Evaluation and Enumeration Problems for Regular Path Queries Wim - - PowerPoint PPT Presentation

Evaluation and Enumeration Problems for Regular Path Queries

Wim Martens and Tina Trautner University of Bayreuth

Practice

QUERYING PATHS IN GRAPH DATABASES

WIM MARTENS AND TINA TRAUTNER

WIM MARTENS AND TINA TRAUTNER

Node- and Edge-labeled directed graph

Graph Database

Bayreuth Salzburg Prague Vienna Passau Munich

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WIM MARTENS AND TINA TRAUTNER

Warm Up Question

Bayreuth Salzburg Prague Vienna Passau Munich

How many paths from Bayreuth to Vienna match the regular path query (road)* ?

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WIM MARTENS AND TINA TRAUTNER

Warm Up Question

Bayreuth Salzburg Prague Vienna Passau Munich

(How many paths from Bayreuth to Vienna only use road-edges?) How many paths from Bayreuth to Vienna match the regular path query (road)* ?

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d road road road r

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d road road road

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WIM MARTENS AND TINA TRAUTNER

Warm Up Question

[Theoreticians]: ∞ [SPARQL 2018]: 1 [SPARQL 2012]: 3 [Cypher]: 5

Bayreuth Salzburg Prague Vienna Passau Munich

(How many paths from Bayreuth to Vienna only use road-edges?) How many paths from Bayreuth to Vienna match the regular path query (road)* ?

r

• a

d road road road r

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d road road road

r a i l

WIM MARTENS AND TINA TRAUTNER

Warm Up Question

all paths is there at least one path? paths without node repetition paths without edge repetition [Theoreticians]: ∞ [SPARQL 2018]: 1 [SPARQL 2012]: 3 [Cypher]: 5

Bayreuth Salzburg Prague Vienna Passau Munich

(How many paths from Bayreuth to Vienna only use road-edges?) How many paths from Bayreuth to Vienna match the regular path query (road)* ?

r

• a

d road road road r

• a

d road road road

r a i l

WIM MARTENS AND TINA TRAUTNER

Warm Up Question

all paths is there at least one path? paths without node repetition paths without edge repetition [Theoreticians]: ∞ [SPARQL 2018]: 1 [SPARQL 2012]: 3 [Cypher]: 5

Bayreuth Salzburg Prague Vienna Passau Munich

(How many paths from Bayreuth to Vienna only use road-edges?) How many paths from Bayreuth to Vienna match the regular path query (road)* ?

r

• a

d road road road r

• a

d road road road

r a i l

WIM MARTENS AND TINA TRAUTNER

Warm Up Question

all paths is there at least one path? paths without node repetition paths without edge repetition [Theoreticians]: ∞ [SPARQL 2018]: 1 [SPARQL 2012]: 3 [Cypher]: 5

Bayreuth Salzburg Prague Vienna Passau Munich

(How many paths from Bayreuth to Vienna only use road-edges?) How many paths from Bayreuth to Vienna match the regular path query (road)* ?

r

• a

d road road road r

• a

d road road road

r a i l

WIM MARTENS AND TINA TRAUTNER

Warm Up Question

all paths is there at least one path? paths without node repetition paths without edge repetition [Theoreticians]: ∞ [SPARQL 2018]: 1 [SPARQL 2012]: 3 [Cypher]: 5

Bayreuth Salzburg Prague Vienna Passau Munich

(How many paths from Bayreuth to Vienna only use road-edges?) How many paths from Bayreuth to Vienna match the regular path query (road)* ?

r

• a

d road road road r

• a

d road road road

r a i l

WIM MARTENS AND TINA TRAUTNER

Warm Up Question

all paths is there at least one path? paths without node repetition paths without edge repetition [Theoreticians]: ∞ [SPARQL 2018]: 1 [SPARQL 2012]: 3 [Cypher]: 5

Bayreuth Salzburg Prague Vienna Passau Munich

(How many paths from Bayreuth to Vienna only use road-edges?) How many paths from Bayreuth to Vienna match the regular path query (road)* ?

r

• a

d road road road r

• a

d road road road

r a i l

WIM MARTENS AND TINA TRAUTNER

Warm Up Question

all paths is there at least one path? paths without node repetition paths without edge repetition [Theoreticians]: ∞ [SPARQL 2018]: 1 [SPARQL 2012]: 3 [Cypher]: 5

Bayreuth Salzburg Prague Vienna Passau Munich

(How many paths from Bayreuth to Vienna only use road-edges?) How many paths from Bayreuth to Vienna match the regular path query (road)* ?

r

• a

d road road road r

• a

d road road road

r a i l

WIM MARTENS AND TINA TRAUTNER

Warm Up Question

all paths is there at least one path? paths without node repetition paths without edge repetition [Theoreticians]: ∞ [SPARQL 2018]: 1 [SPARQL 2012]: 3 [Cypher]: 5

Bayreuth Salzburg Prague Vienna Passau Munich

(How many paths from Bayreuth to Vienna only use road-edges?) How many paths from Bayreuth to Vienna match the regular path query (road)* ?

r

• a

d road road road r

• a

d road road road

r a i l

WIM MARTENS AND TINA TRAUTNER

The Point

There are different ways of matching paths in graphs and any of them can make sense

WIM MARTENS AND TINA TRAUTNER

The Point

There are different ways of matching paths in graphs and any of them can make sense

But which variant do you want to use in a system?

Theory

ON QUERYING PATHS IN GRAPH DATABASES

WIM MARTENS AND TINA TRAUTNER

WIM MARTENS AND TINA TRAUTNER

Computational Problems

graph (the data) Input Regular expression !

called regular path query (RPQ)

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Computational Problems

graph (the data) Input Problem Is there a path from to that matches !? Path existence Regular expression !

called regular path query (RPQ)

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Computational Problems

graph (the data) Input Problem Regular expression !

called regular path query (RPQ)

How many paths from to match !? Path counting

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Computational Problems

graph (the data) Input Problem Regular expression !

called regular path query (RPQ)

Enumerate the paths from to that match ! Path enumeration

Arbitrary paths Paths without node repetitions

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Considering Different Paths

Boolean paths Paths without edge repetitions

Arbitrary paths Paths without node repetitions

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Considering Different Paths

Boolean paths Paths without edge repetitions

Arbitrary paths Paths without node repetitions

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Considering Different Paths

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Considering Different Paths

Simple paths Arbitrary paths

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Complexity of RPQ Evaluation

Existence Counting Enumeration Arbitrary paths Simple paths in P in FP polynomial delay NP-hard #P-hard too much delay

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Complexity of RPQ Evaluation

Existence Coun,ng Enumeration Arbitrary paths Simple paths in P in FP polynomial delay NP-hard #P-hard too much delay “user happy”: “user unhappy”:

WIM MARTENS AND TINA TRAUTNER

Complexity of RPQ Evaluation

Existence Counting Enumeration Arbitrary paths Simple paths in P in FP polynomial delay NP-hard #P-hard too much delay “user happy”: “user unhappy”:

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Complexity of RPQ Evaluation

similar to counting words in language of regular expression #P-complete [Kannan et al., SODA 1995] Existence Counting Enumeration Arbitrary paths Simple paths

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Complexity of RPQ Evaluation

Existence Counting Enumeration Arbitrary paths Simple paths Is there a simple path matching !∗#!∗? NP-complete [Mendelzon, Wood, SICOMP 1995] essentially because „simple path via a node“ is NP-hard [Fortune et al., TCS 1980] Is there a simple path matching !! ∗? NP-complete [Lapaugh, Papadimitriou, Networks 1984]

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Complexity of RPQ Evaluation

Existence Counting Enumeration Arbitrary paths Simple paths Is there a simple path matching !∗#!∗? NP-complete [Mendelzon, Wood, SICOMP 1995] essentially because „simple path via a node“ is NP-hard [Fortune et al., TCS 1980] Is there a simple path matching !! ∗? NP-complete [Lapaugh, Papadimitriou, Networks 1984] [Bagan, Bonifati, Groz PODS 2013] Dichotomy for which expressions the data complexity of this problem is in P or NP-complete

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Theory VS Systems

Theory: Systems:

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Theory VS Systems

„Simple paths are computationally difficult, even for very small RPQs“ Theory: Systems:

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Theory VS Systems

„Simple paths are computationally difficult, even for very small RPQs“ „But we use simple paths and we‘re fine“ Theory: Systems:

What is going on with these

sim simple ple pa paths ths?

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WIM MARTENS AND TINA TRAUTNER

RPQs in SPARQL Query Logs

Extracted 247,404 RPQs from SPARQL query logs (2009 - 2017) (from DBPedia, biological databases, British museum, Wikidata, …) [Bonifati, M., Timm, PVLDB 2017]

WIM MARTENS AND TINA TRAUTNER

RPQs in SPARQL Query Logs

Extracted 247,404 RPQs from SPARQL query logs (2009 - 2017) (from DBPedia, biological databases, British museum, Wikidata, …) [Bonifati, M., Timm, PVLDB 2017] Only very few different kinds of RPQs (± 17)

Single symbols: !, #, $, !%, … Disjunction

• f symbols:

', '%, …

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RPQs in SPARQL Query Logs

Expression Type Relative Expression Type Relative '∗ 48.76% !∗#? <0.01% ' 32.10% !#$∗ <0.01% !% ⋯ !+ 8.66% '% ⋯ '+ <0.01% !∗# 7.73% !#∗ + $ <0.01% '- 1.54% !∗ + # <0.01% !%? ⋯ !+? 1.15% ! + #- <0.01% !'? 0.01% !- + #- <0.01% !%!.? ⋯ !+? 0.01% !# ∗ <0.01% '? <0.01% / ≤ 6 Data from [Bonifati et al., PVLDB 2017]

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Simple Transitive Expressions

disjunction ("# + ⋯ + "&) of symbols (denote this by (, (),...) Atomic Expression

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Simple Transitive Expressions

„follow a path of length k“ „follow a path of length at most k“

disjunction ("# + ⋯ + "&) of symbols (denote this by (, (),...) (# ⋯ (*

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(#? ⋯ (*? Local Expression Atomic Expression

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Simple Transitive Expressions

„follow a path of length k“ „follow a path of length at most k“

disjunction ("# + ⋯ + "&) of symbols (denote this by (, (),...) (# ⋯ (*

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(#? ⋯ (*? Local Expression ,# (∗,. where ,#, ,. are local expressions Simple Transitive Expression (STE) Atomic Expression

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Simple Transitive Expressions

! " #$ %∗#' where #$, #' are local expressions Simple Transitive Expression (STE) (we allow % = ∅)

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Simple Transitive Expressions

!" ⋯ !$

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!"? ⋯ !$? & ' (" !∗(* where (", (* are local expressions Simple Transitive Expression (STE) (we allow ! = ∅)

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Simple Transitive Expressions

!∗ !# ⋯ !%

• r

!#? ⋯ !%? ' ( )# !∗)* where )#, )* are local expressions Simple Transitive Expression (STE) (we allow ! = ∅)

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Simple Transitive Expressions

!∗ !# ⋯ !%

• r

!#? ⋯ !%? !#

' ⋯ !( '

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!#

' ? ⋯ !( '?

) * +# !∗+, where +#, +, are local expressions Simple Transitive Expression (STE) (we allow ! = ∅)

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RPQs in SPARQL Query Logs

Expression Type Relative Expression Type Relative !∗ 48.76% #∗$? <0.01% ! 32.10% #$&∗ <0.01% #' ⋯ #) 8.66% !' ⋯ !) <0.01% #∗$ 7.73% #$∗ + & <0.01% !+ 1.54% #∗ + $ <0.01% #'? ⋯ #)? 1.15% # + $+ <0.01% #!? 0.01% #+ + $+ <0.01% #'#,? ⋯ #)? 0.01% #$ ∗ <0.01% !? <0.01% STE Union of STEs something else

• ≤ 6

Data from [Bonifati et al., PVLDB 2017]

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RPQs in SPARQL Query Logs

Expression Type Relative Expression Type Relative !∗ 48.76% #∗$? <0.01% ! 32.10% #$&∗ <0.01% #' ⋯ #) 8.66% !' ⋯ !) <0.01% #∗$ 7.73% #$∗ + & <0.01% !+ 1.54% #∗ + $ <0.01% #'? ⋯ #)? 1.15% # + $+ <0.01% #!? 0.01% #+ + $+ <0.01% #'#,? ⋯ #)? 0.01% #$ ∗ <0.01% !? <0.01% STE Union of STEs something else

• ≤ 6

Data from [Bonifati et al., PVLDB 2017]

99.99% are STEs!

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Main Theorem Warm-Up

Given graph !, nodes " and #, and RPQ $, is there a simple path from " to # that matches $? Simple path existence

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Main Theorem Warm-Up

Given graph !, nodes " and #, and RPQ $ ∈ &, is there a simple path from " to # that matches $? Simple path existence for & Example classes &:

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Main Theorem Warm-Up

Given graph !, nodes " and #, and RPQ $ ∈ &, is there a simple path from " to # that matches $? Simple path existence for & Example classes &: '' … ' for ) ∈ ℕ denote this by {') | ) ∈ ℕ} )

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Main Theorem Warm-Up

Given graph !, nodes " and #, and RPQ $ ∈ &, is there a simple path from " to # that matches $? Simple path existence for & Example classes &: '' … ' for ) ∈ ℕ denote this by {') | ) ∈ ℕ} ) '' … ''∗ for ) ∈ ℕ denote this by {')'∗| ) ∈ ℕ}

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Main Theorem Warm-Up

Given graph !, nodes " and #, and RPQ $ ∈ &, is there a simple path from " to # that matches $? Simple path existence for & Example classes &: '' … ' for ) ∈ ℕ denote this by {') | ) ∈ ℕ} These are non-trivial problems! ) '' … ''∗ for ) ∈ ℕ denote this by {')'∗| ) ∈ ℕ}

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Main Theorem Warm-Up

Given graph !, nodes " and #, and RPQ $ ∈ &, is there a simple path from " to # that matches $? Simple path existence for & Example classes &: '' … ' for ) ∈ ℕ denote this by {') | ) ∈ ℕ} “Simple path existence for {') | ) ∈ ℕ} is in FPT“ Theorem [Alon, Yuster, Zwick, JACM 1995] Color coding technique ) '' … ''∗ for ) ∈ ℕ denote this by {')'∗| ) ∈ ℕ}

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Main Theorem Warm-Up

Given graph !, nodes " and #, and RPQ $ ∈ &, is there a simple path from " to # that matches $? Simple path existence for & Example classes &: '' … ' for ) ∈ ℕ denote this by {') | ) ∈ ℕ} “Simple path existence for {') | ) ∈ ℕ} is in FPT“ Theorem [Alon, Yuster, Zwick, JACM 1995] Color coding technique “Simple path existence for {')'∗| ) ∈ ℕ} is in FPT“ Theorem [Technique from Fomin et al., JACM 2016] communicated to us by Holger Dell Representative sets technique ) '' … ''∗ for ) ∈ ℕ denote this by {')'∗| ) ∈ ℕ}

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Main Theorem

(*) satisfying a mild condition,

needed for W[1] hardness Let ! be a class(*) of STEs: if ! is cuttable, then simple path existence for ! is in FPT

• therwise, simple path existence for ! is W["]-hard.

Main Theorem Given graph #, nodes $ and %, and RPQ & ∈ !, is there a simple path from $ to % that matches &? Simple path existence for !

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Main Theorem

(*) satisfying a mild condition,

needed for W[1] hardness Let ! be a class(*) of STEs: if ! is cuttable, then simple path existence for ! is in FPT

• therwise, simple path existence for ! is W["]-hard.

Main Theorem parameter: size of RPQ Given graph #, nodes $ and %, and RPQ & ∈ !, is there a simple path from $ to % that matches &? Simple path existence for !

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Intuition behind Cuttability

! " Path that matches #

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Intuition behind Cuttability

! " Path that matches # ? Simple

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Intuition behind Cuttability

! " Path that matches # Does the simple path still match #? § “Easy” to check for $$$$$∗ (check length) § “Hard” to check for &&&&$∗ (check length+label) ? Simple

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Intuition behind Cuttability

! " Path that matches # Does the simple path still match #? § “Easy” to check for $$$$$∗ (check length) § “Hard” to check for &&&&$∗ (check length+label) ? ≥ 4 Simple

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Intuition behind Cuttability

! " Path that matches # Does the simple path still match #? § “Easy” to check for $$$$$∗ (check length) § “Hard” to check for &&&&$∗ (check length+label) ? ≥ 4 cut border for &&&&$∗ Simple

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Formalizing this Idea

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Formalizing this Idea

Consider STE ! = #$ ⋯ #& #∗ Its cut border ℓ is the largest number such that # ⊈ #ℓ (and ℓ = 0 if no such #ℓ exists) Examples

++++∗ ℓ = 0 because + ⊆ + ++-+∗ ℓ = 3 because + ⊈ - (+ + 1)+-(+ + -)∗ ℓ = 3 because +, - ⊈ -

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Formalizing this Idea

Consider STE ! = #$ ⋯ #& #∗ Its cut border ℓ is the largest number such that # ⊈ #ℓ (and ℓ = 0 if no such #ℓ exists) Examples

§ ++++∗ ℓ = 0 because + ⊆ + § ++-+∗ ℓ = 3 because + ⊈ - § (+ + 1)+-(+ + -)∗ ℓ = 3 because +, - ⊈ -

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Formalizing this Idea

Consider STE ! = #$ ⋯ #& #∗ Its cut border ℓ is the largest number such that # ⊈ #ℓ (and ℓ = 0 if no such #ℓ exists) Examples

§ ++++∗ ℓ = 0 because + ⊆ + § ++-+∗ ℓ = 3 because + ⊈ - § (+ + 1)+-(+ + -)∗ ℓ = 3 because +, - ⊈ -

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Formalizing this Idea

Consider STE ! = #$ ⋯ #& #∗ Its cut border ℓ is the largest number such that # ⊈ #ℓ (and ℓ = 0 if no such #ℓ exists) Examples

§ ++++∗ ℓ = 0 because + ⊆ + § ++-+∗ ℓ = 3 because + ⊈ - § (+ + 1)+-(+ + -)∗ ℓ = 3 because +, - ⊈ - A class 4 of STEs is cuttable, if there is a constant 1 such that all its expressions have cut border ≤ 1 Definition

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Formalizing this Idea

A class ! of STEs is cuttable, if there is a constant " such that all its expressions have cut border ≤ " Definition Let ! be a class(*) of STEs: if ! is cuttable, then simple path existence for ! is in FPT

• therwise, simple path existence for ! is W[$]-hard.

Main Theorem parameter: size of RPQ For the FPT upper bound, the complexity in the parameter is single exponential

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Upper Bound Idea

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Upper Bound Idea

Finding simple cycles of length at least k is in FPT Theorem [Fomin et al., JACM 2016] Finding simple paths of length at least k is in FPT Theorem [Technique from Fomin et al., JACM 2016] communicated to us by Holger Dell Finding simple paths of length exactly k is in FPT Theorem [Alon, Yuster, Zwick, JACM 1995] Color coding technique Representative sets technique Representative sets technique

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Upper Bound Idea

! " Find a simple path matching #$ ⋯ #& #∗

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Upper Bound Idea

Brute force ! " #$ ⋯ #& Find a simple path matching #$ ⋯ #' #∗

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Upper Bound Idea

Brute force ! " #$ ⋯ #& Simple Path matching #&'$ ⋯ #( #∗ and avoiding the brute force part Find a simple path matching #$ ⋯ #( #∗

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Upper Bound Idea

Brute force ! " #$ ⋯ #& #&'$ ⋯ #( #∗ Simple Path matching #&'$ ⋯ #( #∗ and avoiding the brute force part Find a simple path matching #$ ⋯ #( #∗

Upper Bound Idea

! " #∗ #% ⋯ #' Find a simple path matching #% ⋯ #( #∗

Upper Bound Idea

! " #∗ #%&' ⋯ #) #' ⋯ #% Since # ⊆ #+ Find a simple path matching #' ⋯ #) #∗

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Lower Bound Idea

!" #$ #" !$ Parameterized Two Disjoint Paths

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Lower Bound Idea

!" #$ #" !$ Given graph %, nodes !" and #" and !$ and #$ and a parameter k Are there node-disjoint paths from !" to #" from !$ to #$ Parameterized Two Disjoint Paths

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Lower Bound Idea

!" #$ #" !$ Given graph %, nodes !" and #" and !$ and #$ and a parameter k Are there node-disjoint paths from !" to #" from !$ to #$ Parameterized Two Disjoint Paths

• f length at most k

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Lower Bound Idea

Parameterized Two Disjoint Paths is W[1]-hard Theorem (Main Technical Result) Building on proofs from [Slivkins, SIDMA 10; Grohe&Grüber ICALP 07] Given graph !, nodes "# and $# and "% and $% and a parameter k Are there node-disjoint paths from "# to $# from "% to $% Parameterized Two Disjoint Paths

• f length at most k

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Lower Bound Idea

Parameterized Two Disjoint Paths is W[1]-hard Theorem (Main Technical Result) Let ! be a class(*) of STEs: if ! is not cuttable, then simple path existence for ! is W["]-hard. Lemma

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Lower Bound Idea

Parameterized Two Disjoint Paths is W[1]-hard Theorem (Main Technical Result) !" #$ #" !$ Let % be a class(*) of STEs: if % is not cuttable, then simple path existence for % is W["]-hard. Lemma

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Lower Bound Idea

Parameterized Two Disjoint Paths is W[1]-hard Theorem (Main Technical Result) !" #$ #" !$ % Let & be a class(*) of STEs: if & is not cuttable, then simple path existence for & is W["]-hard. Lemma Warning: drastic oversimplification

Extensions

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Extensions

The main result extends to: § Enumeration problems FPT time becomes FPT delay using [Yen 1971] § Edge-disjoint paths But the dichotomy slightly changes [ArXiv 2017]

Let ! be a class(*) of STEs: if ! is cuttable, then simple path existence for ! is in FPT

• therwise, simple path existence for ! is W["]-hard.

Main Theorem

Taking a Step Back

WHAT DID WE LEARN HERE?

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So what does all this mean?

Cuttable STEs (ℓ ≤ 2) Thus in FPT Union of STEs something else k ≤ 6

• These expressions have cut border ≤ 2
• The FPT algorithms have parameter % ≤ 6
• But even naive algorithms are expected to work reasonably well

(brute-force checks for paths of lengh 2 and simple paths of length 6) Expression Type Relative Expression Type Relative &∗ 48.76% (∗)? <0.01% & 32.10% ()+∗ <0.01% (, ⋯ (. 8.66% &, ⋯ &. <0.01% (∗) 7.73% ()∗ + + <0.01% &0 1.54% (∗ + ) <0.01% (,? ⋯ (.? 1.15% ( + )0 <0.01% (&? 0.01% (0 + )0 <0.01% (,(1? ⋯ (.? 0.01% () ∗ <0.01% &? <0.01%

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Take Home Messages

§ Looking in query logs can pay off and inspire new research questions! §99.99% of RPQs found in a practical study are Simple Transitive Expressions (STEs) § Dichotomy for simple path evaluation of STEs

§ Another one for no-repeated-edge semantics is similar

§ If “cut borders are bounded”, evaluation of STEs is FPT

§ Cut borders in the real data are at most 2 § “FPT parameters” in the real data are 6 (for exact length) and 2 (for minimum length)

Thank you!

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