Extracting Constrained 2-Interval Subsets in 2-Interval Sets

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Extracting Constrained 2-Interval Subsets in 2-Interval

Sets

Guillaume Blin, Guillaume Fertin, Stéphane Vialette

To cite this version:

Guillaume Blin, Guillaume Fertin, Stéphane Vialette. Extracting Constrained 2-Interval Subsets in

2-Interval Sets. Theoretical Computer Science, Elsevier, 2007, 385 (1-3), pp.241-263. �hal-00417717�

Extra ting Constrained

2

-Interval Subsets

in

2

-Interval Sets

⋆

Guillaume Blin

IGM-LabInfo - UMRCNRS 8049

Université deMarne-la-Vallée

77 454Marne-la-Vallée Cedex 2 - FRANCE

Guillaume Fertin

LINA,FRE CNRS 2729

Université deNantes, 2 rue de la Houssinière

BP 92208 44322 Nantes Cedex 3 - FRANCE

Stéphane Vialette

Laboratoire deRe her he enInformatique (LRI), UMRCNRS8623

Université Paris-Sud, 91405 Orsay Cedex - FRANCE

Abstra t

2

-interval sets were used in [28,29℄ for establishing a general representation for ma ros opi des ribers of RNAse ondary stru tures. In this ontext, we have a

2

-intervalforea hlegallo alfoldinagivenRNAsequen e,anda onstrainedpattern

madeofdisjoint

2

-intervalsrepresentsaputativeRNAse ondarystru ture.Wefo us hereontheproblemofextra tinga onstrained patterninasetof

2

-intervals.More pre isely, given a set of

2

-intervals

D

and a model

R

des ribing if two disjoint

2

-intervalsinasolution anbeinpre eden eorder(

<

),beallowedtonest(

⊏

)and/or be allowed to ross

(≬

), we onsider the problemof nding amaximum ardinality subset

D

′

_{⊆ D}

ofdisjoint

2

-intervals su h thatanytwo

2

-intervals in

D

′

agreewith

R

. The dierent ombinations of restri tions on model

R

alter the omputational omplexity ofthe problem, and need tobe examinedseparately.

Inthispaper,weimprovethetime omplexityof[29℄formodel

R

=

{⊏}

bygiving an optimal

O(n log n)

time algorithm, where

n

is the ardinality of the

2

-interval set

D

. We also give a graph-like relaxation for model

R

=

{⊏, ≬}

that is solvable in

O(n

2

√

_n)

time. Finally, we prove that the onsidered problem is NP- omplete

for model

R

=

{<, ≬}

even for same-length intervals, and give a xed-parameter tra tability resultbasedon the rossing stru tureof

D

.

The problem of establishing a general representation of stru tured patterns,

i.e., ma ros opi des ribers of RNA se ondary stru tures, was onsidered in

[28,29℄.The approa histoset up ageometri des ription ofheli es by means

of a natural generalization of intervals, namely a

2

-interval. A

2

-interval is

the disjoint union of two intervals on the line. The geometri properties of

2

-intervalsprovideapossibleguideforunderstandingthe omputational

om-plexity of nding stru tured patterns in RNA sequen es. Using a model to

represent non sequential informationallows usfor varying restri tions onthe

omplexity of the pattern stru ture. Indeed, two disjoint

2

-intervals,i.e., two

2

-intervalsthat donot interse tinany point, an be inpre eden e order(

<

), be allowed to nest (

⊏

) or be allowed to ross (

≬

). Furthermore, the set of

2

-intervals and the pattern an have dierent restri tions, e.g., all intervals

have the same length or all the intervals are disjoint. These dierent

om-binations of restri tions alter the omputational omplexity of the problems,

andneedtobeexaminedseparately.Thisexaminationprodu ese ient

algo-rithmsformore restri tivestru tured patterns, andhardness resultsfor those

less restri tive.

In this paper, we onsider the problem of nding a onstrained patternin a

set of

2

-intervals.More pre isely, given a set of

2

-intervals

D

and a model

R

des ribing if two disjoint

2

-intervals in a solution an be inpre eden e order (

<

), be allowed to nest (

⊏

) and/or be allowed to ross

(≬

), we onsider the

problemofndingamaximum ardinalitysubset

D

′

_{⊆ D}

ofdisjoint

2

-intervals su hthat any two

2

-intervalsin

D

′

agree with

R

. The problem of ndingthe

largest

2

-interval pattern in a set of

2

-intervals

D

with respe t to a given

abstra t model, referredhereafter asthe

2

-IntervalPatternproblem,has

been introdu edby Vialette [28,29℄. Vialettedivided the problemindierent

lasses based on the stru ture of the modeland gave for most of them either

NP- ompletenessresultsorpolynomial-timealgorithms.Dividingtheproblem

in several lasses was later proved to be extremely useful for approximating

of the

2

-IntervalPattern problem[8℄.

⋆

An extended abstra t of this work appeared in Pro eedings of the

15

th

Annual

Symposium on Combinatorial Pattern Mat hing (CPM 2004) [5℄. This work was

partiallysupported by theCNRS ACIMasse deDonnï¾

1

2

es NavGraphe proje t. Email addresses: gblinuniv-mlv.fr(Guillaume Blin),

fertinlina.univ-nantes.fr(Guillaume Fertin), vialettelri.fr (Stéphane

Inthe present paper, wefo us onthree spe ial ases of the

2

-Interval P

at-tern problem:

(1) The

2

-intervalsof the solution subsetneed to be pairwisenested,

(2) Two

2

-intervals in a solution an only be nested or rossing, and allthe intervalsinvolved inthe

2

-interval set

D

are disjoint,and

(3) Two

2

-intervalsinasolution anonlybe nestedorinpre eden e, andall the intervals involved in the

2

-intervalset

D

have the same length. We givepre ise results for these threeproblems. Those three problems are of

importan e sin e ea h one is a straightforward extension of the problem of

nding a given

2

-interval set in another

2

-interval set introdu ed in [29℄ and furtherstudiedin[19℄and[23℄,andhen eisstronglyrelated,inthe ontextof

mole ularbiology,topatternmat hing overRNA se ondary stru tures.More

pre isely, in this paper, we improve the time omplexity of the best known

algorithmfor

R

=

{⊏}

by givinganoptimal

O(n log n)

timealgorithm.Also,

we give a graph-like relaxation for

R

=

{⊏, ≬}

that is solvable in

O(n

2

√

_n)

time.Finally,weprovethattheproblemisNP- ompletefor

R

=

{<, ≬}

,and,

we give a xed-parameter tra tability result based on the rossing stru ture

of

D

. Those results almost omplete the table proposed by Vialette [29℄ (see

Table1)andprovideanimportantsteptowards abetterunderstandingofthe

pre ise omplexity of

2

-interval patternmat hing problems.

There are basi ally two main lines of resear h our results refer to: (i)

ar -annotated sequen es and protein topologies,and (ii)

t

-intervals ombinatori s.

•

Fora sequen e

S

,an ar -annotation of

S

is a set of unordered pairs of po-sitionsin

S

. In this ontext, given two ar -annotated sequen es

S

1

and

S

2

,

theAr -PreservingSubsequen e(APS)problemaskstondan

o ur-ren eof

S

1

in

S

2

, and the Longest Ar -Preserving Common

Subse-quen e(LAPCS)problemsaskstondthelongest ommonar -annotated

sequen e that o urs both in

S

1

and

S

2

. The APS and LAPCS problems

areusefulinrepresentingthe stru turalinformationofRNAand protein

se-quen es[11,21,18,1℄.Thebasi ideaistoprovideameasureforsimilarity,not

onlyonthesequen e level,butalsoonthestru turallevel(anar -annotated

sequen eisviewedasaRNAsequen etogetherwithphosphodiesterbonds).

Furthermore,asimilarproblemto ompare thethree-dimensionalstru ture

of proteins is the Conta t Map Overlap problem des ribed by in[16℄.

Viksnaand Gilbert des ribed algorithmsfor patternmat hing and pattern

learninginTOPS diagram(formaldes ription of protein topologies)[30℄.

•

Our results are also related to the independent set problem in dierent

extensions of

2

-interval graphs. A graph

G

is a

t

-interval graph if there is an interse tion model whose obje ts onsist of olle tions of

t

intervals,

t

_{≥ 1}

, su h that

G

is the interse tion graph of this model [26,20℄. From

Of parti ular interest is the lass of

2

-interval graphs. For example, line graphs,trees and ir ular-ar graphs are

2

-intervalgraphs. However, West

and Shmoys [31℄ have shown that the re ognition problem for

t

-interval

graphsisNP- ompleteforevery

t

≥ 2

(this hastobe omparedwith linear

time re ognition of

1

-interval graphs). In the ontext of sequen e similar-ity, [22℄ ontains an appli ation of graphs having interval number at most

two. In [3℄, the authors onsidered the problem of s heduling jobs that are

given as groups of non-interse ting segments onthe real line. Of parti ular

importan e, they showed that the maximum weighted independent set for

t

-interval graphs (

t

≥ 2

) is APX-hard even for highly restri ted instan es

Also, they gave a

2t

-approximation algorithm for general instan es based

onafra tional version ofthe Lo alRatio Te hnique [2℄.Finally,some

om-plexityissuesofstandardoptimizationproblems for

t

-intervalsetsare given in[6℄.

The remainder of the paper is organized as follows. In Se tion 2 we briey

review the terminologyintrodu ed in [29℄. In Se tion 3, we improve the time

omplexity of the best known algorithmfor model

R

=

{⊏}

.In Se tion4, we

give a graph-like relaxation for model

{⊏, ≬}

that is solvable in

polynomial-time. In Se tion 5, we prove that the

2

-interval pattern problem for model

R

=

{<, ≬}

is NP- omplete even when all intervals involved in the input

2

-interval set have the same length. Finally, we give in Se tion 6 a

xed-parameter tra tabilityresult based on the rossingstru ture of

D

.

2 Preliminaries

An interval and a 2-interval represent respe tively a sequen e of ontiguous

basesandpairingsbetweentwointervals,i.e.,stems,inRNAse ondary

stru -tures. Thus,

2

-intervals an be seen as ma ros opi des ribers of RNA

stru -tures.

Formally, a

2

-interval is the disjoint union of two intervals on a line. We denote it by

D

= (I

1

, J

1

)

where

I

1

and

J

1

are intervals su h that

I

1

< J

1

(here

<

is the stri t pre eden e order between intervals) ; in that ase we

also write

Left(D) = I

1

and

Right(D) = J

1

. If

[x : y]

and

[x

′

_{: y}

′

_]

are two

intervals su h that

[x : y] < [x

′

_{: y}

′

_]

, we will sometimes write

D

= ([x :

y], [x

′

_{: y}

′

_])

to emphasize on the pre ise denition of the

2

-interval

D

. Let

D

1

= (I

1

, J

1

)

and

D

2

= (I

2

, J

2

)

betwo

2

-intervals.They are alled disjoint if

(I

1

∪J

1

)

∩(I

2

∪J

2

) =

∅

(i.e.,involvedintervalsdonotinterse t). The overing intervalofa

2

-interval

D

,written

Cover

(D)

,istheleast interval overingboth

Left

(D)

and

Right

(D)

.

(I

1

, J

1

)

and

D

2

= (I

2

, J

2

)

. We will write

D

1

< D

2

if

I

1

< J

1

< I

2

< J

2

,

D

1

⊏ D

2

if

I

2

< I

1

< J

1

< J

2

and

D

1

≬ D

2

if

I

1

< I

2

< J

1

< J

2

. Two

2

-intervals

D

1

and

D

2

are

τ

- omparable for some

τ

∈ {<, ⊏, ≬}

if

D

1

τ D

2

or

D

2

τ D

1

. Let

D

be a set of

2

-intervals and

R

⊆ {<, ⊏, ≬}

be non-empty. The set

D

isR- omparableifanytwodistin t

2

-intervalsof

D

are

τ

- omparablefor

some

τ

∈ R

.Throughoutthepaper, thenon-emptysubset

R

is alledamodel.

Clearly, if a set of

2

-intervals

D

is

R

- omparable then

D

is a set of disjoint

2

-intervals.The ground set of aset of

2

-intervals

D

,written

GS(

D)

, isthe set ofallsimpleintervalsinvolved in

D

,i.e.,

GS

(

D) =

S

D

∈D

(Left(D)

∪ Right(D))

. The leftmost (resp. rightmost)elementof a set of disjoint

2

-intervals

D

isthe

2

-interval

D

i

∈ D

su hthat

Left(D

i

) < Left(D

j

)

(resp.

Right(D

j

) < Right(D

i

)

) for all

D

j

∈ D − D

i

. Observe that it ould be the ase that

D

i

is both the

leftmost and rightmost element of

D

(this is indeed the ase if

|D| = 1

or if

D

j

⊏ D

i

for all

D

j

∈ D − D

i

).

We denehereafter two additionalparameterson

D

.The depthof

D

,written

Depth(

_D)

, is the size of a maximum ardinality

{≬}

- omparable subset of

D

(a ording to [29℄, this parameter is polynomial-time omputable). The

for-ward rossingnumberof

D

,written

FCrossing(

D)

,isdenedby

FCrossing(

D) =

max

D

i

∈D

|{D

j

: D

i

≬ D

j

}|

. Clearly,

FCrossing(

D) ≥ Depth(D) − 1

for any set

D

of 2-intervals.

Following[11℄,Vialetteproposedin[29℄,twonaturalrestri tionsontheground

set of

D

(a third restri tion, i.e., balan ed

2

-intervals, well-suited for

investi-gating RNA se ondary stru tures spa e was introdu ed in[8℄):

(1) allthe intervals of the ground set

GS

(

D)

are of the same length,

(2) alltheintervalsof thegroundset

GS(

D)

are disjoint,i.e.,if twointervals

I, I

′

_{∈ GS(D)}

overlap, then

I

= I

′

.

Using restri tions on the ground set allows us for varying restri tions on the

omplexity ofthe

2

-intervalset stru ture, andhen e onthe omplexity ofthe problems. These two restri tions involve three levels of omplexity:

•

unlimited:no restri tions

•

unit: restri tion 1

•

disjoint: restri tions 1and 2

Given a set of

2

-intervals

D

, a model

R

⊆ {<, ⊏, ≬}

and a positive integer

k

, the

2

-Interval Patternproblem onsists in ndinga subset

D

′

_{⊆ D}

of

ardinalityatleast

k

su hthat

D

′

is

R

- omparable.Forthesakeofbrevity,the

2

-Interval Pattern problemwith respe t to a model

R

over anunlimited

(resp. unit and disjoint) ground set is abbreviated in

2

-IP-Unl-

R

(resp.

2

-IP-Unit-

R

and

2

-IP-Dis-

R

).

with respe t tothe models

{<}

,

{⊏}

,

{≬}

and

{<, ⊏}

( f. Table 1).

Inthisarti le,weanswerthree openproblemsand weimprovethe omplexity

of another one, as shown inTable 1. Moreover, we show that

2

-IP-Unit-

{<

,

≬

_}

is xed parameter tra table when parameterized by the forward rossing

number of

D

.

2

-IntervalPatternProblem

GroundSet

Model Unlimited Unit Disjoint

{<, ⊏, ≬}

NP- omplete

O(n

√

n)

[24℄

{⊏, ≬}

NP- omplete

O(n

2

√

n) ⋆

{<, ⊏}

O(n

2

)

{<, ≬}

NP- omplete

⋆

?

{<}

O(n log n)

{⊏}

O

(n log n) ⋆ •

{≬}

O(n

2

log n)

Table 1

2

-interval pattern problem omplexity where

n

=

|D|

.When not spe ied, the omplexity omes from[29℄.

⋆

ontributions of thepresent paper.

•

improvement of theexisting omplexity (whi h was

O(n

2

₎

in[29℄).

3 Improving the omplexity of

2

-IP-Unl-

{⊏}

The problem of nding the largest

{⊏}

- omparable subset in a set of

2

-intervals was onsidered in [29℄. Observing that this problem is equivalent

to nding a largest lique in a omparability graph (a linear time solvable

problem[17℄), an

O(n

2

₎

time algorithmwasthusproposed. We improvethat

result by givingan optimal

O(n log n)

time algorithm.

The ine ien y of the algorithm proposed in [29℄ lies in the ee tive

on-stru tion of a omparability graph. We show that this onstru tion an be

avoided by onsidering trapezoids instead of

2

-intervals. Re all that a

trape-zoid graph is the interse tion graph of a nite set of trapezoids between two

parallellines[9℄(itiseasilyseenthattrapezoidgraphsgeneralizebothinterval

graphs and permutation graphs). Analogously to

2

-intervals, we will denote

by

T

= ([x : y], [x

′

_{: y}

′

_])

the trapezoid with top interval

[x : y]

and bottom

interval

[x

′

_{: y}

′

_]

.

Proposition 1

2

-IP-Unl-

{⊏}

issolvable in

O(n log n)

time.

line. Constru t a olle tion of trapezoids

T = {T

1

, T

2

, . . . , T

n

}

between two parallellinesasfollows.Forea h

2

-interval

D

i

= ([x : y], [x

′

_{: y}

′

_])

_{∈ D}

,weadd

the trapezoid

T

i

= ([x : y], [

−y

′

_:

_−x

′

_])

to

T

.

Claim 2 Forall

1

≤ i ≤ j ≤ n

,the

2

-intervals

D

i

and

D

j

are

{⊏}

- omparable if and only if the trapezoids

T

i

and

T

j

are non-interse ting.

PROOF. [of Claim℄Let

D

i

= ([x

i

: y

i

], [x

′

i

: y

′

i

])

and

D

j

= ([x

j

: y

j

], [x

′

j

: y

′

j

])

be two

2

-intervals of

D

, and

T

i

= ([x

i

: y

i

], [

−y

′

i

:

−x

′

i

])

and

T

j

= ([x

j

:

y

j

], [

−y

j

′

:

−x

′

j

])

be the two orresponding trapezoids in

T

. Suppose that

D

i

and

D

j

are

{⊏}

- omparable. Without loss of generality, we may assume

D

j

⊏ D

i

. Thus, we have

y

i

< x

j

and

y

′

j

< x

′

i

. It follows immediately that

−x

′

i

<

−y

j

′

,and hen e the two trapezoids

T

i

and

T

j

are non-interse ting.The

proof of the onverse isidenti al.

2

Clearly, the olle tion

T

an be onstru ted in

O(n)

time. Based on a

geo-metri representationof trapezoidgraphsby boxes intheplane, Felsneret al.

[12℄ have designed a

O(n log n)

algorithm for nding a maximum ardinality

sub olle tionof non-interse ting trapezoids in a olle tionof trapezoids, and

the propositionfollows.

2

Based on Fredman's bound for the number of omparisons needed to

om-putemaximumin reasingsubsequen esinpermutation[13℄,Felsneret al.[12℄

argued that their

O(n log n)

time algorithm for ndinga maximum

ardinal-ity sub olle tion of non-interse ting trapezoids in a olle tion of trapezoids

is optimal. Then itfollows from Proposition 1 that our

O(n log n)

time

algo-rithm for nding a maximum ardinality

{⊏}

- omparable subset in a set of

2

-intervalsisoptimal aswell.

4 A polynomial-time algorithm for

2

-IP-Dis-

{⊏, ≬}

In this se tion, we give an

O(n

2

√

_n)

time algorithmfor the

2

-IP-Dis-

{⊏, ≬}

problem, where

n

is the ardinality of the set of

2

-intervals

D

. Re all that given aset of

2

-intervals

D

overadisjointgroundset, theproblemaskstond

thesize ofamaximum ardinality

{⊏, ≬}

- omparablesubset

D

′

_{⊆ D}

.Observe

that the

2

-IP-Dis-

{⊏, ≬}

problem has aninteresting formulationin terms of

onstrained mat hings in general graphs: Given a graph

G

together with a

linearordering

π

of itsverti es,the

2

-IP-Dis-

{⊏, ≬}

problemis equivalentto

any two distin t edges

{u, v}

and

{u

′

_{, v}

′

_}

of

M

, neither

max

{π(u), π(v)} <

min

_{π(u

′

_{), π(v}

′

₎

_}

nor

max

{π(u

′

_{), π(v}

′

₎

_{} < min{π(u), π(v)}}

o ur.

Roughly speaking, our algorithmis athree-step pro edure. First,the interval

graphof allthe overingintervalsof the

2

-intervalsin

D

is onstru ted. Next, allthemaximal liquesofthatgrapharee iently omputed.Finally,forea h

maximal liquewe onstru tanewgraphandndasolutionusingamaximum

ardinalitymat hing algorithm.The size of abest solution found inthe third

step is thus returned. Clearly, the e ien y of our algorithm relies upon an

e ientalgorithmforndingallthemaximal liquesintheinterse tionofthe

overing intervals.We now pro eedwith the detailsof our algorithm.

Let

D = {D

i

: 1

≤ i ≤ n}

beasetof

2

-intervals.Considertheset

C

D

omposed

ofallthe overingintervalsofthe

2

-intervalsin

D

,i.e.,

C

D

=

{Cover(D) : D ∈

D}

. Now, let

Ω(

C

D

)

be the interval graph asso iated with

C

D

. The graph

Ω(

_C

_D

)

has a vertex

v

i

for ea h interval

Cover(D

i

)

in

C

D

and two verti es

v

i

and

v

j

of

Ω(

C

D

)

arejoined byanedgeifthetwoasso iatedintervals

Cover

(D

i

)

and

Cover(D

j

)

interse t. An illustration of

C

D

and

Ω(

C

D

)

for a given set of

2

-intervals

D

is given in Figure 1. Most in the interest in the interval graph

Ω(

C

D

)

stems from the followinglemma.

Fig. 1. Illustration of

C

D

and

Ω(

C

D

)

for a given set of

2

-intervals

D

on a disjoint groundset.

Lemma 3 Let

D

be a setof

2

-intervalsand

D

′

be a

{⊏, ≬}

- omparablesubset of

D

. Then,

{v

i

: D

i

∈ D

′

_}

indu es a omplete graph in

Ω(

C

D

)

.

PROOF. Let

D

i

and

D

j

be two distin t

2

-intervals of

D

′

. Sin e

D

i

and

D

j

are

{⊏, ≬}

- omparable then it follows that either intervals

Cover(D

i

)

and

Cover(D

j

)

overlaporoneintervalis ompletely ontainedintheother.Inboth ases, intervals

Cover

(D

i

)

and

Cover

(D

j

)

interse t, and hen e verti es

v

i

and

v

j

arejoinedbyanedgein

Ω(

C

D

)

.Therefore

{v

i

: D

i

∈ D

′

_}

indu esa omplete

graph in

Ω(

C

D

)

.

2

Observethat the onverse isfalsesin e theinterse tionoftwo

2

-intervalsin

D

resultsinanedgein

Ω(

C

D

)

,andhen etwo

2

-intervalsasso iatedtotwodistin t

3 we now only need to fo us on maximal liques of

Ω(

C

D

)

. Several problems

thatare NP- ompleteongeneralgraphshavepolynomial-timealgorithmsfor

interval graphs. The problemof nding allthe maximal liques of a graph is

onesu hexample.Indeed,anintervalgraph

G

= (V, E)

isa hordalgraphand

as su h has at most

|V |

maximal liques [14℄. Furthermore, all the maximal

liques ofa hordalgraph an befound in

O(n + m)

time,where

n

=

|V |

and

m

=

_|E|

, by a modi ation of Maximum Cardinality Sear h (MCS) [25,4℄.

Let

C

be a maximal lique of

Ω(

C

D

)

. As observed above, any two

2

-intervals asso iatedtotwodistin tverti esinthe maximal lique

C

maynotbe

{⊏, ≬}

- omparable. Let

D

′

_{⊆ D}

be the set of all

2

-intervals asso iated to verti es in themaximal lique

C

.Basedon

C

, onsiderthe graph

G

C

= (V

C

, E

C

)

dened by

V

C

= GS(

D

′

₎

and

E

C

=

{{I, J} : D = (I, J) ∈ D

′

_}

. Inother words,the set

of verti es of

G

C

is the groundset of

D

′

and the edges of

G

C

isthe

2

-interval

subset

D

′

itselfviewed asasetof subsetsofsize

2

.Notethatthe onstru tion of

G

C

ispossibleonlybe ause

D

′

hasdisjointgroundset.Thefollowinglemma

is animmediate onsequen e of the denition of

G

C

and Lemma3.

Lemma 4 Let

C

be a lique in

Ω(

C

D

)

and

G

C

= (V

C

, E

C

)

be the graph on-stru tedasdetailedabove.Then,

{(I

i

1

, J

i

1

), (I

i

2

, J

i

2

), . . . , (I

i

k

, J

i

k

)

}

isa

{⊏, ≬}

- omparablesubsetifandonlyif

{{I

i

1

, J

i

1

}, {I

i

2

, J

i

2

}, . . . , {I

i

k

, J

i

k

}}

isa mat h-ing in

G

C

.

Proposition 5 The

2

-IP-Dis-

{⊏, ≬}

problem is solvable in

O(n

2

√

_n)

time,

where

n

is the number of

2

-intervals in

D

.

PROOF. Considerthe algorithmgiven inFigure2.Corre tnessof this

algo-rithm follows from Lemmas 3 and 4. What is left is to prove the time

om-plexity. Clearly, the interval graph

Ω(

C

D

)

an be onstru ted in

O(n

2

₎

time.

Allthe maximal liques of

Ω(

C

D

)

an be found in

O(n + m)

time,where

m

is

the number of edges in

Ω(

C

D

)

[25,4℄. Summingup, the rst two steps an be

done in

O(n

2

₎

timesin e

m < n

2

.Wenow turnto the time omplexityof the

loop(infa tthedominanttermofouranalysis).Forea hmaximal lique

C

of

Ω(

C

D

)

,the graph

G

C

an be onstru tedin

O

(n)

time sin e

|C| ≤ n

.Wenow

onsider the omputationof a maximal mat hing in

G

C

. Mi aliand Vazirani

[24℄ (see also[27℄) gave an

O(

q

|V ||E|)

time algorithmfor nding a maximal

mat hing in a graph

G

= (V, E)

. But

G

C

has at most

n

edges (as ea h edge

orresponds to a

2

-interval) and hen e has at most

2n

verti es. Then it

fol-lowsthatamaximummat hing

M

in

G

C

anbefound in

O(n

√

n)

time.Sin e

Ω(

C

D

)

is an interval graph with

n

verti es, it has at most

n

maximal liques

[14℄, we on lude that the algorithmas awhole runs in

O(n

2

√

_n)

Max

{⊏, ≬}

-Comparable

2

-Interval Pattern

Input: A set of

2

-intervals

D

with disjoint groundset

Output: The size of a maximum ardinality

{⊏, ≬}

- omparable subset of

D

1. Constru t the interval graph

Ω(

C

D

)

2. Compute all maximal liques in

Ω(

C

D

)

3. For ea h maximal lique

C

in

Ω(

C

D

)

3.1. Constru t the graph

G

C

3.2. Compute a maximalmat hing

M

in

G

C

3.3. Store the ardinality of

M

in

m(C)

4. Return

max

{m(C) : C

is amaximal lique of

Ω(

C

D

)

}

Fig.2.AlgorithmMax

{⊏, ≬}

-Comparable

2

-Interval Pattern.

5

2

-IP-Unit-

{<, ≬}

is NP- omplete

Theorem6below ompletestheanalysis of

2

-IP-Unit-

R

and

2

-IP-Unl-

R

for

any model

R

⊆ {<, ⊏, ≬}

(see Table 1).

Theorem 6 The

2

-IP-Unit-

{<, ≬}

problem is NP- omplete.

PROOF. First,we willpresent the two de ision problems we willdeal with

(Exa t 3-CNF-Sat and

2

-IP-Unit-

{<, ≬}

). Then, we will give several

in-termediate lemmasthat willnallybeused inProposition 14 tovalidatethe

proof of the NP- ompleteness of the

2

-IP-Unit-

{<, ≬}

problem.

We provide a polynomial-timeredu tion from the Exa t 3-CNF-Sat

prob-lem: Given a set

V

n

of

n

variables and a set

C

q

of

q

lauses (ea h omposed

of three literals) over

V

n

, the problem asks to nd a truth assignment for

V

n

that satises all lauses of

C

q

. It is well-known that the Exa t 3-CNF-Sat

problem is NP- omplete [15℄. For the sake of larity, we now state formally

the

2

-IP-Unit-

{<, ≬}

problem: Given a set of

2

-intervals

D

, and a positive integer

k

,the problemaskstondasubset

D

′

_{⊆ D}

of ardinalitygreaterthan

orequal to

k

,su h that

D

′

is

{<, ≬}

- omparable.

Clearly,

2

-IP-Unit-

{<, ≬}

problemisinNP.Weshowthatgivenanyinstan e

ofExa t3-CNF-Satwith

q

lausesonaset of

n

variables,we an onstru t

in polynomial-time an instan e of the

2

-IP-Unit-

{<, ≬}

problem with

k

=

(7n

− 2)q

su hthat there exists a satisfyingtruthassignmentfor the boolean formula i there exists a

{<, ≬}

- omparablesubset

D

′

_{∈ D}

of size at least

k

. We detail this onstru tion hereafter.

Let

V

n

=

{x

1

, x

2

, ...x

n

}

be a set of

n

variables and

C

q

=

{c

1

, c

2

, . . . , c

q

}

be a olle tionof

q

lauses. Forthe sake of larity, letus dene

D

on the integral linesu h that any interval of the groundset is of size four. Letus start with

the pre ise denition of the representation of a single lause

c

i

of

C

q

as illus-tratedinFigure4.The dottedre tangle ontheleft (resp.right)ispart ofthe

representation of lause

c

i

−1

(resp.

c

i+1

). The pre ise adjustment of the rep-resentation of two onse utive lauses is illustrated in Figure 3 and formally

dened afterwards. For onvenien e, wewillsplit the representation of

c

i

into seven groups(representedingray):

A

i

,

B

i

,

C

i

L

,

C

i

R

,

D

i

,

E

i

and

F

i

.Ea hgroup

inturnisdivided intoblo ks (represented inwhite).There are

11 + 2n

blo ks for ea h lause:

n

blo ks for

A

i

;

3

blo ks for

B

i

;

1

blo k for

C

i

L

;

n

blo ks for

C

i

R

;

2

blo ks for

D

i

;

3

blo ks for

E

i

;

2

blo ks for

F

i

.

Fig.3.Jun tion between therepresentation of lauses

c

i

−1

and

c

i

For example, in Figure 4 we use three boolean variables and hen e we have

seventeen blo ks. For the sake of larity, in the gures of this se tion, the

intervalsof the ground set might bedrawn ondierent levels.

We now turn to give a pre ise denition of ea h group in the representation

ofa given lause

c

i

.In thefollowing,wewillrefertoanintervalofthe ground set as a simple interval. Let

F P

(c

i

)

denote the smallest starting position of any simpleintervalof the representation of lause

c

i

.Weset, for onvenien e,

F P

(c

1

) = 0

. For any

1 < i

≤ q

, we have

F P

(c

i

) = F P (c

i

−1

) + 104n

−

21

. Moreover, let

F P

(α)

denote the smallest starting position of any simple

intervalof group

α

∈ {C

i

L

, A

i

, B

i

, C

R

i

, D

i

, E

i

, F

i

|1 ≤ i ≤ q}

.

Group

C

i

L

is omposed of one blo k ontaining

2n

simple intervals (as

illus-tratedinFigure5):

{[F P (C

i

L

) + 7k, F P (C

L

i

) + 7k + 4]

|0 ≤ k ≤ 2n − 1}

,where

F P

(C

i

L

) = F P (c

i

)

.The

2n

simpleintervalsoftheblo kofgroup

C

i

L

represent

inthe left torightorder(

x

1

, x

1

, x

2

, x

2

. . . x

n

, x

n

).Bydenition, the simple in-terval representing

x

m

in

C

i

L

is dened by

[F P (C

i

L

) + 14(m

− 1), F P (C

L

i

) +

14(m

_{− 1) + 4]}

. And onsequently, the simple interval representing

x

m

in

C

i

L

is dened by

[F P (C

i

squares)ofgroup

C

i

L

.

Group

D

i

is omposedoftwoblo ks(

D

i

1

and

D

i

2

),ea h ontaining

2n

−1

simple intervals (as illustrated in Figure 6):

{[F P (D

i

_{) + 5k, F P (D}

i

_{) + 5k + 4]}

_{|0 ≤}

k

_{≤ 4n − 3}}

where

F P

(D

i

_{) = F P (c}

i

) + 34n

− 10

. By onstru tion,blo k

D

i

1

is omposed of the following simpleintervals:

{[F P (D

i

_{) + 5k, F P (D}

i

_{) + 5k +}

4]

_{|0 ≤ k ≤ 2n −2}}

andblo k

D

i

2

is omposedofthefollowingsimpleintervals:

{[F P (D

i

_{) + 5k, F P (D}

i

_{) + 5k + 4]}

_{|2n − 1 ≤ k ≤ 4n − 3}}

.

Fig. 6.Des riptionof thesimple intervals ofgroup

D

i

.

Group

A

i

is omposed of

n

blo ks (one blo k for ea h boolean variable),

ea h ontaining foursimple intervals(as illustrated inFigure7):

{[F P (A

i

_{) +}

7k, F P (A

i

_{) + 7k + 4], [F P (A}

i

_{) + 2 + 14l, F P (A}

i

_{) + 6 + 14l], [F P (A}

i

_{) + 5 +}

14l, F P (A

i

_{) + 9 + 14l]}

_{|0 ≤ k ≤ 2n − 1, 0 ≤ l ≤ n − 1}}

where

F P

(A

i

_{) =}

F P

(c

i

) + 54n

− 20

. The

4n

simple intervals of group

A

i

represent in the left

to right order (

x

1

, x

1

, x

1

, x

1

, x

2

, x

2

, x

2

, x

2

, . . . x

n

, x

n

, x

n

, x

n

). By onstru tion,

in any blo k of group

A

i

the se ond (resp. third) simple interval overlaps

both the rst and the third(resp. the se ond and the fourth) simpleinterval.

By denition, the two simple intervals representing

x

m

in

A

i

are dened by

[F P (A

i

_{) + 14(m}

_{−1) + 7, F P (A}

i

_{) + 14(m}

_{−1) + 11]}

and

[F P (A

i

_{) + 14(m}

_{−1) +}

2, F P (A

i

_{) + 14(m}

_{− 1) + 6]}

.And onsequently, the two simpleintervals

repre-senting

x

m

in

A

i

are dened by

[F P (A

i

_{) + 14(m}

_{− 1), F P (A}

i

_{) + 14(m}

_{− 1) + 4]}

and

[F P (A

i

_{) + 14(m}

_{− 1) + 5, F P (A}

i

_{) + 14(m}

_{− 1) + 9]}

.

Fig. 7.Des riptionof thesimple intervals ofgroup

A

i

.

Group

B

i

is omposed of three blo ks (one for ea h literal in a lause),

ea h ontaining

2n

simple intervals (as illustrated in Figure 8):

{[F P (B

i

1

) +

6k, F P (B

i

1

)+6k +4], [F P (B

2

i

)+6k, F P (B

2

i

)+6k +4], [F P (B

3

i

)+6k, F P (B

3

i

)+

6k + 4]

_{|0 ≤ k ≤ 2n − 1}}

where

F P

(B

i

1

) = F P (c

i

) + 68n

− 20

,

F P

(B

i

2

) =

F P

(c

i

)+80n

−20

,

F P

(B

i

3

) = F P (c

i

)+92n

−20

.The

2n

simpleintervalsofea h

blo k of group

B

i

represent inthe left toright order (

x

1

, x

1

, x

2

, x

2

. . . x

n

, x

n

). By denition, the simple intervalrepresenting

x

m

in

B

i

j

, with

j

∈ {1, 2, 3}

, is

dened by

[F P (B

i

j

) + 12(m

− 1), F P (B

j

i

) + 12(m

− 1) + 4]

.And onsequently,

the simple interval representing

x

m

in

B

i

j

, with

j

∈ {1, 2, 3}

, is dened by

[F P (B

i

j

) + 12(m

− 1) + 6, F P (B

j

i

) + 12(m

− 1) + 10]

.

Fig.8.Des riptionofthesimpleintervalsofgroup

B

i

.Duetospa e onsiderations,

the des ription is divided in three lines. Ea h line starts with the end part of the

previous lineinorder to indi atethe onguration ofthewholedes ription.

Group

E

i

is omposedof threeblo ks, ea h ontaining

2n

− 1

simpleintervals (as illustrated in Figure 9):

{[F P (E

i

1

) + 6k, F P (E

1

i

) + 6k + 4], [F P (E

2

i

) +

6k, F P (E

i

2

) + 6k + 4], [F P (E

3

i

) + 6k, F P (E

3

i

) + 6k + 4]

|0 ≤ k ≤ 2n − 2}

where

F P

(E

i

1

) = F P (c

i

) + 68n

− 17

,

F P

(E

i

2

) = F P (c

i

) + 80n

− 17

,

F P

(E

i

3

) =

F P

(c

i

)+92n

−17

.Therefore,ea hsimpleintervalofblo k

E

i

j

interse tsexa tly two simple intervalsof blo k

B

i

j

, for

1

≤ j ≤ 3

.

Group

C

i

R

is omposed of

n

blo ks (one blo k for ea h boolean variable),

ea h ontainingtwosimpleintervals(asillustratedinFigure10):

{[F P (C

i

R

) +

14k, F P (C

i

R

) + 14k + 4], [F P (C

R

i

) + 14k + 3, F P (C

R

i

) + 14k + 7]

|0 ≤ k ≤

n

_{− 1}}

where

F P

(C

i

R

) = F P (c

i

) + 104n

− 19

.The

2n

simpleintervalsofgroup

C

i

R

represent in the left to right order (

x

1

, x

1

, x

2

, x

2

. . . x

n

, x

n

). By denition, the simple interval representing

x

m

in

C

i

R

is dened by

[F P (C

i

R

) + 14(m

−

1), F P (C

i

R

)+14(m

−1)+4]

.And onsequently,thesimpleintervalrepresenting

x

m

in

C

i

R

isdened by

[F P (C

i

R

) + 14(m

− 1) + 3, F P (C

R

i

) + 14(m

− 1) + 7]

.

Therefore, by onstru tion,inany blo kof group

C

i

R

the two simpleintervals

omposing this blo k are overlapping.

Finally, group

F

i

is omposed of two blo ks, ea h ontaining

2n

− 1

simple

intervals (as illustrated in Figure 11):

{[F P (F

i

_{) + 5k, F P (F}

i

_{) + 5k + 4]}

_{|0 ≤}

k

_{≤ 4n−3}}

where

F P

(F

i

_{) = F P (c}

i

) + 118n

−21

.By onstru tion,blo k

F

i

1

is

Fig.9.Des riptionof the simpleintervalsof group

E

i

.AsinFigure8,due to spa e

onsiderations,the des ription isdividedinthree lines.

Fig.10.Des ription of the simpleintervals ofgroup

C

i

R

.

omposedofthefollowingsimpleintervals:

{[F P (F

i

_{)+5k, F P (F}

i

_)+5k+4]

_{|0 ≤}

k

_{≤ 2n − 2}}

and blo k

F

i

2

is omposed of the following simple intervals:

{[F P (F

i

_{) + 5k, F P (F}

i

_{) + 5k + 4]}

_{|2n − 1 ≤ k ≤ 4n − 3}}

.

Fig.11.Des ription of thesimpleintervalsof group

F

i

.

Theset ofsimple intervalsof theinstan e of

2

-IP-Unit-

{<, ≬}

isobtainedby assembling together in order the representation of the lauses

c

1

to

c

q

. It is easy to he k the following properties (whi hare represented inFigure12):

•

for any

1 < i

≤ q

, the smallest position of any simple intervalof group

C

i

L

is greater than the biggest position of any simple interval of groups

E

i

₋₁

and

B

i

−1

;

•

forany

1 < i

≤ q

,the smallestpositionofanysimpleintervalofgroup

F

i

−1

isgreater than the biggestpositionof any simple interval of group

C

i

L

;

•

forany

1 < i

≤ q

, the biggest position of any simpleinterval of group

F

i

−1

isless thanthe smallestposition of any simple intervalof group

D

i

•

for any

1

≤ i ≤ q

, the smallest position of any simple interval of group

A

i

isgreater than the biggestpositionof any simple interval of group

D

i

;

•

forany

1

≤ i ≤ q

, the biggest positionof any simple intervalof group

A

i

is

less than the smallest positionof any simpleinterval of groups

B

i

and

E

i

;

•

for any

1

≤ i ≤ q

, the smallestpositionof any simple intervalof group

C

i

R

isgreaterthan the biggest positionof any simple intervalof groups

B

i

and

E

i

;

•

forany

1

≤ i ≤ q

,the biggestpositionof anysimple intervalofgroup

C

i

R

is less than the smallest positionof any simpleinterval of group

F

i

.

Now that we have dened the ground set of

D

, let us dene formally the

2-intervalsof

D

(partially illustratedin Figure4).

Forea h lause

c

i

,

D

is omposed of

2n

2-intervalsbuiltwithasimpleinterval

of group

C

i

L

and asimple intervalof group

A

i

:

• {([F P (C

i

L

) + r, F P (C

L

i

) + r + 4], [F P (A

i

) + s, F P (A

i

) + s + 4]),

• ([F P (C

i

L

) + s, F P (C

L

i

) + s + 4], [F P (A

i

) + r, F P (A

i

) + r + 4])

}

with

r

= 14(k

− 1), s = r + 7, 1 ≤ k ≤ n

For ea h lause

c

i

,

D

is omposed of

4n

− 2

2-intervals built with a simple intervalof group

D

i

and a simpleinterval of group

E

i

:

• {([F P (D

i

_{) + 5k, F P (D}

i

_{) + 5k + 4], [F P (E}

i

1

) + 6k

′′

, F P

(E

1

i

) + 6k

′′

+ 4]),

• ([F P (D

i

_{) + 5k}

′

_{, F P}

_(D

i

_{) + 5k}

′

_{+ 4], [F P (E}

i

2

) + 6k

′′

, F P

(E

2

i

) + 6k

′′

+ 4])

}

with

0

≤ k ≤ 2n − 2, 2n − 1 ≤ k

′

_{≤ 4n − 3, 0 ≤ k}

′′

_{≤ 2n − 2}

.

Forea h lause

c

i

,

D

is omposed of

6n

2-intervalsbuiltwithasimpleinterval

of group

B

i

and a simpleinterval of group

C

i

R

:

• {([F P (B

i

j

) + r, F P (B

j

i

) + r + 4], [F P (C

R

i

) + s, F P (C

R

i

) + s + 4]),

• ([F P (B

i

j

) + r + 6, F P (B

i

j

) + r + 10], [F P (C

R

i

) + s + 3, F P (C

R

i

) + s + 7])

}

with

r

= 12(k

− 1), s = 14(k − 1), j ∈ {1, 2, 3}, 1 ≤ k ≤ n

.

For ea h lause

c

i

,

D

is omposed of

4n

− 2

2-intervals built with a simple intervalof group

E

i

and a simpleinterval of group

F

i

:

• {([F P (E

i

2

) + 6k

′

, F P

(E

2

i

) + 6k

′

+ 4], [F P (F

i

) + 5k, F P (F

i

) + 5k + 4]),

• ([F P (E

i

3

) + 6k

′

, F P

(E

3

i

) + 6k

′

+ 4], [F P (F

i

) + 5k

′′

, F P

(F

i

) + 5k

′′

+ 4])

}

with

2n

− 2 ≤ k ≤ 4n − 3, 0 ≤ k

′

_{≤ 2n − 2, 4n − 2 ≤ k}

′′

_{≤ 6n − 4}}

.

Forea h lause

c

i

,

D

is omposed of

6n

2-intervalsbuiltwithasimpleinterval

of group

A

i

and asimple interval ofgroup

B

i

:

• {([F P (A

i

_{) + r + 2, F P (A}

i

_{) + r + 6], [F P (B}

i

j

) + s, F P (B

j

i

) + s + 4]),

• ([F P (A

i

_{) + r + 5, F P (A}

i

_{) + r + 9], [F P (B}

i

j

) + s + 6, F P (B

j

i

) + s + 10])

}

with

r

= 14(k

− 1), s = 12(k − 1), j ∈ {1, 2, 3}, 1 ≤ k ≤ n

.

For ea h lause

c

i

, in order to represent the lause

c

i

, we delete from

D

the 2-interval

([F P (A

i

_{) + r + 2, F P (A}

i

_{) + r + 6], [F P (B}

i

j

) + s, F P (B

j

i

) + s + 4])

with

r

= 14(m

− 1)

,

s

= 12(m

− 1)

if

x

m

is thevalueof the

j

th

literal of

c

i

. In a similar way, if

x

m

is the value of the

j

th

literal of

c

i

, we delete from

D

the 2-interval

([F P (A

i

_{) + r + 5, F P (A}

i

_{) + r + 9], [F P (B}

i

j

) + s + 6, F P (B

j

i

) + s + 10])

with

r

= 14(m

− 1)

,

s

= 12(m

− 1)

.

Clearly,this onstru tion an be arriedout inpolynomial-time.Wenowgive

anintuitivedes ription of the dierent elements of the set of

2

-intervalsthat

we have built. Blo k

B

i

1

(resp.

B

i

2

and

B

i

3

) represents the value of the rst (resp. se ond and third) literal, say

x

m

(or

x

m

), of the lause

c

i

; for this, the

2-interval between the simple interval of the

m

th

blo k of group

A

i

and the simpleinterval of

B

i

1

(resp.

B

i

2

and

B

i

3

) orrespondingto

x

m

(or

x

m

) isnot in

D

(stillthesimpleintervalsare in

GS(

D)

).Forinstan e,inFigure13,the fa t that there is no2-intervalbetween the simple interval orrespondingto

x

1

in

B

i

1

andasimpleintervalof group

A

i

indi atesthat the rstliteralof lause

c

i

is

x

1

.Similarly,the fa tthat thereisno2-intervalbetween thesimpleinterval orrespondingto

x

2

(resp.

x

3

)in

B

i

2

(resp.

B

i

3

)and asimple intervalof group

A

i

indi atesthat the se ond (resp. third) literalof lause

c

i

is

x

2

(resp.

x

3

).

Fig. 13.Zoomon group

B

i

oftherepresentation ofa lause

c

i

= (x

1

∨ x

2

∨ x

3

)

The sequen e of blo ks

(C

i

−1

R

,

C

i

L

,

A

i

,

B

i

,

C

i

R

)

orresponds to a me hanism

whi h propagates the value of ea h variableof

V

n

. Blo ks

(D

i

,

E

i

,

F

i

₎

orre-spond to a literal sele ting me hanism that indi ates, for ea h lause

c

i

, the literal (i.e., the rst, se ond or third) whi h satises

c

i

. Noti e that the two previous intuitivenotions willbe detailedand laried afterwards.

We start the proof by giving some properties (Lemmas 8 to 13) about the

maximal ardinality of a set of

{<, ≬}

- omparable

2

-intervals in

D

in our

onstru tion. Then, these results will be used in Proposition 14to prove the

validity of the redu tion. In the rest of this paper, we will use the following

notations:

where

I

isa simpleintervalbelongingtoblo k

X

and

J

is asimpleinterval belongingtoblo k

Y

;

•

for any

1

≤ i ≤ q

and any set of groups

α

⊆ {C

i

L

, A

i

, B

i

, C

R

i

, D

i

, E

i

, F

i

}

,

D(α)

denotesasetof

{<, ≬}

- omparable

2

-intervalsbetweenblo ksofgroups

belongingto

α

(forexample,

D(D

i

_{, E}

i

_{, F}

i

₎

denotesasetof

{<, ≬}

- omparable

2

-intervalsbetween blo ks

D

i

1

, D

2

i

, E

1

i

, E

2

i

, E

3

i

, F

1

i

and

F

i

2

);

•

for any

1

≤ i ≤ q

,

D(c

i

)

denotes a set of

{<, ≬}

- omparable

2

-intervals in the representation of lause

c

i

.

Lemma 7 For any set of groups

α

and

β

,

|D(α)| + |D(β)| ≥ |D(α

S

β)

_|

.

PROOF. The union of the sets

α

and

β

ould result in one of the following

ases:

(a)

D(α)

and

D(β)

have atleast a

2

-interval in ommon;

(b) atleast a

2

-interval of

D(α)

and a

2

-interval of

D(β)

are not disjoint; ( ) at least a

2

-interval of

D(α)

and a

2

-interval of

D(β)

are not

{<, ≬}

- omparable.

In ase (a) it is lear that the dupli ated

2

-interval willnot be ounted more

than on e in

|D(α

S

β)

|

. In ase (b), only one of the two

2

-intervals whi h

are not disjoint an be in

D(α

S

β)

. In ase ( ), only one of the two

2

-intervals whi h are not

{<, ≬}

- omparable an be in

D(α

S

β)

. If none of

those three ases o ur then,

D(α)

S

D(β)

is

{<, ≬}

- omparable, and thus,

|D(α)| + |D(β)| = |D(α

S

β)

_|

.Therefore,

|D(α)| + |D(β)| ≥ |D(α

S

β)

_|

.

2

By onstru tion, a

2

-interval an only exist between two blo ks that

orre-spond to a single lause ( f. Figure 4). Thus, the maximum ardinality of

a set of

{<, ≬}

- omparable

2

-intervals of

D

(i.e., the full representation of

the booleanformula) an bededu ed from the maximum ardinalityof

D(c

i

)

where

c

i

is a lause of

C

q

, for any

1

≤ i ≤ q

. Pre isely, the maximum ardi-nalityofaset of

{<, ≬}

- omparable

2

-intervalsinthe representationofallthe lauses isless than orequalto

q

· max

i

∈[1,q]

|D(c

i

)

|

.

Werst omputethemaximum ardinalityofaset

D(c

i

)

of

{<, ≬}

- omparable

2

-intervalsbetween blo ks orresponding toa single lause

c

i

.

Lemma 8

|D(α)| ≤ 3n

for

α

=

{C

i

L

, A

i

, B

i

, C

R

i

}

.

PROOF. Bythedisjun tion onstraint,atmostonesimpleintervalperblo k

of

A

i

anbeinvolvedina

2

-intervalbetweenblo ksof

A

i

and

B

i

.Asthereare

n

blo ksin

A

i

,wehave

|D(A

i

_{, B}

i

₎

_{| ≤ n}

.Similarly,bythedisjun tion onstraint,

at most one simple interval per blo k of

C

i

between blo ks of

B

i

and

C

i

R

. As there are

n

blo ks in

C

i

R

,

|D(B

i

_{, C}

i

R

)

| ≤ n

.

Thus, a ording to Lemma 7,

|D(A

i

_{, B}

i

_{, C}

i

R

)

| ≤ |D(A

i

, B

i

)

| + |D(B

i

, C

R

i

)

| ≤

2n

.

Moreover, at most one simple interval per blo k of

A

i

an be involved in a

2

-interval between blo ks of

A

i

and

C

i

L

sin e the two

2

-intervals between a

given blo k of

A

i

and

C

i

L

are

{⊏}

- omparable. As there are

n

blo ks in

A

i

,

|D(C

i

L

, A

i

)

| ≤ n

. Thus, by Lemma 7,

|D(C

i

L

, A

i

, B

i

, C

R

i

)

| ≤ |D(A

i

, B

i

, C

R

i

)

| +

|D(C

i

L

, A

i

)

| ≤ 3n

.

2

Inthe following,

θ(i, j)

willdenotetheset ofallthesimpleintervalsin

B

i

j

and

E

i

j

,with

1

≤ j ≤ 3

.Theset

δ(i, j)

⊆ θ(i, j)

willdenoteaset ofdisjointsimple intervalsand

k(E, i, j)

(resp.

k(B, i, j)

)willbethe numberof simpleintervals ofblo k

E

i

j

(resp.

B

i

j

)whi harein

δ(i, j)

.By onstru tion,ea hsimpleinterval inblo k

E

i

j

interse ts twosimple intervalsof blo k

B

i

j

( f.Figure14and page 14).

Observation 1 (a) If

k(E, i, j) > 0

thenatleast

k(E, i, j)+1

simpleintervals of blo k

B

i

j

annot belong to

δ(i, j)

. Thus,

k(B, i, j)

≤ 2n − (k(E, i, j) + 1)

. Hen e,

|δ(i, j)| ≤ k(B, i, j) + k(E, i, j) ≤ 2n − (k(E, i, j) + 1) + k(E, i, j) ≤

2n

_{− 1}

.

(b)If

k(E, i, j) = 0

thenallthesimpleintervals(i.e.,

2n

)ofblo k

B

i

j

anbelong to

δ(i, j)

. Thus,

k(B, i, j)

≤ 2n

. Hen e,

|δ(i, j)| ≤ k(B, i, j) + k(E, i, j) ≤ 2n

.

Fig. 14. If two simple intervals of blo k

E

i

j

are part of

δ(i, j)

then at least three simple intervals ofblo k

B

i

j

annotbelong to

δ(i, j)

,and thus

|δ(i, j)| ≤ 2n − 1

.

Lemma 9 If

|D(D

i

_{, E}

i

_{, F}

i

₎

_{| > 4n − 2}

then

|D(c

i

)

| < 7n − 2

.

PROOF. Assumethat

|D(D

i

_{, E}

i

_{, F}

i

₎

_{| = 4n−2+γ}

with

γ >

0

.Asea hblo k

of group

E

i

(i.e.,

E

i

1

, E

2

i

, E

3

i

) is omposed of

2n

− 1

simple intervals,there is atleast one simple intervalin ea h blo k of group

E

i

involved in a

2

-interval

of

D(D

i

_{, E}

i

_{, F}

i

₎

.

Thus, onsidering only the simple intervalsin groups

B

i

and

E

i

, there are at

most

6n

− 3

(i.e.,

3

· (2n − 1)

by Observation 1 (a)) disjoint simpleintervals.

By onstru tion, any

2

-interval of

D(A

i

_{, B}

i

_{, C}

i

R

, D

i

, E

i

, F

i

)

is omposed of a

simple interval of either group

B

i

or

E

i

. Thus, as there are at most

6n

− 3

disjoint simple intervals in groups

B

i

and

E

i

, there are at most

6n

− 3 2

-intervalsin

D(A

i

_{, B}

i

_{, C}

i

R

, D

i

, E

i

, F

i

)

. As

|D(C

i

L

, A

i

)

| ≤ n

( f.proof of Lemma

8), by Lemma 7, we an on lude that

|D(C

i