Topological Sorting under Regular Constraints

Topological Sorting under Regular Constraints

Antoine Amarilli¹,Charles Paperman² December 7th, 2018

1Télécom ParisTech

2Université de Lille

Constrained Topological Sorting

•

^{Fix an}alphabet: e.g.,Σ ={a,b}

•

^{Fix alanguage: e.g.,L= (ab)∗}

•

^{We study}constrained topological sorting:

• Input:directed acyclic graph(DAG) with vertices labeled withΣ

• Output: is there atopological sort that falls inL?

•

^Question:when is this problemtractable?

a

b a b

b a

a b a b a b ... inL!

Constrained Topological Sorting

•

^{Fix an}alphabet: e.g.,Σ ={a,b}

•

^{Fix a}language: e.g.,L= (ab)^∗

•

^{We study}constrained topological sorting:

• Input:directed acyclic graph(DAG) with vertices labeled withΣ

• Output: is there atopological sort that falls inL?

•

^Question:when is this problemtractable?

a

b a b

b a

a b a b a b ... inL!

Constrained Topological Sorting

•

^{Fix an}alphabet: e.g.,Σ ={a,b}

•

^{Fix a}language: e.g.,L= (ab)^∗

•

^{We study}constrained topological sorting:

• Input:directed acyclic graph(DAG) with vertices labeled withΣ

• Output: is there atopological sort that falls inL?

•

^Question:when is this problemtractable?

a

b a b

b a

a b a b a b ... inL!

Constrained Topological Sorting

•

^{Fix an}alphabet: e.g.,Σ ={a,b}

•

^{Fix a}language: e.g.,L= (ab)^∗

•

^{We study}constrained topological sorting:

• Input:directed acyclic graph(DAG) with vertices labeled withΣ

• Output: is there atopological sort that falls inL?

•

^Question:when is this problemtractable?

a

b a b

b a

a b a b a b ... inL!

Constrained Topological Sorting

•

^{Fix an}alphabet: e.g.,Σ ={a,b}

•

^{Fix a}language: e.g.,L= (ab)^∗

•

^{We study}constrained topological sorting:

• Input:directed acyclic graph(DAG) with vertices labeled withΣ

• Output: is there atopological sort that falls inL?

•

^Question:when is this problemtractable?

a

b a b

b a

a b a b a b ... inL!

Constrained Topological Sorting

•

^{Fix an}alphabet: e.g.,Σ ={a,b}

•

^{Fix a}language: e.g.,L= (ab)^∗

•

^{We study}constrained topological sorting:

• Input:directed acyclic graph(DAG) with vertices labeled withΣ

• Output: is there atopological sort that falls inL?

•

^Question:when is this problemtractable?

a

b a b

b a

a

b a b b a ... not inL!

Constrained Topological Sorting

•

^{Fix an}alphabet: e.g.,Σ ={a,b}

•

^{Fix a}language: e.g.,L= (ab)^∗

•

^{We study}constrained topological sorting:

• Input:directed acyclic graph(DAG) with vertices labeled withΣ

• Output: is there atopological sort that falls inL?

•

^Question:when is this problemtractable?

a

b a b

b a

a b

a b b a ... not inL!

Constrained Topological Sorting

•

^{Fix an}alphabet: e.g.,Σ ={a,b}

•

^{Fix a}language: e.g.,L= (ab)^∗

•

^{We study}constrained topological sorting:

• Input:directed acyclic graph(DAG) with vertices labeled withΣ

• Output: is there atopological sort that falls inL?

•

^Question:when is this problemtractable?

a

b a b

b a

a b a

b b a ... not inL!

Constrained Topological Sorting

•

^{Fix an}alphabet: e.g.,Σ ={a,b}

•

^{Fix a}language: e.g.,L= (ab)^∗

•

^{We study}constrained topological sorting:

• Input:directed acyclic graph(DAG) with vertices labeled withΣ

• Output: is there atopological sort that falls inL?

•

^Question:when is this problemtractable?

a

b a b

b a

a b a b

b a ... not inL!

Constrained Topological Sorting

•

^{Fix an}alphabet: e.g.,Σ ={a,b}

•

^{Fix a}language: e.g.,L= (ab)^∗

•

^{We study}constrained topological sorting:

• Input:directed acyclic graph(DAG) with vertices labeled withΣ

• Output: is there atopological sort that falls inL?

•

^Question:when is this problemtractable?

a

b a b

b a

a b a b b

a ... not inL!

Constrained Topological Sorting

•

^{Fix an}alphabet: e.g.,Σ ={a,b}

•

^{Fix a}language: e.g.,L= (ab)^∗

•

^{We study}constrained topological sorting:

• Input:directed acyclic graph(DAG) with vertices labeled withΣ

• Output: is there atopological sort that falls inL?

•

^Question:when is this problemtractable?

a

b a b

b a

a b a b b a

... not inL!

Constrained Topological Sorting

•

^{Fix an}alphabet: e.g.,Σ ={a,b}

•

^{Fix a}language: e.g.,L= (ab)^∗

•

^{We study}constrained topological sorting:

• Input:directed acyclic graph(DAG) with vertices labeled withΣ

• Output: is there atopological sort that falls inL?

•

^Question:when is this problemtractable?

a

b a b

b a

a b a b b a ... not inL!

Constrained Topological Sorting

•

^{Fix an}alphabet: e.g.,Σ ={a,b}

•

^{Fix a}language: e.g.,L= (ab)^∗

•

^{We study}constrained topological sorting:

• Input:directed acyclic graph(DAG) with vertices labeled withΣ

• Output: is there atopological sort that falls inL?

•

^Question:when is this problemtractable?

a

b a b

b a

a b a b a b ... inL!

Constrained Topological Sorting

•

^{Fix an}alphabet: e.g.,Σ ={a,b}

•

^{Fix a}language: e.g.,L= (ab)^∗

•

^{We study}constrained topological sorting:

• Input:directed acyclic graph(DAG) with vertices labeled withΣ

• Output: is there atopological sort that falls inL?

•

^Question:when is this problemtractable?

a

b a b

b a

a

b a b a b ... inL!

Constrained Topological Sorting

•

^{Fix an}alphabet: e.g.,Σ ={a,b}

•

^{Fix a}language: e.g.,L= (ab)^∗

•

^{We study}constrained topological sorting:

• Input:directed acyclic graph(DAG) with vertices labeled withΣ

• Output: is there atopological sort that falls inL?

•

^Question:when is this problemtractable?

a

b a b

b a

a b

a b a b ... inL!

Constrained Topological Sorting

•

^{Fix an}alphabet: e.g.,Σ ={a,b}

•

^{Fix a}language: e.g.,L= (ab)^∗

•

^{We study}constrained topological sorting:

• Input:directed acyclic graph(DAG) with vertices labeled withΣ

• Output: is there atopological sort that falls inL?

•

^Question:when is this problemtractable?

a

b a b

b a

a b a

b a b ... inL!

Constrained Topological Sorting

•

^{Fix an}alphabet: e.g.,Σ ={a,b}

•

^{Fix a}language: e.g.,L= (ab)^∗

•

^{We study}constrained topological sorting:

• Input:directed acyclic graph(DAG) with vertices labeled withΣ

• Output: is there atopological sort that falls inL?

•

^Question:when is this problemtractable?

a

b a b

b a

a b a b

a b ... inL!

Constrained Topological Sorting

•

^{Fix an}alphabet: e.g.,Σ ={a,b}

•

^{Fix a}language: e.g.,L= (ab)^∗

•

^{We study}constrained topological sorting:

• Input:directed acyclic graph(DAG) with vertices labeled withΣ

• Output: is there atopological sort that falls inL?

•

^Question:when is this problemtractable?

a

b a b

b a

a b a b a

b ... inL!

Constrained Topological Sorting

•

^{Fix an}alphabet: e.g.,Σ ={a,b}

•

^{Fix a}language: e.g.,L= (ab)^∗

•

^{We study}constrained topological sorting:

• Input:directed acyclic graph(DAG) with vertices labeled withΣ

• Output: is there atopological sort that falls inL?

•

^Question:when is this problemtractable?

a

b a b

b a

a b a b a b

... inL!

Constrained Topological Sorting

•

^{Fix an}alphabet: e.g.,Σ ={a,b}

•

^{Fix a}language: e.g.,L= (ab)^∗

•

^{We study}constrained topological sorting:

• Input:directed acyclic graph(DAG) with vertices labeled withΣ

• Output: is there atopological sort that falls inL?

•

^Question:when is this problemtractable?

a

b a b

b a

a b a b a b ... inL!

Constrained Topological Sorting

•

^{Fix an}alphabet: e.g.,Σ ={a,b}

•

^{Fix a}language: e.g.,L= (ab)^∗

•

^{We study}constrained topological sorting:

• Input:directed acyclic graph(DAG) with vertices labeled withΣ

• Output: is there atopological sort that falls inL?

•

^Question:when is this problemtractable?

a

b a b

b a

a b a b a b ... inL!

Constrained Topological Sorting

•

^{Fix an}alphabet: e.g.,Σ ={a,b}

•

^{Fix a}language: e.g.,L= (ab)^∗

•

^{We study}constrained topological sorting:

• Input:directed acyclic graph(DAG) with vertices labeled withΣ

• Output: is there atopological sort that falls inL?

•

^Question:when is this problemtractable?

a

b a b

b a

a b a b a b ... inL!

Motivation

•

^{How we}really ended upstudying this problem:

2011 2012 2013 2014 2015 2016 2017 2018

Probabilistic XML XML versioning

•

^Whicha-posteriori motivationdid we invent for the problem?

→Schedulingwith constraints! →Verification forconcurrent code!

→Computational biology! →Blockchain! (joke)

•

But why do weactually care?

→ Naturalproblem and apparentlynothingwas known about it!

Motivation

•

^{How we}really ended upstudying this problem:

2011 2012 2013 2014 2015 2016 2017 2018

Probabilistic XML XML versioning

•

^Whicha-posteriori motivationdid we invent for the problem?

→Schedulingwith constraints! →Verification forconcurrent code!

→Computational biology! →Blockchain! (joke)

•

But why do weactually care?

→ Naturalproblem and apparentlynothingwas known about it!

Motivation

•

^{How we}really ended upstudying this problem:

2011 2012 2013 2014 2015 2016 2017 2018

Probabilistic XML XML versioning

Order-uncertain databases Top-k Aggregate queries

•

^Whicha-posteriori motivationdid we invent for the problem?

→Schedulingwith constraints! →Verification forconcurrent code!

→Computational biology! →Blockchain! (joke)

•

But why do weactually care?

→ Naturalproblem and apparentlynothingwas known about it!

Motivation

•

^{How we}really ended upstudying this problem:

2011 2012 2013 2014 2015 2016 2017 2018

Probabilistic XML XML versioning

Order-uncertain databases Top-k Aggregate queries Possible answers

•

^Whicha-posteriori motivationdid we invent for the problem?

→Schedulingwith constraints! →Verification forconcurrent code!

→Computational biology! →Blockchain! (joke)

•

But why do weactually care?

→ Naturalproblem and apparentlynothingwas known about it!

Motivation

•

^{How we}really ended upstudying this problem:

2011 2012 2013 2014 2015 2016 2017 2018

Probabilistic XML XML versioning

Order-uncertain databases Top-k Aggregate queries Possible answers

DAGs Algebraic language theory?!

•

^Whicha-posteriori motivationdid we invent for the problem?

→Schedulingwith constraints! →Verification forconcurrent code!

→Computational biology! →Blockchain! (joke)

•

But why do weactually care?

→ Naturalproblem and apparentlynothingwas known about it!

Motivation

•

^{How we}really ended upstudying this problem:

2011 2012 2013 2014 2015 2016 2017 2018

Probabilistic XML XML versioning

Order-uncertain databases Top-k Aggregate queries Possible answers

DAGs Algebraic language theory?!

•

^Whicha-posteriori motivationdid we invent for the problem?

→Schedulingwith constraints! →Verification forconcurrent code!

→Computational biology! →Blockchain! (joke)

•

But why do weactually care?

→ Naturalproblem and apparentlynothingwas known about it!

Motivation

•

^{How we}really ended upstudying this problem:

2011 2012 2013 2014 2015 2016 2017 2018

Probabilistic XML XML versioning

Order-uncertain databases Top-k Aggregate queries Possible answers

DAGs Algebraic language theory?!

•

^Whicha-posteriori motivationdid we invent for the problem?

→Schedulingwith constraints! →Verification forconcurrent code!

→Computational biology! →Blockchain! (joke)

•

But why do weactually care?

→ Naturalproblem and apparentlynothingwas known about it!

Motivation

•

^{How we}really ended upstudying this problem:

2011 2012 2013 2014 2015 2016 2017 2018

Probabilistic XML XML versioning

Order-uncertain databases Top-k Aggregate queries Possible answers

DAGs Algebraic language theory?!

•

^Whicha-posteriori motivationdid we invent for the problem?

→Schedulingwith constraints! →Verification forconcurrent code!

→Computational biology! →Blockchain!

(joke)

•

But why do weactually care?

→ Naturalproblem and apparentlynothingwas known about it!

Motivation

•

^{How we}really ended upstudying this problem:

2011 2012 2013 2014 2015 2016 2017 2018

Probabilistic XML XML versioning

Order-uncertain databases Top-k Aggregate queries Possible answers

DAGs Algebraic language theory?!

•

^Whicha-posteriori motivationdid we invent for the problem?

→Schedulingwith constraints! →Verification forconcurrent code!

→Computational biology! →Blockchain! (joke)

•

But why do weactually care?

→ Naturalproblem and apparentlynothingwas known about it!

Motivation

•

^{How we}really ended upstudying this problem:

2011 2012 2013 2014 2015 2016 2017 2018

Probabilistic XML XML versioning

Order-uncertain databases Top-k Aggregate queries Possible answers

DAGs Algebraic language theory?!

•

^Whicha-posteriori motivationdid we invent for the problem?

→Schedulingwith constraints! →Verification forconcurrent code!

→Computational biology! →Blockchain! (joke)

•

But why do weactually care?

→ Naturalproblem and apparentlynothingwas known about it!

Motivation

•

^{How we}really ended upstudying this problem:

2011 2012 2013 2014 2015 2016 2017 2018

Probabilistic XML XML versioning

Order-uncertain databases Top-k Aggregate queries Possible answers

DAGs Algebraic language theory?!

•

^Whicha-posteriori motivationdid we invent for the problem?

→Schedulingwith constraints! →Verification forconcurrent code!

→Computational biology! →Blockchain! (joke)

•

Formal problem statement

•

^{Fix a}regular languageLon anfinite alphabetΣ

•

Constrained topological sortproblemCTS(L):

• Input:aDAGGwith vertices labeled by letters ofΣ

• Output:is there atopological sortofGsuch that the sequence of vertex labels is awordofL

•

Special case: theconstrained shuffle problemCSh(L):

• Input:a set ofwordsw₁, . . . ,w_nofΣ^∗

• Output:is there ashuffleofw1, . . . ,wnwhich is inL

•

This is likeCTSbut the input DAG is anunion of paths

→ Question:What is thecomplexityofCTS(L)andCSh(L), depending on the fixed languageL?

Formal problem statement

•

^{Fix a}regular languageLon anfinite alphabetΣ

•

Constrained topological sortproblemCTS(L):

• Input:aDAGGwith vertices labeled by letters ofΣ

• Output:is there atopological sortofGsuch that the sequence of vertex labels is awordofL

a

b a b

b a

•

Special case: theconstrained shuffle problemCSh(L):

• Input:a set ofwordsw₁, . . . ,w_nofΣ^∗

• Output:is there ashuffleofw1, . . . ,wnwhich is inL

•

This is likeCTSbut the input DAG is anunion of paths

→ Question:What is thecomplexityofCTS(L)andCSh(L), depending on the fixed languageL?

Formal problem statement

•

^{Fix a}regular languageLon anfinite alphabetΣ

•

Constrained topological sortproblemCTS(L):

• Input:aDAGGwith vertices labeled by letters ofΣ

• Output:is there atopological sortofGsuch that the sequence of vertex labels is awordofL

a

b a b

b a

•

Special case: theconstrained shuffle problemCSh(L):

• Input:a set ofwordsw₁, . . . ,w_nofΣ^∗

• Output:is there ashuffleofw1, . . . ,wnwhich is inL

•

This is likeCTSbut the input DAG is anunion of paths

→ Question:What is thecomplexityofCTS(L)andCSh(L), depending on the fixed languageL?

Formal problem statement

•

^{Fix a}regular languageLon anfinite alphabetΣ

•

Constrained topological sortproblemCTS(L):

• Input:aDAGGwith vertices labeled by letters ofΣ

• Output:is there atopological sortofGsuch that the sequence of vertex labels is awordofL

a

b a b

b a

•

Special case: theconstrained shuffle problemCSh(L):

• Input:a set ofwordsw₁, . . . ,w_nofΣ^∗

• Output:is there ashuffleofw1, . . . ,wnwhich is inL

•

This is likeCTSbut the input DAG is anunion of paths

a a b

b b a

→ Question:What is thecomplexityofCTS(L)andCSh(L), depending on the fixed languageL?

Formal problem statement

•

^{Fix a}regular languageLon anfinite alphabetΣ

•

Constrained topological sortproblemCTS(L):

• Input:aDAGGwith vertices labeled by letters ofΣ

• Output:is there atopological sortofGsuch that the sequence of vertex labels is awordofL

a

b a b

b a

•

Special case: theconstrained shuffle problemCSh(L):

• Input:a set ofwordsw₁, . . . ,w_nofΣ^∗

• Output:is there ashuffleofw1, . . . ,wnwhich is inL

•

This is likeCTSbut the input DAG is anunion of paths

a a b

b b a

→ Question:What is thecomplexityofCTS(L)andCSh(L), depending on the fixed languageL?

Dichotomy

Conjecture

Conjecture

For everyregular language L, exactly one of the following holds:

•

^Lhas [some nice property] andCTS(L)isin NL

•

^Lhas [some nasty property] andCTS(L)isNP-hard

Here’s what we actually know:

•

^{CTSandCShareNP-hard}for some languages, including(ab)^∗

•

^{They arein NLfor}some language families(monomials, groups)

•

Some languages are tractable forseemingly unrelatedreasons

→ Very mysterious landscape!(to us)

Dichotomy Conjecture

Conjecture

For everyregular language L, exactly one of the following holds:

•

^Lhas [some nice property] andCTS(L)isin NL

•

^Lhas [some nasty property] andCTS(L)isNP-hard

Here’s what we actually know:

•

^{CTSandCShareNP-hard}for some languages, including(ab)^∗

•

^{They arein NLfor}some language families(monomials, groups)

•

Some languages are tractable forseemingly unrelatedreasons

→ Very mysterious landscape!(to us)

Dichotomy Conjecture

Conjecture

For everyregular language L, exactly one of the following holds:

•

^Lhas [some nice property] andCTS(L)isin NL

•

^Lhas [some nasty property] andCTS(L)isNP-hard Here’s what we actually know:

•

^{CTSandCShareNP-hard}for some languages, including(ab)^∗

•

^{They arein NLfor}some language families(monomials, groups)

•

Some languages are tractable forseemingly unrelatedreasons

→ Very mysterious landscape!(to us)

Dichotomy Conjecture

Conjecture

For everyregular language L, exactly one of the following holds:

•

^Lhas [some nice property] andCTS(L)isin NL

•

^Lhas [some nasty property] andCTS(L)isNP-hard Here’s what we actually know:

•

^{CTSandCShareNP-hard}for some languages, including(ab)^∗

•

^{They arein NLfor}some language families(monomials, groups)

•

Some languages are tractable forseemingly unrelatedreasons

→ Very mysterious landscape!(to us)

Dichotomy Conjecture

Conjecture

For everyregular language L, exactly one of the following holds:

•

^Lhas [some nice property] andCTS(L)isin NL

•

^Lhas [some nasty property] andCTS(L)isNP-hard Here’s what we actually know:

•

^{CTSandCShareNP-hard}for some languages, including(ab)^∗

•

^{They arein NLfor}some language families(monomials, groups)

•

Some languages are tractable forseemingly unrelatedreasons

→ Very mysterious landscape!(to us)

Dichotomy Conjecture

Conjecture

For everyregular language L, exactly one of the following holds:

•

^Lhas [some nice property] andCTS(L)isin NL

•

^Lhas [some nasty property] andCTS(L)isNP-hard Here’s what we actually know:

•

^{CTSandCShareNP-hard}for some languages, including(ab)^∗

•

^{They arein NLfor}some language families(monomials, groups)

•

Some languages are tractable forseemingly unrelatedreasons

→ Very mysterious landscape!(to us)

Hardness Results

Existing Hardness Result

... but the target is awordwhich isprovided as input!

→ Does not directly apply for us, because wefixthe target language

Existing Hardness Result

... but the target is awordwhich isprovided as input!

→ Does not directly apply for us, because wefixthe target language

Existing Hardness Result

... but the target is awordwhich isprovided as input!

→ Does not directly apply for us, because wefixthe target language

Hardness for CTS

•

We can reduce their problem toCShfor thelanguage(aA+bB)^∗

•

To determine if the shuffle of aab and bbcontains ababb ...

solve theCSh-problem for aab and bband ABABB

→ CSh((aA+bB)^∗)isNP-hardand the same holds forCTS

•

Similar technique:CSh((ab)^∗)isNP-hard

•

^Customreduction techniqueto show NP-hardness

Hardness for CTS

•

We can reduce their problem toCShfor thelanguage(aA+bB)^∗

•

To determine if the shuffle of aab and bbcontains ababb ...

solve theCSh-problem for aab and bband ABABB

→ CSh((aA+bB)^∗)isNP-hardand the same holds forCTS

•

Similar technique:CSh((ab)^∗)isNP-hard

•

^Customreduction techniqueto show NP-hardness

Hardness for CTS

•

We can reduce their problem toCShfor thelanguage(aA+bB)^∗

•

To determine if the shuffle of aab and bbcontains ababb ...

solve theCSh-problem for aab and bband ABABB

→ CSh((aA+bB)^∗)isNP-hardand the same holds forCTS

•

Similar technique:CSh((ab)^∗)isNP-hard

•

^Customreduction techniqueto show NP-hardness

Hardness for CTS

•

We can reduce their problem toCShfor thelanguage(aA+bB)^∗

•

To determine if the shuffle of aab and bbcontains ababb ...

solve theCSh-problem for aab and bband ABABB

→ CSh((aA+bB)^∗)isNP-hardand the same holds forCTS

•

Similar technique:CSh((ab)^∗)isNP-hard

•

^Customreduction techniqueto show NP-hardness

Hardness for CTS

•

We can reduce their problem toCShfor thelanguage(aA+bB)^∗

•

To determine if the shuffle of aab and bbcontains ababb ...

solve theCSh-problem for aab and bband ABABB

→ CSh((aA+bB)^∗)isNP-hardand the same holds forCTS

•

Similar technique:CSh((ab)^∗)isNP-hard

•

^Customreduction techniqueto show NP-hardness

The reduction technique

G a

b a b

b a

P b c a c b c a c b c a c

•

Say we want to solveCTSfor(ab)^∗(NP-hard)

•

Say we know how to solveCTSfor(abc)^∗

•

^{Take aninstanceGfor(ab)∗, with2nvertices}

•

^{Add thepathP: (bcac)n}

•

A topsort ofG∪Pachieving(abc)^∗ gives a topsort ofGachieving(ab)^∗

•

Conversely, any topsort ofGachieving(ab)^∗ extends to a topsort ofG+Pachieving(abc)^∗

•

^{Hence,CTS((abc)∗)isNP-hard}

The reduction technique

G a

b a b

b a

P b c a c b c a c b c a c

•

Say we want to solveCTSfor(ab)^∗(NP-hard)

•

Say we know how to solveCTSfor(abc)^∗

•

^{Take aninstanceGfor(ab)∗, with2nvertices}

•

^{Add thepathP: (bcac)n}

•

A topsort ofG∪Pachieving(abc)^∗ gives a topsort ofGachieving(ab)^∗

•

Conversely, any topsort ofGachieving(ab)^∗ extends to a topsort ofG+Pachieving(abc)^∗

•

^{Hence,CTS((abc)∗)isNP-hard}

The reduction technique

G a

b a b

b a

P b c a c b c a c b c a c

•

Say we want to solveCTSfor(ab)^∗(NP-hard)

•

Say we know how to solveCTSfor(abc)^∗

•

^{Take aninstanceGfor(ab)∗, with2nvertices}

•

^{Add thepathP: (bcac)n}

•

A topsort ofG∪Pachieving(abc)^∗ gives a topsort ofGachieving(ab)^∗

•

Conversely, any topsort ofGachieving(ab)^∗ extends to a topsort ofG+Pachieving(abc)^∗

•

^{Hence,CTS((abc)∗)isNP-hard}

The reduction technique G

a

b a b

b a

P b c a c b c a c b c a c

•

Say we want to solveCTSfor(ab)^∗(NP-hard)

•

Say we know how to solveCTSfor(abc)^∗

•

^{Take aninstanceGfor(ab)∗, with2nvertices}

•

^{Add thepathP: (bcac)n}

•

A topsort ofG∪Pachieving(abc)^∗ gives a topsort ofGachieving(ab)^∗

•

Conversely, any topsort ofGachieving(ab)^∗ extends to a topsort ofG+Pachieving(abc)^∗

•

^{Hence,CTS((abc)∗)isNP-hard}

The reduction technique G

a

b a b

b a

P b c a c b c a c b c a c

•

Say we want to solveCTSfor(ab)^∗(NP-hard)

•

Say we know how to solveCTSfor(abc)^∗

•

^{Take aninstanceGfor(ab)∗, with2nvertices}

•

^{Add thepathP: (bcac)n}

•

A topsort ofG∪Pachieving(abc)^∗ gives a topsort ofGachieving(ab)^∗

•

Conversely, any topsort ofGachieving(ab)^∗ extends to a topsort ofG+Pachieving(abc)^∗

•

^{Hence,CTS((abc)∗)isNP-hard}

The reduction technique G

a

b a b

b a

P b c a c b c a c b c

•

Say we want to solveCTSfor(ab)^∗(NP-hard)

•

Say we know how to solveCTSfor(abc)^∗

•

^{Take aninstanceGfor(ab)∗, with2nvertices}

•

^{Add thepathP: (bcac)n}

•

A topsort ofG∪Pachieving(abc)^∗ gives a topsort ofGachieving(ab)^∗

•

Conversely, any topsort ofGachieving(ab)^∗ extends to a topsort ofG+Pachieving(abc)^∗

•

^{Hence,CTS((abc)∗)isNP-hard}

The reduction technique G

a

b a b

b a

P b c a c b c a c b c a c

•

Say we want to solveCTSfor(ab)^∗(NP-hard)

•

Say we know how to solveCTSfor(abc)^∗

•

^{Take aninstanceGfor(ab)∗, with2nvertices}

•

^{Add thepathP: (bcac)n}

•

A topsort ofG∪Pachieving(abc)^∗ gives a topsort ofGachieving(ab)^∗

•

Conversely, any topsort ofGachieving(ab)^∗ extends to a topsort ofG+Pachieving(abc)^∗

•

^{Hence,CTS((abc)∗)isNP-hard}

The reduction technique G

a

b a b

b a

P b c a c b c a c b c

•

Say we want to solveCTSfor(ab)^∗(NP-hard)

•

Say we know how to solveCTSfor(abc)^∗

•

^{Take aninstanceGfor(ab)∗, with2nvertices}

•

^{Add thepathP: (bcac)n}

•

A topsort ofG∪Pachieving(abc)^∗ gives a topsort ofGachieving(ab)^∗

•

Conversely, any topsort ofGachieving(ab)^∗ extends to a topsort ofG+Pachieving(abc)^∗

•

^{Hence,CTS((abc)∗)isNP-hard}

The reduction technique G

a

b a b

b a

P b c a c b c a c b c a c

•

Say we want to solveCTSfor(ab)^∗(NP-hard)

•

Say we know how to solveCTSfor(abc)^∗

•

^{Take aninstanceGfor(ab)∗, with2nvertices}

•

^{Add thepathP: (bcac)n}

•

A topsort ofG∪Pachieving(abc)^∗ gives a topsort ofGachieving(ab)^∗

•

Conversely, any topsort ofGachieving(ab)^∗ extends to a topsort ofG+Pachieving(abc)^∗

•

^{Hence,CTS((abc)∗)isNP-hard}

Formalizing the reduction

Definition

A languageLshuffle-reducesto a languageL⁰ if, given anynin unary, we can compute in PTIME a wordP_i having the following property:

for any wordwof lengthn, we havew∈L iff the shuffle ofwandP_i contains a word ofL⁰.

Theorem

IfLshuffle-reduces toL⁰ then:

•

^CSh(L)reduces in PTIME toCSh(L⁰)

•

^CTS(L)reduces in PTIME toCTS(L⁰)

Formalizing the reduction

Definition

A languageLshuffle-reducesto a languageL⁰ if, given anynin unary, we can compute in PTIME a wordP_i having the following property:

for any wordwof lengthn, we havew∈L iff the shuffle ofwandP_i contains a word ofL⁰.

Theorem

IfLshuffle-reduces toL⁰ then:

•

^CSh(L)reduces in PTIME toCSh(L⁰)

•

^CTS(L)reduces in PTIME toCTS(L⁰)

Formalizing the reduction

Definition

A languageLshuffle-reducesto a languageL⁰ if, given anynin unary, we can compute in PTIME a wordP_i having the following property:

for any wordwof lengthn, we havew∈L iff the shuffle ofwandP_i contains a word ofL⁰.

Theorem

IfLshuffle-reduces toL⁰ then:

•

^CSh(L)reduces in PTIME toCSh(L⁰)

•

^CTS(L)reduces in PTIME toCTS(L⁰)

Other hard languages

•

The reduction showshardnessfor:

• (ab+b)^∗(also simpler argument)

• (aa+bb)^∗withP_2i= (ab)ⁱ

• u^∗ifucontains two different letters

•

Conjecture:ifFis finite thenCTS(F^∗)isNP-hard unless it contains a power of each of its letters

• Idea: reason aboutconsumption ratesof letters?

• Not evencompleteforF^∗languages, as(aa+bb)^∗isNP-hard

Other hard languages

•

The reduction showshardnessfor:

• (ab+b)^∗(also simpler argument)

• (aa+bb)^∗withP_2i= (ab)ⁱ

• u^∗ifucontains two different letters

•

Conjecture:ifFis finite thenCTS(F^∗)isNP-hard unless it contains a power of each of its letters

• Idea: reason aboutconsumption ratesof letters?

• Not evencompleteforF^∗languages, as(aa+bb)^∗isNP-hard

Other hard languages

•

The reduction showshardnessfor:

• (ab+b)^∗(also simpler argument)

• (aa+bb)^∗withP_2i= (ab)ⁱ

• u^∗ifucontains two different letters

•

Conjecture:ifFis finite thenCTS(F^∗)isNP-hard unless it contains a power of each of its letters

• Idea: reason aboutconsumption ratesof letters?

• Not evencompleteforF^∗languages, as(aa+bb)^∗isNP-hard

Other hard languages

•

The reduction showshardnessfor:

• (ab+b)^∗(also simpler argument)

• (aa+bb)^∗withP_2i= (ab)ⁱ

• u^∗ifucontains two different letters

•

Conjecture:ifFis finite thenCTS(F^∗)isNP-hard unless it contains a power of each of its letters

• Idea: reason aboutconsumption ratesof letters?

• Not evencompleteforF^∗languages, as(aa+bb)^∗isNP-hard

Tractability Results

Tractability for Monomials

•

^Monomial:language of the formA^∗₁ a1A^∗₂ a2 · · · A^∗_nanA^∗_n+1 wherea₁, . . . ,a_n∈ΣandA₁, . . . ,A_n+1 ⊆Σ

•

Union of monomials:union of finitely many such languages

• Example:pattern matchingΣ^∗word1Σ^∗+ Σ^∗word2Σ^∗

• Logical interpretation:languages definable inΣ₂[<] Theorem

For any union of monomialsL, the problemCTS(L)isin NL

Tractability for Monomials

•

^Monomial:language of the formA^∗₁ a1A^∗₂ a2 · · · A^∗_nanA^∗_n+1 wherea₁, . . . ,a_n∈ΣandA₁, . . . ,A_n+1 ⊆Σ

•

Union of monomials:union of finitely many such languages

• Example:pattern matchingΣ^∗word1Σ^∗+ Σ^∗word2Σ^∗

• Logical interpretation:languages definable inΣ₂[<] Theorem

For any union of monomialsL, the problemCTS(L)isin NL

Tractability for Monomials

•

^Monomial:language of the formA^∗₁ a1A^∗₂ a2 · · · A^∗_nanA^∗_n+1 wherea₁, . . . ,a_n∈ΣandA₁, . . . ,A_n+1 ⊆Σ

•

Union of monomials:union of finitely many such languages

• Example:pattern matchingΣ^∗word1Σ^∗+ Σ^∗word2Σ^∗

• Logical interpretation:languages definable inΣ₂[<]

Theorem

For any union of monomialsL, the problemCTS(L)isin NL

Tractability for Monomials

•

^Monomial:language of the formA^∗₁ a1A^∗₂ a2 · · · A^∗_nanA^∗_n+1 wherea₁, . . . ,a_n∈ΣandA₁, . . . ,A_n+1 ⊆Σ

•

Union of monomials:union of finitely many such languages

• Example:pattern matchingΣ^∗word1Σ^∗+ Σ^∗word2Σ^∗

• Logical interpretation:languages definable inΣ₂[<]

Theorem

For any union of monomialsL, the problemCTS(L)isin NL

Proof Idea for Monomials

•

Tractable languages are clearlyclosed under union

so it suffices to consider a monomial:A^∗₁ a₁A^∗₂ a₂ · · · A^∗_na_nA^∗_n+1 wherea1, . . . ,an∈ΣandA1, . . . ,An+1 ⊆Σ

•

^{We canguess}the positions of the individuala_i

•

Check that the other verticescan fitin theA^∗_i (uses NL = co-NL)

• Check that thedescendantsofa_nare all inA_n+1

• Find the vertices thatmustbe enumerated beforea_n

• Theancestorsof thea_i

• Theancestorsof vertices with a letternot inAn+1

• Inductively solve the problem for these vertices and A^∗₁ a₁A^∗₂ a₂ · · · A^∗_n

Proof Idea for Monomials

•

Tractable languages are clearlyclosed under union

so it suffices to consider a monomial:A^∗₁ a₁A^∗₂ a₂ · · · A^∗_na_nA^∗_n+1 wherea1, . . . ,an∈ΣandA1, . . . ,An+1 ⊆Σ

•

^{We canguess}the positions of the individuala_i

•

Check that the other verticescan fitin theA^∗_i (uses NL = co-NL)

• Check that thedescendantsofa_nare all inA_n+1

• Find the vertices thatmustbe enumerated beforea_n

• Theancestorsof thea_i

• Theancestorsof vertices with a letternot inAn+1

• Inductively solve the problem for these vertices and A^∗₁ a₁A^∗₂ a₂ · · · A^∗_n

Proof Idea for Monomials

•

Tractable languages are clearlyclosed under union

so it suffices to consider a monomial:A^∗₁ a₁A^∗₂ a₂ · · · A^∗_na_nA^∗_n+1 wherea1, . . . ,an∈ΣandA1, . . . ,An+1 ⊆Σ

•

^{We canguess}the positions of the individuala_i

•

Check that the other verticescan fitin theA^∗_i (uses NL = co-NL)

• Check that thedescendantsofa_nare all inA_n+1

• Find the vertices thatmustbe enumerated beforea_n

• Theancestorsof thea_i

• Theancestorsof vertices with a letternot inAn+1

• Inductively solve the problem for these vertices and A^∗₁ a₁A^∗₂ a₂ · · · A^∗_n