via via exa xact and nd he heuri ristic lists of f min inim - - PowerPoint PPT Presentation

Computing tr treewidth via via exa xact and nd he heuri ristic lists of f min inim imal l se separa rators rs

Hisao Tamaki Meiji University

2019/3/7 Shonan Meeting 144

Con Contents

• Overview of the approach
• Experimental results
• Technical details
• Dynamic programming
• Listing minimal separators

2019/3/7 Shonan Meeting 144

Ov Overv rview of

• f the

he appro approach: thr hree com

• mponents

msDP(𝐻, 𝑙, Δ)

𝐻 : graph 𝑙 : positive integer, Δ : a set of minimal separators of 𝐻 Decides if 𝐻 has a tree-decomposition of width ≤ 𝑙 that uses minimal separators only from Δ

listExact(𝐻,𝑙)

Generates Δ𝑙(𝐻) , the set of all minimal separators of cardinality ≤ 𝑙

listHeuristic(𝐻,𝑈, 𝑙)

𝑈: a tree decomposition of 𝐻 Iteratively generates expanding subsets Δ0 ⊂ Δ1 ⊂ Δ2 ⊂ Δ3 … ⊆ Δ𝑙 𝐻

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Thre Three algo algori rithms for

• r com
• mputing the

he treewidth of

• f 𝐻

Ascend

for 𝑙 ascending from a trivial lower bound:

decide if 𝑢𝑥 𝐻 ≤ 𝑙 by calling msDP(𝐻, 𝑙, listExact(𝐻, 𝑙)) if YES then stop

Descend

𝑈 := heuristic tree-decomposition of 𝐻 by a greedy heuristic heuristically improve 𝑈 as log as possible

to improve 𝑈 of width 𝑥, use listHeuristic(𝐻, 𝑈, 𝑥 - 1) to generate Δ0 ⊂ Δ1 ⊂ Δ2 ⊂ Δ3 … ⊆ Δ𝑥−1 𝐻 and try msDP(𝐻, 𝑙, Δ𝑗) for 𝑗 = 0, 1, 2, …

try to show 𝑈 of width 𝑥 is optimal by msDP(𝐻, 𝑥 – 1, listExact(𝐻, 𝑥 – 1))

Alternate

Alternate between Descend and Ascend, with some resource balancing

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Ra Rand ndom in inst stances

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Som Some DI DIMACS gr grap aph col

• lori

ring in inst stances

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Com Computing envi nviro ronment

CPU: Intel Core i7-6700 (4 cores), 3.40GHz, 8192KB cache RAM:32GB Operating system: Ubuntu 18.04.1 LTS Programming language: Java 1.8 JVM: jre1.8.0_111 The maximum heap size: 28GB

Implementations are single threaded, except that multiple threads may be invoked for garbage collection by JVM. The time measured is the elapsed time. To minimize the influence of system processes, the computer is detached from the network and the graphic user interface is disabled.

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Per Perform rman ances of

• f min

min sep sep li list sting algo algori rithms

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Pe Performances of

• f the

the tr treewidth al algorith thms on

• n ran

andom inst stances (1) (1)

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Pe Performances of

• f the

the tr treewidth al algorith thms on

• n ran

andom inst stances (2) (2)

6-hour time-out time (seconds) is that of the last improvement

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Pe Performances of

• f the

the tr treewidth al algorith thms on

• n DIMACS inst

stances

6-hour time-out time (seconds) is that of the last improvement

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Min inimal l se sepa para rators rs

𝑇 ⊆ 𝑊 𝐻 is a separator of 𝐻 if 𝐻 ∖ 𝑇 ≔ 𝐻 𝑊 𝐻 ∖ 𝑇 is disconnected Each connected component of 𝐻 ∖ 𝑇 is called a component associated with separator 𝑇 A component 𝐷 associated with 𝑇 is a full component if 𝑂 𝐷 = 𝑇 𝑇 is a minimal separator if it has at least two full components associated with it

• r, equivalently, if 𝑇 separates a pair of vertices

but no proper subset of 𝑇 does not separator this pair. 𝑇 full components components associated with 𝑇

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Fe Feas asibili ility of

• f a

a con

• nnected se

set: sub subpro roblem for

• r DP

DP

Fix 𝐻 and 𝑙. A connected 𝐷 ⊆ 𝑊 𝐻 is feasible with respect to Δ ⊆ Δ𝑙 𝐻 if there is a tree-decomposition of 𝐻 𝑂 𝐷

• f width ≤ 𝑙 that
• has a bag containing 𝑂(𝐷) and
• uses separators only from Δ

C 𝑂(𝐷)

?

∈ Δ

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Dynamic pr progra ramming for

• r treewidth

Dynamic programming of Bouchitte and Todinca 2001:

1. List minimal separators and potential maximal cliques 2. Decide the feasibility of components associated with minimal separators, through a recurrence involving potential maximal cliques

Positive instance driven (PID) variant (Tamaki 2017)

Does not list minimal separators or potential maximal cliques in advance Generates “on the fly”

• feasible components associated with minimal separators
• potential maximal cliques needed to show their feasibility

New approach

1. List minimal separators, but not potential maximal cliques 2. Decide the feasibility of components associated with minimal separators through a direct recurrence, in which potential maximal cliques are implicit

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Well ll-form rmed ed tree ee-decompositi itions (1) (1)

𝑉 ⊆ 𝑊 𝐻 is baggy if

• there is no connected set 𝐷 such that

𝑂(𝐷) = 𝑉 and

• for every non-empty 𝑌 ⊆ 𝑉,
• there is a connected set 𝐷 containing 𝑌

such that 𝑂 𝐷 = 𝑉 ∖ 𝑌.

Note:

A potential maximal clique is baggy, but not vice versa U X

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Well ll-form rmed ed tree ee-decompositi itions (1) (1)

𝑉 ⊆ 𝑊 𝐻 is baggy if

• there is no connected set 𝐷 such that

𝑂(𝐷) = 𝑉 and

• for every non-empty 𝑌 ⊆ 𝑉,
• there is a connected set 𝐷 containing 𝑌

such that 𝑂 𝐷 = 𝑉 ∖ 𝑌.

Note:

A potential maximal clique is baggy, but not vice versa 𝑂(𝐷) U X C

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Well ll-form rmed ed tree ee-decompositi itions (2) (2)

Tree-decomposition T of G is well-formed if

• every bag of T is baggy and,
• for every connected vertex set C of G such that C is

a component of 𝐻 ∖ 𝑌 for some bag 𝑌 of 𝑈,

• there is a subtree 𝑈′ of 𝑈 and a bag 𝑍 in 𝑈′ such that
• 𝑈′ is a tree-decomposition of 𝐻[𝑂[𝐷]],
• 𝑍 is adjacent to a bag of 𝑈, say 𝑎 , not in 𝑈′ with

𝑍 ∩ 𝑎 = 𝑂 𝐷 .

Proposition

Every graph 𝐻 has a well-formed tree-decomposition of width tw(𝐻).

Reason:

The tree-decomposition corresponding to a minimal triangulation of 𝐻 is well-formed X C 𝑂(𝐷)

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Well ll-feas asibility y of

• f con
• nnected se

sets

A connected 𝐷 ⊆ 𝑊 𝐻 is well-feasible with respect to Δ ⊆ Δ𝑙 𝐻 if there is a well-formed tree-decomposition of 𝐻[𝑂[𝐷]] that

• has a bag containing 𝑂(𝐷) and
• uses separators only from Δ

Note:

For Δ = Δ𝑙(𝐻), 𝐷 is well-feasible if and only if 𝐷 is feasible.

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Or Orie ienting mini minimal l se sepa para rators rs

Assume a total order on the vertices: 𝑊 𝐻 = 1, 2, … , 𝑜 Induced partial order on vertex sets: 𝑉 < 𝑊 if min 𝑉 < min 𝑊 A connected set 𝐷 is inbound if

there is a full component 𝐸 associated with 𝑂(𝐷) such that 𝐸 < 𝐷

• therwise 𝐷 is outbound

𝑇

• utbound

2 4 5 7 6 8 inbound

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Dyn Dynamic pr progra ramming algo algori rithm (1) (1)

Main iteration of msDP: decides the feasibility of each inbound connected set with respect to Δ ⊆ Δ𝑙(𝐻)

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is isFe Feas asible(C) (C)

C 𝑂(𝐷)

?

C 𝑂(𝐷) X

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is isFe Feas asible(C) (C)

C 𝑂(𝐷)

?

𝑂(𝐷) X feasible inbounds

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is isFe Feas asible(C) (C)

C 𝑂(𝐷)

?

X min 𝐷 X try all feasible inbound 𝐸 with 𝑛𝑗𝑜 𝐷 ∈ 𝐸, 𝐸 ⊆ 𝐷, and |𝑂(𝐸) ∪ 𝑂(𝐷)| ≤ 𝑙 + 1 D

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is isFe Feas asible(C) (C)

C 𝑂(𝐷)

?

X min 𝐷 X 𝑇 = 𝑂 𝐷 ∪ {min(𝐷)} D 𝑇 = 𝑂 𝐷 ∪ 𝑂 𝐸 try all feasible inbound 𝐸 with 𝑛𝑗𝑜 𝐷 ∈ 𝐸, 𝐸 ⊆ 𝐷, and |𝑂(𝐸) ∪ 𝑂(𝐷)| ≤ 𝑙 + 1

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is isFe Feas asible(C) (C)

Case 1: 𝑇 does not have a full component 𝑇 inbounds A right bag 𝑌 = 𝑇 is found, if all these inbounds are feasible

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is isFe Feas asible(C) (C)

Case 2: 𝑇 has a full component 𝐸 𝑇 inbounds full component 𝑌 𝐸

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is isFe Feas asible(C) (C)

Case 2: 𝑇 has a full component 𝐸 𝑇 inbounds If any of the inbounds are not marked feasible, then fail Otherwise, call isFeasible(𝐸) full component 𝑌 𝐸

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Dyn Dynamic pr progra ramming algo algori rithm (2) (2)

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The Theore rem: corr

• rrectness of
• f the

he algo algori rithm

Let 𝐷 ⊆ 𝑊 𝐻 be connected with 𝑂 𝐷 ≤ 𝑙.

• If call isFeasible (𝐷) is made during the execution of our

dynamic programming algorithm, and returns true then 𝐷 is feasible with respect to Δ.

• if 𝐷 is inbound with 𝑂 𝐷 ∈ Δ and moreover is well-feasible

with respect to Δ, then the algorithm marks 𝐷 as feasible.

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Str Structure re of

• f Δ 𝐻 : the

he se set of

• f all

all min inimal se sepa para rators of

• f G

Let 𝐵 ⊆ 𝑊 𝐻 be connected. For each component 𝐷 of 𝐻 ∖ 𝐵 , 𝑂(𝐷) is a minimal separator, called a minimal separator close to 𝐵 Digraph Λ 𝐻 on Δ 𝐻 :

has an edge from 𝑇 to 𝑆 if and only if 𝑆 is a separator close to 𝐷 ∪ {𝑤} for some full component 𝐷 of 𝑇 and 𝑤 ∈ 𝑇.

Theorem [Berry et al. 2000]

Every minimal separator of 𝐻 is reachable in Λ 𝐻 from a separator close to a singleton

=> Algorithm for listing all minimal separators in 𝑃 𝑜3 time per each. 𝐵 C 𝑂 𝐵 𝑂(𝐷) 𝐷 𝑇 𝑤

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Str Structure re of

• f Δ 𝐻 : the

he se set of

• f all

all min inimal se sepa para rators of

• f G

Let 𝐵 ⊆ 𝑊 𝐻 be connected. For each component 𝐷 of 𝐻 ∖ 𝐵 , 𝑂(𝐷) is a minimal separator, called a minimal separator close to 𝐵 Digraph Λ 𝐻 on Δ 𝐻 :

has an edge from 𝑇 to 𝑆 if and only if 𝑆 is a separator close to 𝐷 ∪ {𝑤} for some full component 𝐷 of 𝑇 and 𝑤 ∈ 𝑇.

Theorem [Berry et al. 2000]

Every minimal separator of 𝐻 is reachable in Λ 𝐻 from a separator close to a singleton

=> Algorithm for listing all minimal separators in 𝑃 𝑜3 time per each. 𝐵 C 𝑂 𝐵 𝑂(𝐷) 𝑤 𝐷 ∪ {𝑤} 𝑂(𝐷 ∪ 𝑤 )

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Str Structure re of

• f Δ 𝐻 : the

he se set of

• f all

all min inimal se sepa para rators of

• f G

Let 𝐵 ⊆ 𝑊 𝐻 be connected. For each component 𝐷 of 𝐻 ∖ 𝐵 , 𝑂(𝐷) is a minimal separator, called a minimal separator close to 𝐵 Digraph Λ 𝐻 on Δ 𝐻 :

has an edge from 𝑇 to 𝑆 if and only if 𝑆 is a separator close to 𝐷 ∪ {𝑤} for some full component 𝐷 of 𝑇 and 𝑤 ∈ 𝑇.

Theorem [Berry et al. 2000]

Every minimal separator of 𝐻 is reachable in Λ 𝐻 from a separator close to a singleton

=> Algorithm for listing all minimal separators in 𝑃 𝑜3 time per each. 𝐵 C 𝑂 𝐵 𝑂(𝐷) 𝑤 𝐷 ∪ {𝑤} 𝑆

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Ge Gene nera rating Δ𝑙(𝐻)

Based on Takata’s algorithm for generating Δ 𝐻

• backtrack search through Λ 𝐻
• pick S only if 𝑇 ≤ 𝑙
• exploit the cardinality constraint for pruning

For 𝐵 ⊆ 𝑊 𝐻 connected such that 𝑂 𝐵 is an 𝑏-𝑐 minimal separator and 𝐺 ⊆ 𝑂 𝐵 define Δ𝑏𝑐 𝐵, 𝐺 : the set of all 𝑏-𝑐 minimal separators 𝑇 satisfying …

𝐵 𝑂 𝐵 𝐺 𝑏 𝑐

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Ge Gene nera rating Δ𝑙(𝐻)

For 𝐵 ⊆ 𝑊 𝐻 connected such that 𝑂 𝐵 is an 𝑏-𝑐 minimal separator and 𝐺 ⊆ 𝑂 𝐵 define Δ𝑏𝑐 𝐵, 𝐺 : the set of all 𝑏-𝑐 minimal separators 𝑇 satisfying

• 𝑇 ≤ 𝑙
• 𝐺 ⊆ 𝑇
• The component 𝐷𝑏 of 𝐻 ∖ 𝑇 containing 𝑏 contains 𝐵
• The component 𝐷𝑐of 𝐻 ∖ 𝑇 containing 𝑐 is disjoint

from 𝑂[𝐵]

• 𝑏 = min 𝐷𝑏 and 𝑐 = min 𝐷𝑐
• 𝐷𝑏 ≤ 𝐷𝑐

𝐵 𝑂 𝐵 𝐺 𝑏 𝐷𝑏 𝑐 𝐷𝑐 𝑇

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Base Base case ases for

• r Δ𝑏𝑐 𝐵, 𝐺

Δ𝑏𝑐 𝐵, 𝐺 is empty if either

• 𝑇 > 𝐺
• 𝐺 ∖ 𝑂(𝐷𝑐) ≠ ∅ , where 𝐷𝑐 is the component of

𝐻 ∖ 𝑂 𝐵 containing 𝑐

• min 𝐵 ≠ 𝑏 or min 𝐷𝑐 ≠ 𝑐
• |𝐺| = 𝑙 and 𝑂 𝐵 ≠ 𝐺
• 𝐵 >

𝐷𝑐

• 𝑂 𝐵

> 𝑙 and 𝐵 + 𝑂 𝐵 − 𝑙 > min{ 𝐷𝑐 ,

𝑊 𝐻 − 𝑙 2

}

Δ𝑏𝑐 𝐵, 𝐺 = {𝐺} if

𝑂 𝐵 = 𝑂 𝐷𝑐 = 𝐺, 𝐺 ≤ 𝑙, and 𝐵 ≤ 𝐷𝑐 𝐵 𝑂 𝐵 𝐺 𝑏 𝑐 𝐷𝑐

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𝑤

Re Recurr rrence for

• r Δ𝑏𝑐 𝐵, 𝐺

𝚬𝒃𝒄(𝑩, 𝑮) = 𝚬𝒃𝒄(𝑩, 𝑮 ∪ {𝒘}) ) ∪ 𝚬𝒃𝒄(𝑩’, 𝑮) where 𝒘 ∈ 𝑂(𝐵) ∖ 𝐺 is arbitrary and 𝑂(𝐵′) is the minimal 𝐵 ∪ {𝑤} − 𝑐 separator close to 𝐵 ∪ {𝑤 }.

𝐵 𝑂 𝐵 𝐺 𝑏 𝑐 𝑤 𝐵 𝑂 𝐵 𝐺 𝑏 𝑐 𝑤 𝐵 𝑂 𝐵 ∪ {𝑤} 𝐺 𝑏 𝑐

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𝑐 𝑤

Re Recurr rrence for

• r Δ𝑏𝑐 𝐵, 𝐺

𝚬𝒃𝒄(𝑩, 𝑮) = 𝚬𝒃𝒄(𝑩, 𝑮 ∪ {𝒘}) ) ∪ 𝚬𝒃𝒄(𝑩’, 𝑮) where 𝒘 ∈ 𝑂(𝐵) ∖ 𝐺 is arbitrary and 𝑂(𝐵′) is the minimal 𝐵 ∪ {𝑤} − 𝑐 separator close to 𝐵 ∪ {𝑤 }.

𝐵 𝑂 𝐵 𝐺 𝑏 𝑐 𝑤 𝐵 𝑂 𝐵 𝐺 𝑏 𝑐 𝑤 𝐵 𝐺 𝑏

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𝑂 𝐵 ∪ {𝑤}

𝑐 𝑤

Re Recurr rrence for

• r Δ𝑏𝑐 𝐵, 𝐺

𝚬𝒃𝒄(𝑩, 𝑮) = 𝚬𝒃𝒄(𝑩, 𝑮 ∪ {𝒘}) ) ∪ 𝚬𝒃𝒄(𝑩’, 𝑮) where 𝒘 ∈ 𝑂(𝐵) ∖ 𝐺 is arbitrary and 𝑂(𝐵′) is the minimal 𝐵 ∪ {𝑤} − 𝑐 separator close to 𝐵 ∪ {𝑤 }.

𝐵 𝑂 𝐵 𝐺 𝑏 𝑐 𝑤 𝐵 𝑂 𝐵 𝐺 𝑏 𝑐 𝑤 𝐵’ 𝑂 𝐵′ 𝐺 𝑏

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Pr Pruning

𝐵 𝑂 𝐵 𝐺 𝑏 𝑐 𝑙 = 3

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Pr Pruning

Vertex disjoint paths from 𝑂 𝐵 ∖ 𝐺 through 𝑊 𝐻 ∖ 𝑂 𝐵

𝐵 𝑂 𝐵 𝐺 𝑏 𝑐 𝑙 = 3

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Pr Pruning

Vertex disjoint paths from 𝑂 𝐵 ∖ 𝐺 through 𝑊 𝐻 ∖ 𝑂 𝐵

𝐵 𝑂 𝐵 𝑇 𝑏 𝑐 𝑙 = 3

This many (15) vertices must be added to the 𝑏-side of any minimal separator in Δ𝑏𝑐 𝐵, 𝐺 Δ𝑏𝑐 𝐵, 𝐺 is empty if |𝐵| + 15 is too large for a smaller of the two full components of a minimal separator

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𝑏

How How to

• it

itera rate thro hrough (𝑏, 𝑐) pairs pairs?

Observation: 𝑌: an arbitrary separator of 𝐻 A minimal separator 𝑇 of 𝐻 either

crosses 𝑌

• r

is local to some component 𝐷 of 𝐻 ∖ 𝑌 𝑌 𝑐 𝑇 𝑌 𝑇 𝑂[𝐷]

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Divi Divide and and con

• nquer?

r?

Δ𝑙(𝐻) is the union of

• Δ𝑏𝑐 for each pair 𝑏, 𝑐 ∈ 𝑌
• Δ𝑙(𝐼) for local graph 𝐼 that arises from each component of 𝐻 ∖ 𝑌

Should 𝑌 be a balanced separator?

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Divi Divide and and con

• nquer?

r?

Δ𝑙(𝐻) is the union of

• Δ𝑏𝑐 for each pair 𝑏, 𝑐 ∈ 𝑌
• Δ𝑙(𝐼) for local graph 𝐼 that arises from each component of 𝐻 ∖ 𝑌

Should 𝑌 be a balanced separator?

NO (experimental observation) Having small |𝑌| or small number of non-adjacent (𝑏, 𝑐) pairs in X is much more important!

Choose 𝑌 to be 𝑂 𝑤 with the smallest number of non- adjacent pairs: 𝑤 is a minfill vertex Nibling

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He Heuris ristic li list sting

• Used to improve an upper bound of treewidth
• Can assume we have a tree-decomposition 𝑈 of width ≥

𝑙 + 1

• The initial set Δ0 consists of minimal separators each of

which is local to some bag of 𝑈

• The expansion Δ 𝑗+1 of Δ𝑗 is obtained by adding the

successors of 𝑇 in Λ 𝐻 that are in Δ𝑙(𝐻) , for each 𝑇 ∈ Δ𝑗

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Thank you !

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