CCG)
X
1
A
1
0.7
0.5
X
2
1.3
A
2
0.6
X
3
2.2
A
3
0.4 A
4
0.7 A
5
0.3 A
6
0.1
14
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MWVC on the Constraint Composite Graph (CCG)
X
1
A
1
0.7
0.5
X
2
1.3
A
2
0.6
X
3
2.2
A
3
0.4 A
4
0.7 A
5
0.3 A
6
0.1
An MWVC of the CCG encodes an optimal solution of the original WCSP! 15
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Agenda
The Weighted Constraint Satisfaction Problem (WCSP)
The Constraint Composite Graph (CCG)
Computational Techniques Facilitated by the CCG
The Nemhauser-Trotter (NT) Reduction
Min-Sum Message Passing (MSMP)
Conclusion
Slide 20
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Agenda
The Weighted Constraint Satisfaction Problem (WCSP)
The Constraint Composite Graph (CCG)
Computational Techniques Facilitated by the CCG
The Nemhauser-Trotter (NT) Reduction
Min-Sum Message Passing (MSMP)
Conclusion
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The Nemhauser-Trotter (NT) Reduction
A
C D
B
w
4
w
3
w
1
w
2
A(w
1
)
B(w
2
)
C(w
3
)
D(w
4
)
A'(w
1
)
C'(w
3
)
D'(w
4
)
B'(w
2
)
A(w
1
)
B(w
2
)
C(w
3
)
D(w
4
)
A'(w
1
)
C'(w
3
)
D'(w
4
)
B'(w
2
)
A is in the minimum weighted VC
B is not in the minimum weighted VC
C and D are in the Kernel
16
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Experimental Evaluation: Instances
• The UAI 2014 Inference Competition: PR and MMAP benchmark instances
(Up to 10 thousands variables and constraints)
• Converted to WCSP instances by taking negative logarithms normalization.
• WCSP Instances from (Hurley et al. 2016) (Up to less than 1 million
variables and millions of constraints)
• The Probabilistic Inference Challenge 2011
• The Computer Vision and Pattern Recognition OpenGM2 benchmark
• The Weighted Partial MaxSAT Evaluation 2013
• The MaxCSP 2008 Competition
• The MiniZinc Challenge 2012 & 2013
• The CFLib (a library of cost function networks)
• Only instances in which variables have only binary domains are used.
17
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Experimental Evaluation: Results
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Fraction
0
25
50
75
100
125
Number of Instances
Benchmark instances from UAI 2014 Inference Competition:
19 out of 160 benchmark instances solved by the NT reduction 18
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Experimental Evaluation: Results
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Fraction
0
50
100
150
200
250
Number of Instances
Benchmark instances from (Hurley et al. 2016):
53 out of 410 benchmark instances solved by the NT reduction 19
Slide 25
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Agenda
The Weighted Constraint Satisfaction Problem (WCSP)
The Constraint Composite Graph (CCG)
Computational Techniques Facilitated by the CCG
The Nemhauser-Trotter (NT) Reduction
Min-Sum Message Passing (MSMP)
Conclusion
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Min-Sum Message Passing (MSMP) Algorithms
• Min-Sum Message Passing Algorithms
• are variants of belief propagation
• are widely used
• have information passed locally between variables and constraints
• Original MSMP Algorithm
• Perform MSMP on WCSPs directly
• Messages are passed between variables and constraints
• Lifted MSMP Algorithm
• Perform MSMP on the MWVC problem instance of the CCG
• Messages are passed between adjacent vertices
20
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Operations on Tables: Min
minX1
X1
X2 0 1
0 1 2
1 4 3
=
X1
0 1
1 3
21
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Operations on Tables: Sum
X1
X2 0 1
0 1 2
1 4 3
+
X1
0 5
1 6
=
X1
X2 0 1
0 1 + 5 = 6 2 + 5 = 7
1 4 + 6 = 10 3 + 6 = 9
22
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Original MSMP Algorithm: Message Passing for the WCSP
(Xu et al. 2017, Fig. 1)
• A message is a table over
the single variable, which is
the sender or the receiver.
• A vertex of k neighbors
1. applies sum on the
messages from its k − 1
neighbors and internal
constraint table, and
2. applies min on the
summation result and
sends the resulting table
to its kth neighbor. 23
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Original MSMP Algorithm: Example
X1
C12
X2
C23
X3
ν
X1→C12
= 0, 0
−
−
−
−
−
−
−
−
−
−
→
−
−
−
−
−
−
−
−
−
−
→
νX3
→C23
= 0, 0
ˆ
νC12
→X2
= 0, 0
←
−
−
−
−
−
−
−
−
−
−
ν
X2→C23
= 0, 0
−
−
−
−
−
−
−
−
−
−
→
ˆ
νC23
→X3
= 0, 0
←
−
−
−
−
−
−
−
−
−
−
←
−
−
−
−
−
−
−
−
−
−
ˆ
ν
C23→X2
= 0, 0
−
−
−
−
−
−
−
−
−
−
→
νX2
→C12
= 0, 0
←
−
−
−
−
−
−
−
−
−
−
ˆ
ν
C12→X1
= 0, 0
X1
X2 0 1
0 2 3
1 1 2
(a) C12
X2
X3 0 1
0 1 4
1 2 2
(b) C23
24
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Original MSMP Algorithm: Example
X1
C12
X2
C23
X3
ν
X1→C12
= 0, 0
−
−
−
−
−
−
−
−
−
−
→
−
−
−
−
−
−
−
−
−
−
→
νX3
→C23
= 0, 0
ˆ
νC12
→X2
= 0, 1
←
−
−
−
−
−
−
−
−
−
−
ν
X2→C23
= 0, 0
−
−
−
−
−
−
−
−
−
−
→
ˆ
νC23
→X3
= 0, 0
←
−
−
−
−
−
−
−
−
−
−
←
−
−
−
−
−
−
−
−
−
−
ˆ
ν
C23→X2
= 0, 0
−
−
−
−
−
−
−
−
−
−
→
νX2
→C12
= 0, 0
←
−
−
−
−
−
−
−
−
−
−
ˆ
ν
C12→X1
= 0, 0
X1
X2 0 1
0 2 3
1 1 2
(a) C12
X2
X3 0 1
0 1 4
1 2 2
(b) C23
24
Slide 32
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Original MSMP Algorithm: Example
X1
C12
X2
C23
X3
ν
X1→C12
= 0, 0
−
−
−
−
−
−
−
−
−
−
→
−
−
−
−
−
−
−
−
−
−
→
νX3
→C23
= 0, 0
ˆ
νC12
→X2
= 0, 1
←
−
−
−
−
−
−
−
−
−
−
ν
X2→C23
= 0, 1
−
−
−
−
−
−
−
−
−
−
→
ˆ
νC23
→X3
= 0, 0
←
−
−
−
−
−
−
−
−
−
−
←
−
−
−
−
−
−
−
−
−
−
ˆ
ν
C23→X2
= 0, 0
−
−
−
−
−
−
−
−
−
−
→
νX2
→C12
= 0, 0
←
−
−
−
−
−
−
−
−
−
−
ˆ
ν
C12→X1
= 0, 0
X1
X2 0 1
0 2 3
1 1 2
(a) C12
X2
X3 0 1
0 1 4
1 2 2
(b) C23
24
Slide 33
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Original MSMP Algorithm: Example
X1
C12
X2
C23
X3
ν
X1→C12
= 0, 0
−
−
−
−
−
−
−
−
−
−
→
−
−
−
−
−
−
−
−
−
−
→
νX3
→C23
= 0, 0
ˆ
νC12
→X2
= 0, 1
←
−
−
−
−
−
−
−
−
−
−
ν
X2→C23
= 0, 1
−
−
−
−
−
−
−
−
−
−
→
ˆ
νC23
→X3
= 0, 2
←
−
−
−
−
−
−
−
−
−
−
←
−
−
−
−
−
−
−
−
−
−
ˆ
ν
C23→X2
= 0, 0
−
−
−
−
−
−
−
−
−
−
→
νX2
→C12
= 0, 0
←
−
−
−
−
−
−
−
−
−
−
ˆ
ν
C12→X1
= 0, 0
X1
X2 0 1
0 2 3
1 1 2
(a) C12
X2
X3 0 1
0 1 4
1 2 2
(b) C23
24
Slide 34
Slide 34 text
Original MSMP Algorithm: Example
X1
C12
X2
C23
X3
ν
X1→C12
= 0, 0
−
−
−
−
−
−
−
−
−
−
→
−
−
−
−
−
−
−
−
−
−
→
νX3
→C23
= 0, 0
ˆ
νC12
→X2
= 0, 1
←
−
−
−
−
−
−
−
−
−
−
ν
X2→C23
= 0, 1
−
−
−
−
−
−
−
−
−
−
→
ˆ
νC23
→X3
= 0, 2
←
−
−
−
−
−
−
−
−
−
−
←
−
−
−
−
−
−
−
−
−
−
ˆ
ν
C23→X2
= 0, 1
−
−
−
−
−
−
−
−
−
−
→
νX2
→C12
= 0, 0
←
−
−
−
−
−
−
−
−
−
−
ˆ
ν
C12→X1
= 0, 0
X1
X2 0 1
0 2 3
1 1 2
(a) C12
X2
X3 0 1
0 1 4
1 2 2
(b) C23
24
Slide 35
Slide 35 text
Original MSMP Algorithm: Example
X1
C12
X2
C23
X3
ν
X1→C12
= 0, 0
−
−
−
−
−
−
−
−
−
−
→
−
−
−
−
−
−
−
−
−
−
→
νX3
→C23
= 0, 0
ˆ
νC12
→X2
= 0, 1
←
−
−
−
−
−
−
−
−
−
−
ν
X2→C23
= 0, 1
−
−
−
−
−
−
−
−
−
−
→
ˆ
νC23
→X3
= 0, 2
←
−
−
−
−
−
−
−
−
−
−
←
−
−
−
−
−
−
−
−
−
−
ˆ
ν
C23→X2
= 0, 1
−
−
−
−
−
−
−
−
−
−
→
νX2
→C12
= 0, 1
←
−
−
−
−
−
−
−
−
−
−
ˆ
ν
C12→X1
= 0, 0
X1
X2 0 1
0 2 3
1 1 2
(a) C12
X2
X3 0 1
0 1 4
1 2 2
(b) C23
24
Slide 36
Slide 36 text
Original MSMP Algorithm: Example
X1
C12
X2
C23
X3
ν
X1→C12
= 0, 0
−
−
−
−
−
−
−
−
−
−
→
−
−
−
−
−
−
−
−
−
−
→
νX3
→C23
= 0, 0
ˆ
νC12
→X2
= 0, 1
←
−
−
−
−
−
−
−
−
−
−
ν
X2→C23
= 0, 1
−
−
−
−
−
−
−
−
−
−
→
ˆ
νC23
→X3
= 0, 2
←
−
−
−
−
−
−
−
−
−
−
←
−
−
−
−
−
−
−
−
−
−
ˆ
ν
C23→X2
= 0, 1
−
−
−
−
−
−
−
−
−
−
→
νX2
→C12
= 0, 1
←
−
−
−
−
−
−
−
−
−
−
ˆ
ν
C12→X1
= 1, 0
X1
X2 0 1
0 2 3
1 1 2
(a) C12
X2
X3 0 1
0 1 4
1 2 2
(b) C23
24
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Original MSMP Algorithm: Example
X1
C12
X2
C23
X3
ν
X1→C12
= 0, 0
−
−
−
−
−
−
−
−
−
−
→
−
−
−
−
−
−
−
−
−
−
→
νX3
→C23
= 0, 0
ˆ
νC12
→X2
= 0, 1
←
−
−
−
−
−
−
−
−
−
−
ν
X2→C23
= 0, 1
−
−
−
−
−
−
−
−
−
−
→
ˆ
νC23
→X3
= 0, 2
←
−
−
−
−
−
−
−
−
−
−
←
−
−
−
−
−
−
−
−
−
−
ˆ
ν
C23→X2
= 0, 1
−
−
−
−
−
−
−
−
−
−
→
νX2
→C12
= 0, 1
←
−
−
−
−
−
−
−
−
−
−
ˆ
ν
C12→X1
= 1, 0
• X1
= 1 minimizes
ˆ
νC12→X1
(X1
)
• X2
= 0 minimizes
ˆ
νC12→X2
(X2
) + ˆ
νC23→X2
(X2
)
• X3
= 0 minimizes
ˆ
νC23→X3
(X3
)
• Optimal solution:
X1
= 1, X2
= 0, X3
= 0
24
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Lifted MSMP Algorithm: Finding an MWVC on the CCG
• Treat MWVC problems on the CCG as WCSPs and apply the MSMP
algorithm on it.
• Messages are simplified passed between adjacent vertices.
25
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Experimental Evaluation: Setup
• Use the same benchmark instances as before.
• Solutions are reported if the MSMP algorithms do not terminate in 5 min.
• Optimal solutions are computed using toulbar2 (Hurley et al. 2016) or
integer linear programming.
• Experiments were performed on a GNU/Linux workstation with an Intel
Xeon processor E3-1240 v3 (8MB Cache, 3.4GHz) and 16GB RAM.
26
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Experimental Evaluation: Results — Solution Quality
100 107 1014 1021
The Lifted MSMP Solution Quality
100
105
1010
1015
1020
The Original MSMP Solution Quality
(a) Benchmark instances from the UAI
2014 Inference Competition: 126/9/18
above/below/close to the diagonal dashed line
100 104 108 1012
The Lifted MSMP Solution Quality
100
103
106
109
1012
The Original MSMP Solution Quality
(b) Benchmark instances from (Hurley et al.
2016): 222/68/19 above/below/close to the di-
agonal dashed line 27
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Experimental Evaluation: Results — Solution Quality
0 < 10% ≥ 10%, < 20% ≥ 20%, < 30% > 30%
(MSMP solution - optimal solution) / optimal solution
0
25
50
75
100
125
Number of Instances
Lifted MSMP
Original MSMP
UAI 2014 Inference Competition: Compare qualities of solution with the optimal
solutions. 28
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Experimental Evaluation: Results — Solution Quality
0 < 10% ≥ 10%, < 20% ≥ 20%, < 30% > 30%
(MSMP solution - optimal solution) / optimal solution
0
25
50
75
100
Number of Instances
Lifted MSMP
Original MSMP
Benmark instances from (Hurley et al. 2016): Compare qualities of solution with the
optimal solutions. 29
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Experimental Evaluation: Results — Convergence
Benchmark Instance Set Neither Both Original Lifted
UAI 2014 Inference Competition 25 4 124 0
(Hurley et al. 2016) 258 7 44 0
(Xu et al. 2017, Tab. 1)
• Neither: Neither of the MSMP algorithms terminates in 5 min.
• Both: Both of the MSMP algorithms terminate in 5 min.
• Original: Only the original MSMP algorithm terminates in 5 min.
• Lifted: Only the lifted MSMP algorithm terminates in 5 min.
30
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Agenda
The Weighted Constraint Satisfaction Problem (WCSP)
The Constraint Composite Graph (CCG)
Computational Techniques Facilitated by the CCG
The Nemhauser-Trotter (NT) Reduction
Min-Sum Message Passing (MSMP)
Conclusion
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Conclusion
• NT reduction on the CCG is effective for many benchmark instances.
• The NT reduction could determine the optimal values of all variables for
about 1/8 of the benchmark instances without search.
• We revived the MSMP algorithm for solving the WCSP by applying it on its
CCG instead of its original form.
• The lifted MSMP algorithm produced solutions that are significantly better
than the original MSMP algorithm in general.
• The lifted MSMP algorithm produced solutions that are close to optimal for
a large fraction of benchmark instances.
• However, the lifted MSMP algorithm is less advantageous in terms of
convergence.
• (Future work) Both MSMP algorithms can be easily adjusted to distributed
settings. 31
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References I
Barry Hurley, Barry O’Sullivan, David Allouche, George Katsirelos, Thomas Schiex, Matthias Zytnicki,
and Simon de Givry. “Multi-language evaluation of exact solvers in graphical model discrete
optimization”. In: Constraints 21.3 (2016), pp. 413–434.
Vladimir Kolmogorov. Primal-dual Algorithm for Convex Markov Random Fields. Tech. rep.
MSR-TR-2005-117. Microsoft Research, 2005.
T. K. Satish Kumar. “A framework for hybrid tractability results in boolean weighted constraint
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Slide 47
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References II
Hong Xu, T. K. Satish Kumar, and Sven Koenig. “The Nemhauser-Trotter Reduction and Lifted
Message Passing for the Weighted CSP”. In: the 14th International Conference on Integration of
Artificial Intelligence and Operations Research Techniques in Constraint Programming (CPAIOR). 2017.
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Matthias Zytnicki, Christine Gaspin, and Thomas Schiex. “DARN! A Weighted Constraint Solver for
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