[PPT] - Efficiently Enumerating Minimal Triangulations Nofar Carmeli Batya PowerPoint Presentation

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Nofar Carmeli Batya Kenig Benny Kimelfeld

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Efficiently Enumerating Minimal Triangulations

Recent Trends in Knowledge Compilation 09/2017

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Outline

Short background
TDs in probabilistic inference and knowledge compilation
The enumeration algorithm
Case study
An open problem

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Tree Decompositions

Every edge is contained in some bag Tree Every node

ccurs in a

connected subtree

Graph Tree decomposition

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Generate a tree decomposition

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1. Triangulate the graph
(, )

Every cycle of length > 3 has a chord Now ( + ) is chordal, and is called a triangulation of A triangulation is minimal if no edge can be removed while maintaining chordality

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Generate a tree decomposition

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2. Extract the maximal cliques = , … , in (, + )
(, )

, , , , , ,

A chordal graph has at most || maximal cliques [Fulkerson+1965]
Can be easily and efficiently found
3. Connect the maximal cliques into a graph (, ℰ) where ,

∈ ℰ iff ∩ ≠ ∅ = 2 = 2 = 1

4. Let (, ℰ′) be a maximum weight spanning tree of (, ℰ)

(, ℰ′) is a TD of (, )

The weight of each edge ,
∈ ℰ is = | ∩

|

(, ℰ)

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Outline

Short background
TDs in probabilistic inference and knowledge compilation
The enumeration algorithm
Case study
An open problem

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Tree Decompositions in KC and Probabilistic Inference

Find a set of variables ! that decompose the problem into one
r more independent subproblems
For every possible assignment to the variables in !
Independently solve smaller subproblems
Merge results (product)
Aggregate results corresponding to different assignments (sum,

max)

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Guided by a TD We want ! to be small

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Concrete Examples

C2D compiler [Darwiche’01, Darwiche ‘04, Chavira+Darwiche ’08,…]
Guided by a d-tree
Min-fill heuristic
Repeatedly apply hypergraph partitioning (hmetis)
Choose one with smallest width
MiniC2D compiler [Oztok+Darwiche ‘15]
Guided by a Decision-Vtree
Min-fill heuristic
Repeatedly apply of Hypergraph partitioning (hmetis)
Choose one with smallest width
AND/OR search [Dechter ‘13, Marinescu+ ‘14, Otten+ ’14, Lam+ ’16,…]
Guided by a Pseudo-Tree
Various heuristics: MCS, Min-Degree, Min-Fill,3

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A form of tree-decomposition

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Which TD to use ??

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width time

BN: 100 vars Choosing the right TD can have a profound impact

n running time.

What makes these so much better for the problem?

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Which TD to use ??

Improving the Efficiency of Dynamic Programing on Tree

Decompositions via Machine Learning [Abseher+ IJCAI ‘15]

Even TDs of the same width may yield extremely diverging runtimes
Other parameters influence the runtime at least to the same extent as width
Main idea: generate a pool of TDs and then choose the one with the “greatest

promise”

Authors generate 10 TDs for every problem instance using min-fill
Apply ML algorithms for ranking the TDs according to their expected performance

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Required: Large amounts of diverse TDs in

rder to extract important features

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Which TDs not to use?

Graph Better tree decomposition

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Tree decomposition Better tree decomposition

Avoid:

1. Redundant bags
2. Redundant edges

(bag equivalent) Proper

tree decompositions Minimal triangulations

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Outline

Short background
TDs in probabilistic inference and knowledge compilation
The enumeration algorithm
Case study
An open problem

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Goal

Problem: Enumerating all minimal triangulations of a graph

1. Complexity guarantees
2. Effective practical solution

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Exponentially many minimal triangulations, what is an “efficient” algorithm?

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Efficiency of enumeration algorithms

[Johnson,Papadimitriou,Yannakakis 88]

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incremental polynomial time

Delay before answer i is polynomial in input + i start time start time

polynomial delay

Delay between successive answers is poly(input)

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The main theoretical result Main Theorem:

Given a graph, it is possible to enumerate in incremental polynomial time:

The minimal triangulations
The proper TDs

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Goal: Enumerate Minimal Triangulations

A bijection [Parra&Scheffler97]:

minimal triangulations ↔ maximal sets of saturated non-crossing minimal vertex separators

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Goal: Enumerate Maximal Independent Sets

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Problem: The graph may be of exponential size! Challenge: Solve without generating the graph

Enumerating max independent sets can be done in polynomial delay [Johnson+88]

… …

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The Enumeration Algorithm

Separator graph !, ℇ of (, ) may be of exponential size but3
We can efficiently enumerate the minimal separator set !
Polynomial-delay iterator [Berry+99]
We can efficiently test whether a pair #, $ ∈ ℇ
Size of each maximal independent set % ⊆ ! of is less than ||
A triangulation of has at most − 1 minimal vertex separators [Fulkerson+1965]
We can efficiently (P-time) extend any independent set to one that is

maximal

Extension Triangulation
We can apply any known triangulation algorithm for extension
MCS-M [Berry+02]
LB_TRIANG [Berry+06]

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Shifting a Min-Triangulation with a Min-Separator

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s

1. Add s to T
2. Extract max. subset of non-crossing min-separators containing s
3. Extend to a minimal triangulation

s-shifting T: s s

s-shift of T

Min-Fill MCS …

T

Maximal Independent set of non-crossing minimal separators Node

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The Enumeration Algorithm

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1. Generate a minimal triangulation T0
2. Insert T0 into Q

P Q

Algorithm Min-Sep Iterator next()

(

Repeat until Q is empty:

1. Move some T from Q to P
2. Print T
3. Insert into Q an s-shift of T for all

minimal separators s in (

4. Repeat until Q is non-empty or !Iterator.hasNext()
1. + ← Iterator. -./0()
2. Insert into Q an s-shift of T for all minimal

triangulations T in P Validate that the s-extension is not already in Q or P

Output:

+-extension

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Experiments

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Is the algorithm effective? Does it help improve standard measures?

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Experiments

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C++ implementation
Triangulation algorithms: MCS-M [Berry+02]

LB-Triang [Berry+06] with min fill heuristics

Datasets:
Database queries, TPC-H (LogicBlox translation)

2-19 nodes, 1-46 edges

Probabilistic graphical models, UAI inference challenge

60-1039 nodes, 135-1696 edges

Random

30-200 nodes, 131-13955 edges

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Experiments

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A single run (UAI, 414 nodes, 801 edges, MCS-M, 30 minutes)

1000 2000 3000 4000 5000 6000 7000 5 10 15 20 25 30 35 40 45 50 5 10 15 20 25 30

fill width time (minutes)

500 1000 1500 2000 2500 3000 3500 4000 4500 5 10 15 20 25 30

number of results time (minutes)

46 39 6232 3934 all results Min width results ≤ results

SLIDE 24

Outline

Short background
TDs in probabilistic inference and knowledge compilation
The enumeration algorithm
Case study
An open problem

24

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Case Study

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On the Impact of Pseudo-Trees on the Performance of Optimization Algorithms over AND/OR Search Spaces

Héctor Otero Mediero Rina Dechter

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AND/OR Search Tree [Dechter and Mateescu 2007, Dechter 20133]

OR AND OR AND OR OR AND AND

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1 1 1

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A B C D E F G

Solution tree: corresponds to a complete configuration of all variables primal graph

A B C F G D E

pseudo tree

[Freuder and Quinn 1985] ℎ ≤ ∗ ⋅ log8

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Experiments

Use enumeration algorithm to generate pseudo trees of

varying width and height ℎ

Test the influence of , ℎ, and their ratio 1 ≤

9 : ≤ 8 on

execution times.

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Experiments

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width width time time

BN: 100 vars, domain size: 2 Pedigree: 385 vars

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Experiments

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width time time

Promedas: 1005 vars WCSP: 143 vars,

width time

SLIDE 30

Outline

Short background
TDs in probabilistic inference and knowledge compilation
The enumeration algorithm
Case study
An open problem

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Polynomial Delay ??

We could enumerate minimal triangulations in p-delay if we could

answer the following decision problem efficiently

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Given (, ) and two sets ;, < of vertex pairs X, Y ⊆ × Is there a minimal triangulation ℎ(, + ) such that: 1.; ⊆ 2.< ∩ = ∅

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The Decision Problem

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Does there exist a minimal triangulation ℎ(, + ) of (, ) such that:

1. ; ⊆
2. < ∩ = ∅

Does there exist a chordal supergraph of , ℎ(, ℇ) such that: ∪ ; ⊆ ℇ ⊆ ( × ) ∖ < The chordal graph sandwich problem is NP-complete [Golumbick+1994]

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Future Work

Theoretical
Polynomial delay
Practical
Optimizing / parallelizing the algorithm
Heuristically ranked enumeration

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Thank you

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Tree Decompositions

Databases
Query evaluation [Yannakakis81],[Chekuri&Rajaraman97]
Bioinformatics
prediction of RNA secondary structure [Zhao+06]
Probabilistic graphical models
statistical inference [Lauritzen&Spiegelhalter88]
Constraint-satisfaction problems [Kolaitis&Vardi00]
Weighted model counting [Li+08]
...

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Future Work

Theoretical
Polynomial delay
Ranked enumeration
Approximately ranked enumeration (w/ approx. guarantee)
Practical
Optimizing / parallelizing the algorithm
Heuristically ranked enumeration

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Generate a tree decomposition

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1. Triangulate the graph
(, )

Every cycle of length > 3 has a chord Now ( + ) is chordal, and is called a triangulation of A triangulation is minimal if no edge can be removed while maintaining chordality

SLIDE 38

Generate a tree decomposition

38

2. Extract the maximal cliques = , … , in (, + )
(, )

, , , , , ,

A chordal graph has at most || maximal cliques [Fulkerson+1965]
Can be easily and efficiently found
3. Connect the maximal cliques into a graph (, ℰ) where ,

∈ ℰ iff ∩ ≠ ∅ = 2 = 2 = 1

4. Let (, ℰ′) be a maximum weight spanning tree of (, ℰ)

(, ℰ′) is a TD of (, )

The weight of each edge ,
∈ ℰ is = | ∩

|

(, ℰ)

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A graph can have many TDs
Common – minimize the cardinality of the largest bag (smallest width)
Smallest width is NP-hard [Arnborg+87]

Graph Tree decompositions

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Which TD to use?

a) b)

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Which TD to use?

Common: Use heuristics
Width isn’t enough
Different applications – different requirements
Goals:
TDs with lower width
Many TDs with low width

Flexible Caching in Trie Joins [Kalinsky+16]

Query TD1 TD2

TD1 runs 100 times faster!

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TD enumeration is needed

Related work:
Query plans using generalized hypertree decompositions [Tu&Ré15]
Generate all, choose one
No complexity guarantees
Works for small graphs
Improving the efficiency of dynamic programing on tree

decompositions using machine learning [Abseher+15]

Generate a pool, choose using machine learning
Heuristically generated pool, may not contain the best
Can we enumerate the TDs with efficiency guarantees?

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Goal

Problem: Enumerating all TDs of a graph

1. Complexity guarantees
2. Effective practical solution

?

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all

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Which TDs to Generate?

Graph Better tree decomposition

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Tree decomposition Better tree decomposition

Avoid:

1. Redundant bags
2. Redundant edges

(bag equivalent) Proper

tree decompositions Minimal triangulations

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Goal: Enumerate Proper TDs

A bijection:

classes of bag equivalent proper TDs ↔ min triangulations

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Every cycle of length>3 has a chord

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The Algorithm (Enumerating max independent sets)

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Redesign of an algorithm for hereditary

graph properties [Cohen+08]

Assuming:
Efficiently enumerating nodes
Efficiently checking edges
Polynomial size of max independent sets
Efficiently extending an independent set
Steps of extending an independent set
We can use any triangulation or TD algorithm
First result = algorithm’s result

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Solution Summary

Enumerate proper TDs Enumerate min triangulations Enumerate max independent sets

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Minimal Triangulation algorithms