Covering graphs with few complete bipartite subgraphs

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Theoretical Computer Science

journal homepage:www.elsevier.com/locate/tcs

Covering graphs with few complete bipartite subgraphs

I

Herbert Fleischner

a

, Egbert Mujuni

b

, Daniël Paulusma

c,∗

, Stefan Szeider

c a_{Department of Computer Science, Vienna Technical University, A-1040 Vienna, Austria}

b_{Mathematics Department, University of Dar es Salaam, PO Box 35062, Dar es Salaam, Tanzania} c_{Department of Computer Science, Durham University, Durham DH1 3LE, United Kingdom}

a r t i c l e i n f o Article history:

Received 12 August 2008

Received in revised form 22 December 2008 Accepted 25 December 2008

Communicated by G. Ausiello

Keywords:

Bicliques

Parameterized complexity Covering and partitioning problems

a b s t r a c t

We consider computational problems on covering graphs with bicliques (complete bipartite subgraphs). Given a graph and an integerk, thebiclique cover problemasks whether the edge-set of the graph can be covered with at mostkbicliques; thebiclique partition problemis defined similarly with the additional condition that the bicliques are required to be mutually edge-disjoint. Thebiclique vertex-cover problemasks whether the vertex-set of the given graph can be covered with at mostkbicliques, thebiclique vertex-partition problemis defined similarly with the additional condition that the bicliques are required to be mutually vertex-disjoint. All these four problems are known to be NP-complete even if the given graph is bipartite. In this paper, we investigate them in the framework of parameterized complexity: do the problems become easier ifkis assumed to be small? We show that, consideringkas the parameter, the first two problems are fixed-parameter tractable, while the latter two problems are not fixed-fixed-parameter tractable unless P=NP.

©2008 Elsevier B.V. All rights reserved.

1. Introduction

LetGbe a simple undirected graph and letSbe a set of (not necessarily vertex-induced) subgraphs ofG. The setSis a

coverofGofsize

|

S

|

if every edge ofGis contained in at least one of the subgraphs inS. The setSis avertex-coverofGif every vertex ofGis contained in at least one of the subgraphs inS. If all subgraphs inSarebicliques, that is, complete connected bipartite graphs, then we speak of abiclique coveror abiclique vertex-cover, respectively.

We consider the following four problems.

Biclique Cover

Instance:A graphGand a positive integerk.

Question:DoesGhave a biclique cover of size at mostk?

Biclique Partition

Instance:A graphGand a positive integerk.

Question:DoesGhave a biclique cover of size at mostkconsisting of mutuallyedge-disjointbicliques?

I _{A preliminary and shortened version of this paper appeared in the Proceedings of the27th International Conference of Foundations of Software} Technology and Theoretical Computer Science (FSTTCS 2007).

∗_{Corresponding author. Tel.: +44 1913341723.}

E-mail addresses:fleisch@dbai.tuwien.ac.at(H. Fleischner),emujuni@maths.udsm.ac.tz(E. Mujuni),daniel.paulusma@durham.ac.uk(D. Paulusma), stefan.szeider@durham.ac.uk(S. Szeider).

0304-3975/$ – see front matter©2008 Elsevier B.V. All rights reserved. doi:10.1016/j.tcs.2008.12.059

Biclique Vertex-Cover

Instance:A graphGand positive integerk.

Question:DoesGhave a biclique vertex-cover of size at mostk?

Biclique Vertex-Partition

Instance:A graphGand positive integerk.

Question:DoesGhave a biclique vertex-cover of size at mostkconsisting of mutuallyvertex-disjointbicliques? We observe that the Biclique Vertex-Coverand Biclique Vertex-Partition problem are equivalent, since we can

always make the bicliques of a biclique vertex-cover disjoint without increasing the size of the cover (and we can do so without introducing trivial bicliques, that is, bicliques having one vertex only). However,Biclique Cover and

Biclique Partitionare not equivalent. Take, for example, the bipartite graph with vertex setU1

∪

U2,U1

= {

x1

,

x2

,

x3

}

,

U2

=

{

y1

,

y2

,

y3

}

, and all possible edges between vertices in U1 and U2 except for the edges x1y3 and x3y1. This

graph has a biclique cover of size 2, namely the biclique cover with bicliques

(

{

x1

,

y1

,

x2

,

y2

}

,

{

x1y1

,

x1y2

,

x2y1

,

x2y2

}

)

and

(

{

x2

,

y2

,

x3

,

y3

}

,

{

x2y2

,

x2y3

,

x3y2

,

x3y3

}

)

. However, any biclique cover that consists of mutually edge-disjoint bicliques has

size at least 3.

One can consider variants of the above problems where solutions must consist of nontrivial bicliques only. However, minimal solutions forBiclique CoverandBiclique Partitionclearly do not contain trivial bicliques, and it is easy to see that

forBiclique Vertex-CoverandBiclique Vertex-Partitionthere is always a minimal solution where only isolated vertices

are contained in trivial bicliques. Hence, the computational complexities of the four problems do not change if solutions must avoid trivial bicliques.

TheBiclique Coverproblem arises in both theoretical and practical areas for more than thirty years. From a theoretical point of view, theBiclique Coverproblem is equivalent to the set basis problem [22] and related to boolean algebraic forms

associated with graphs and combinatorial optimization problems. There, the minimum number of bicliques necessary to cover all the edges of a graphGis also called thebipartite dimensionofG, which is considered to be an interesting graph property on its own. For more details we refer to [1,3,7]. From a more practical perspective, bicliques are used to model the rectangle cover problem that asks if a rectilinear polygon can be expressed as the union of a minimum number of rectangles [16]. Both theBiclique Coverand theBiclique Partitionproblem play a significant role in the analysis of so-called HLA reaction matrices used in biology [19]. Other practical applications lie in artificial intelligence and data mining. In Formal Concept Analysis, context is structured into a set of concepts with binary relations. It turns out that each concept corresponds to a so-called closed item set in data mining and, by representing the binary relations as bipartite graphs, to a maximal biclique. See [24] for more details. Applications ofBiclique Vertex-Coverinclude data mining, e-commerce, information retrieval and network management. In all these applications, large bipartite graphs are analyzed in order to discover so-called cross associations corresponding to bicliques [13].

All four problems are computationally hard problems:Biclique Coveris NP-complete and remains NP-hard for chordal

bipartite graphs [18,21]. TheBiclique Partitionproblem is also already NP-complete for bipartite graphs [15]. Very recently,

Heydari et al. [13] showed thatBiclique Vertex-Partition, and consequently,Biclique Vertex-Coverare NP-complete for bipartite graphs.

In this paper, we investigate the questions of whether these problems become easier if the given upper boundkon the number of bicliques in the cover is assumed to be small. We undertake this investigation in the framework of parameterized complexity as developed by Downey and Fellows [6], considering the upper boundkon the number of bicliques in the cover as the parameter. We give some basic background of parameterized complexity in Section2.2. In principle, the problems under consideration can fall into any of the following three categories.

1. For every fixedkthe problem can be solved in polynomial time where the order of the polynomial is independent ofk; in this case we say that the problem isfixed-parameter tractable.

2. For every fixedkthe problem can be solved in polynomial time but the order of the polynomial grows withk. 3. For some fixedkthe problem is NP-hard.

Problems that fall into the second category can be further categorized by means of the complexity classes W[1], W[2]

, . . . ,

XP (see Section2.2). In the literature, a similar study has been performed for the problemsClique CoverandClique Partition. These NP-complete problems ask if a given graph has a cover consisting of at mostkcliques orkmutually edge-disjoint cliques, respectively. BothClique Cover[10,12] andClique Partition[17] are fixed-parameter tractable. The problem Clique Vertex-Cover(orPartition into Cliques) asks if the vertices of a given graph can be covered by at mostkcliques. This problem is NP-complete for each fixedk

≥

3 and polynomial-time solvable fork

≤

2 [9, GT15].

New results

Our results show that the problems under consideration fall into all three of the above categories, spanning a wide range of parameterized complexities.

1. ProblemsBiclique CoverandBiclique Partitionare fixed-parameter tractable.

We show these results in Section3. We make use ofkernelization, that is, we give an algorithm that reduces an instance ofBiclique CoverorBiclique Partitionin polynomial time into an equivalent instance where the number of vertices is

2. For k

≤

2the problemBiclique Vertex-Covercan be solved in polynomial time for bipartite graphs. For every fixed k

≥

3

the problemBiclique Vertex-Coveris NP-complete and remains NP-hard for bipartite graphs.

As the problemBiclique Vertex-Coveris equivalent to the problemBiclique Vertex-Partition, the above result is also

valid for the latter problem. In Section4.1we establish the NP-completeness result by a reduction from an NP-complete variant of theList-Coloringproblem. Note that our NP-completeness result is stronger than the one in [13] as we assume thatk

≥

3 is a constant and not part of the input. In Section4.2we show the polynomial casek

=

2. The result for this case follows directly from a stronger result on a graph homomorphism problem defined on the complement graph (we explain this in detail in Section4.2).

In view of the NP-hardness it makes sense to study the more restricted problemb-Biclique Vertex-Coverwhere the

bicliques in the cover are bicliques where at least one of the bipartite sets contains at mostbvertices. Indeed, in Section4.3, we show that this restriction moves the problem from the third to the second of the above categories:

3. For every fixed b

≥

1the problem b-Biclique Vertex-Coveris W[2]-complete and remains W[2]-hard for bipartite graphs.

2. Preliminaries

2.1. Graph theoretic terminology

For graph theoretic terminology not defined in this paper, we refer the reader to standard text books [2,5]. In this paper, we consider connected simple graphsG

=

(

V

,

E

)

. The set of neighbors of a vertex

v

in a graphGis denoted byNG

(v)

, and we

setNG

(

T

)

=

S

v∈TNG

(v)

forT

⊂

V(we often omit the subscriptGif it is clear from the context which graphGis considered).

A setD

⊆

Vis adominating setofGif every vertex ofGis either inDor has a neighbor inD. Thedistance dG

(

u

, v)

between

two verticesuand

v

inGis the number of edges of a shortest path fromuto

v

. Thediameterdiam

(

G

)

ofGis the maximum distance over all pairs of vertices ofG; diam

(

G

)

= ∞

ifGis disconnected. IfV0

⊆

V, we denote byG

[

V0

]

the subgraph ofG

induced byV0_{. We writeG}

₌

₍₍

_V

1

,

V2

),

E

)

for a bipartite graphG

=

(

V

,

E

)

having the vertex bipartitionV

=

V1

∪

V2. We

say thatG

=

((

V1

,

V2

),

E

)

is abicliqueifGis connected andEcontains all possible edges between vertices inV1and vertices

inV2. A biclique

((

U1

,

U2

),

E

)

is astar centered at a vertex uifU1

= {

u

}

orU2

= {

u

}

.

2.2. Parameterized complexity

We give some basic background on parameterized complexity; for a detailed discussion we refer the reader to other sources [6,20]. In parameterized complexity theory, we consider the problem input as consisting of two parts; that is, a pair

(

I

,

k

)

, whereIis the main part andk(usually an integer given in unary) is the parameter. We say a problem isfixed parameter tractableif an instance

(

I

,

k

)

can be solved in timeO

(

f

(

k

)

+

nc

₎

_orO

₍

_f

₍

_k

₎

_nc

₎

_{, wherefdenotes a computable function andc}

denotes a constant that is independent of the parameterk. Therefore, such an algorithm may provide an efficient solution to the problem if the parameter is reasonably small. We denote by FPT the class of all fixed-parameter tractable decision problems.

A well known technique to show that a parameterized problemΠis fixed-parameter tractable, is to find areduction to a problem kernel(this is also calledkernelization). It replaces an instance

(

I

,

k

)

ofΠwith a reduced instance

(

I0

_,

_k0

₎

_ofΠ(called problem kernel) such that

(i) k0

≤

kand

|

I0

| ≤

g

(

k

)

for some computable functiong;

(ii) the reduction from

(

I

,

k

)

to

(

I0

,

k0

)

is computable in polynomial time; (iii)

(

I

,

k

)

is a yes-instance ofΠif and only if

(

I0

_,

_k0

₎

_{is a yes-instance ofΠ.}

It is well known that a parameterized problem is fixed-parameter tractable if and only if it is kernelizable [12,14,20]. Parameterized complexity offers a completeness theory, similar to the theory of NP-completeness, that allows the accumulation of strong theoretical evidence that some parameterized problems are not fixed-parameter tractable. This completeness theory is based on a hierarchy of complexity classes W[1]

,

W[2]

, . . . ,

XP. Each class contains all parameterized decision problems that can be reduced to a certain fixed parameterized decision problem underfpt-reductions. An fpt-reduction from a parameterized problemΠto a parameterized problemΠ0is an algorithm that computes for every instance

(

I

,

k

)

ofΠan instance

(

I0

_,

_k0

₎

_ofΠ0_{in at mostf}

₍

_k

₎

_|

_I

_|

c_{time for some computable functionfand constantcsuch that}

(i) k0

_≤

_h

₍

_k

₎

_{for some computable functionh, and}

(ii)

(

I

,

k

)

is a yes-instance ofΠif and only if

(

I0

_,

_k0

₎

_{is a yes-instance ofΠ}0_.

This means that ifΠ0_{belongs to some parameterized complexity classWthenΠalso belongs toW. For instance, the class}

W[1] contains all parameterized problems that can be reduced toWeighted3-CNF-Satisfiabilityby an fpt-reduction. The latter problem asks for a given instanceF of 3CNF and a positive integerk, whetherF can be satisfied by setting exactly

kvariables to true. The class XP consists of parameterized decision problemsΠsuch that for each instance

(

I

,

k

)

, it can be decided inO

(

f

(

k

)

|

I

|

g(k)

)

time whether

(

I

,

k

)

∈

Π, wheref

,

gare computable functions depending only onk. That is, XP consists of parameterized decision problems which can be solved in polynomial time if the parameter is considered as a constant. The above classes form the chain FPT

⊆

W[1]

⊆

W[2]

⊆ · · · ⊆

XP where all inclusions are conjectured to be proper; FPT

6=

XP is known [6,8].

3. Covering the edges

As mentioned in the introduction, the decision problem corresponding to Biclique Coveris NP-complete even for bipartite graphs [21]. In this section, we establish fixed-parameter tractability.

We start with two simple reduction rules that can be easily applied to simplify an instance of theBiclique Coveror

Biclique Partitionproblem.

Rule 1. Given an instance

(

G

,

k

)

and a vertex

v

∈

V

(

G

)

of degree 0, then

(

G

,

k

)

is a yes-instance if and only if

(

G

−

v,

k

)

is a yes-instance.

Rule 2. Given an instance

(

G

,

k

)

and a pair of (non-adjacent) verticesu

, v

such thatN

(

u

)

=

N

(v)

, then

(

G

,

k

)

is a yes-instance if and only if

(

G

− {

v

}

,

k

)

is a yes-instance.

Clearly, the following is true.

Lemma 1. Rules 1 and 2 are correct for both problemsBiclique CoverandBiclique Partition, and can be applied in polynomial time.

We say that an instance

(

G

,

k

)

isreduced(with respect to Rules 1 and 2) if these rules cannot be applied.

Theorem 2 (Kernelization). If

(

G

,

k

)

is a reduced yes-instance ofBiclique CoverorBiclique Partitionthen G has at most3k

vertices. Furthermore, if G is bipartite, then it has at most2k+1vertices.

Proof. Let

(

G

,

k

)

be a reduced instance of theBiclique CoverorBiclique Partitionproblem with biclique coverS

=

{

C1

, . . . ,

Cl

}

of sizel

≤

k, whereCi

=

((

Xi

,

Yi

),

Ei

)

. We will argue similarly as Gramm et al. [10]. We assign to each vertex

v

∈

V

(

G

)

a vectorb_v

∈ {

0

,

1

,

2

}

l_{where thei-th componentb}

v,i

=

1 if

v

is contained inXi,bv,i

=

2 if

v

is contained in

Yi, andbv,i

=

0 otherwise. Since

(

G

,

k

)

is reduced, each vertex belongs to at least one biclique ofS. Consider an arbitrary

but fixed vectorb

∈ {

0

,

1

,

2

}

l. LetVbbe the set of vertices ofGsuch thatbu

=

bfor allu

∈

Vb. SupposeVbcontains two

distinct verticesx

,

y. Sincebx

=

by

,

it follows thatxandybelong to the same partition classes of the same bicliques. Then

N

(

x

)

=

N

(

y

)

and we obtain a contradiction. Hence

|

Vb

| =

1. Therefore we conclude thatGhas at most

|{

0

,

1

,

2

}

l

| ≤

3k

vertices. IfGis bipartite, we can defineb_v,i

=

1 if

v

is contained inCiandbv,i

=

0 otherwise. Then we find that each set

Vbmust be complete (as otherwise we could apply rule 2 for two verticesx

,

ywithbx

=

by), and thus contains at most two

vertices. This means thatGhas at most 2

× |{

0

,

1

}

l

| ≤

2k+1vertices if it is bipartite.

A direct consequence ofTheorem 2is thatBiclique CoverandBiclique Partitionare fixed-parameter tractable.

Corollary 3. Both theBiclique Coverand theBiclique Partitionproblem can be solved in O

(

f

(

k

)

+

n3

)

time where f

(

k

)

=

32k2+3k_{for non-bipartite graphs and f}

₍

_k

₎

₌

₂2k2+3k_{for bipartite graphs.}

Proof. We represent a graphG

=

(

V

,

E

)

on

|

V

| =

nvertices by itsadjacency matrix, i.e., then

×

nmatrixA

=

(

aij

)

with

rows and columns indexed by the vertices ofVsuch thatauv

=

1 ifu

v

∈

Eandauv

=

0 otherwise. Then it takesO

(

n2

)

time

to detect and remove all isolated vertices inG(Rule 1) andO

(

n3

)

time to verify ifN

(

u

)

=

N

(v)

for any two verticesuand

v

(Rule 2). Then, byTheorem 2, we find a reduced graphG0_{with 3}k_{vertices and consequentlyO}

₍

₉k

₎

_{edges if it is non-bipartite}

and 2k+1_{vertices and consequentlyO}

₍

₄k

₎

_{edges if it is bipartite inO}

₍

_n3

₎

_time.

A brute force algorithm that solves theBiclique Partitionproblem with input

(

H

,

k

)

, whereHis a graph withmedges, guesses for each edge ofHto which biclique it belongs and verifies if the resulting partition ofE

(

H

)

yields a biclique cover of size at mostk. This takesO

(

mk

₎

_{time. As any partition ofE}

₍

_H

₎

_{fixes the vertices of both bipartition classes of each biclique,}

we only have to verify if all the fixed sets of vertices indeed induce mutually edge-disjoint bicliques. This verification process takesO

(

|

V

(

H

)

|

3

₎

_{time. We can do exactly the same for the}

Biclique Coverproblem except that here we do not

care if the bicliques are mutually edge-disjoint. Hence, for both problems, we findf

(

k

)

=

32k2+3k_ifG0_{is non-bipartite and} f

(

k

)

=

22k2+3kotherwise. This finishes the proof ofCorollary 3.

4. Covering the vertices

As we observed in Section1thatBiclique Vertex-CoverandBiclique Vertex-Partitionare equivalent, we will only

consider the former problem in this section.

4.1. NP-hardness

We now proceed to show thatBiclique Vertex-Coveris NP-complete for fixedk

≥

3, even if the given graph is bipartite.

We present a polynomial-time reduction from the following problem.

List-Coloring

Instance:A graphG

=

(

V

,

E

)

and a mappingLthat assigns to every

v

∈

Va listL

(v)

of colors allowed for

v

.

Question:Is there a coloringcofV

(

G

)

such thatc

(v)

∈

L

(v)

for each

v

∈

Vandc

(

u

)

6=

c

(v)

for eachu

v

∈

E? If such a coloringcexists, then we callcanL-coloring of G, and we say thatGisL-colorable. If the number of available colors

k

= |

S

v∈VL

(v)

|

is fixed, then the problem is calledk-List-Coloring. This problem is known to be NP-complete for bipartite

Fig. 1.A graphGwith list assignmentLand the graphHobtained fromG. TheL-coloringcofGwithc(x1)=c(x3)=c(y1)=1 andc(x2)=c(y2)=

c(y3)=2 and the corresponding biclique vertex-partition ofHare indicated with black and white vertices.

Our reduction proceeds as follows. Let

(

G

,

L

)

be an instance ofk-List-ColoringwhereG

=

((

U

,

V

),

E

)

is a bipartite graph.

We assume that

S

v∈VL

(v)

= {

1

,

2

, . . . ,

k

}

. We construct a graphHas follows (seeFig. 1for an example):

1. LetG∗be the bipartite complement ofG; i.e.,V

(

G∗

)

=

V

(

G

)

=

U

∪

VandE

(

G∗

)

= {

u

v

:

u

∈

U

, v

∈

V

,

u

v /

∈

E

(

G

)

}

. 2. Fori

=

1

, . . . ,

k, introduceknew edgesui

v

ifori

=

1

, . . . ,

kusing 2knew verticesui

, v

i

∈

/

V

(

G∗

)

.

3. Now takeG∗and thekedgesui

v

i. For everyx

∈

Uandi

∈ {

1

, . . . ,

k

}

, ifi

∈

L

(

x

)

add an edgex

v

i. For everyy

∈

Vand

i

∈ {

1

, . . . ,

k

}

, ifi

∈

L

(

y

)

add an edgeyui. Call the resulting graphH. Thus,His a bipartite graph containingG∗as a proper

subgraph (note thatV

(

H

)

=

(

U

∪ {

ui

:

1

≤

i

≤

k

}

)

∪

(

V

∪ {

v

i

:

1

≤

i

≤

k

}

))

.

In general, it is clear thatHcan be constructed in polynomial time and

|

V

(

H

)

| = |

V

(

G

)

| +

2k. Furthermore, the following can be established (as illustrated inFig. 1).

Lemma 4. G is L-colorable if and only if V

(

H

)

can be covered by k bicliques.

Proof. Suppose thatGhas anL-coloringc. Define a partition ofV

(

H

)

as follows. Fori

=

1

, . . . ,

k, define

Ci

:= {

v

∈

V

(

G

)

:

c

(v)

=

i

} ∪ {

ui

, v

i

}

.

LetS

:= {

H

[

C1

]

, . . . ,

H

[

Ck

]}

. Note that each vertex ofHbelongs to precisely one element ofS. We claim thatSis a biclique

vertex-cover ofH. Choose an arbitrary elementH

[

Ci

] ∈

S. Letx

,

y

∈

V

(

G

)

be two vertices inH

[

Ci

]

belonging to different

classes in the vertex bipartition ofHinduced by the vertex bipartition ofG. Clearlyxy

∈

/

E

(

G

)

becausec

(

x

)

=

c

(

y

)

. Thus

xy

∈

E

(

G∗

₎

_{which in turn implies thatxy}

_∈

_E

₍

_H

_[

_C

i

]

)

. Moreover, by definition ofH,uiy

∈

E

(

H

[

Ci

]

)

for everyy

∈

Ci

∩

V, and

v

ix

∈

E

(

H

[

Ci

]

)

for everyx

∈

Ci

∩

U. Thus,H

[

Ci

]

is a biclique ofH. Thus, we conclude the setSis a biclique vertex-cover ofG.

Now suppose thatH has a biclique vertex-coverS

= {

G1

, . . . ,

Gk

}

. The edgesui

v

i,i

=

1

, . . . ,

kbelong to distinct

bicliques sinceui

v

j

∈

/

E

(

H

)

,i

,

j

∈ {

1

, . . . ,

k

}

,i

6=

j. Hence we may assume, w.l.o.g., thatui

v

i

∈

E

(

Gi

)

,i

=

1

, . . . ,

k. Let

Ci

:=

V

(

Gi

)

− {

ui

, v

i

}

. The set

{

C1

, . . . ,

Ck

}

defineskdisjoint independent sets inGsinceH

[

Ci

]

is a biclique or a subset ofU

or a subset ofV. Now define a function

γ

:

V

(

G

)

→

S

v∈V(G)L

(v)

as follows:

γ (v)

:=

i if and only if

v

∈

Ci

.

For every

v

∈

Ciwe have, by definition ofH,i

∈

L

(v)

, and as we deduced above

γ (

x

)

6=

γ (

y

)

forxy

∈

E

(

G

)

. Thus

γ

defines

anL-coloring ofG.

For every fixedkthe problemBiclique Vertex-Coverbelongs to NP. Since, as mentioned above,k-List-Coloringis

NP-complete for bipartite graphs fork

≥

3, the reduction inLemma 4yields following the result.

Theorem 5.Biclique Vertex-Coveris NP-complete for every fixed k

≥

3. This also holds if only bipartite graphs are considered.

Corollary 6. Biclique Vertex-Coveris not fixed-parameter tractable unless P

=

NP.

4.2. Polynomial cases

Next we study the question of whetherk

≥

3 is an optimal bound for the NP-hardness ofBiclique Vertex-Cover. Ifk

=

1 the problem is trivially solvable in polynomial time:Ghas a biclique vertex-cover of size one if and only if the complement graphG(which has vertex setV

(

G

)

=

V

(

G

)

and edgesu

v

wheneveru

v /

∈

E

(

G

)

) is disconnected or

|

V

| =

1. The casek

=

2 is still open. However, we can establish polynomial-time results for a special graph class that includes all bipartite graphs.

For this purpose we transform Biclique Vertex-Cover for k

=

2 into an equivalent problem involving graph

homomorphisms. We need the following definitions. LetG

,

Hbe two simple graphs. A mappingh

:

V

(

G

)

→

V

(

H

)

is a

homomorphism from G to the reflexive closure of Hif for every edgeu

v

∈

E

(

G

)

we have eitherh

(

u

)

=

h

(v)

orh

(

u

)

h

(v)

∈

E

(

H

)

. The homomorphismhisvertex-surjectiveif for eachc

∈

V

(

H

)

there is some

v

∈

V

(

G

)

withh

(v)

=

c. LetCkdenote the cycle

onkverticesc1

, . . . ,

ckwhereciandcjare adjacent if and only if

|

i

−

j

| ≡

1

(

modk

)

. We make the following observation,

which is easy to see.

Observation 7. A graph G has a biclique vertex-cover consisting of two non-trivial vertex-disjoint bicliques if and only if there is a vertex-surjective homomorphism from the complement graph G to the reflexive closure of C4.

Theorem 8. We can check in polynomial time whether a graph G allows a vertex-surjective homomorphism to the reflexive closure of C4if

(i) G has a dominating edge, or

(ii)G has diameter not equal to two, or

(iii) G has bounded maximum degree, or

(iv)G is triangle-free.

Proof. LetG

=

(

V

,

E

)

be a graph. The following terminology is useful. Lethbe a vertex-surjective homomorphism fromG

to the reflexive closure ofC4. Ifhmaps a vertex

v

∈

Vtoci, we say that

v

hascolor i. This wayhinduces a coloring with

exactly four different colors 1, 2, 3, 4 such that neither color pair

(

1

,

3

)

nor

(

2

,

4

)

is used on the end vertices of an edge. We callhadiagonal coloring. Since any diagonal coloring corresponds to a vertex-surjective homomorphism fromGto the reflexive closure ofC4as well, we are done if we can decide in polynomial time ifGhas a diagonal coloring for cases (i)–(iv).

We prove each case separately.

(i) Supposexyis a dominating edge ofG. Clearly,

{

x

,

y

}

will be assigned two different colors by any diagonal coloringhofG. Suppose such a coloringhexists. Then we may, w.l.o.g., assume thatxhas got color 1 andyhas got color 2. We will show how we can check in polynomial time whether this precoloring can be extended to a full diagonal coloring ofG. We call a setU

⊆

V coloredif every vertex inUhas received a color. In a precoloring, we denote the set of all colored neighbors of a vertexubyNc

₍

_u

₎

_{, and we call a colored setU j-chromaticif the number of different colors inUequalsj.}

We proceed as follows. First we guess an uncolored vertexsnot adjacent toxthat we assign color 3, and an uncolored vertext not adjacent toythat we assign color 4. Note that the number of guesses is bounded byO

(

|

V

|

2

₎

_{. We apply the}

following rule as long as possible: if there exists an uncolored vertexuwith 3-chromaticNc

(

u

)

thenucan only get one possible color, which we then assign tou. Afterwards, we check if there exists a vertex

w

with a 4-chromatic colored neighbor set. If so, then pair

(

s

,

t

)

was a wrong guess, because we cannot assign an appropriate color to

w

. We then guess another pair

(

s0

,

t0

)

that we assign color 3

,

4 respectively, and so on.

Suppose that for a particular pair

(

s

,

t

)

we have applied the above rule as long as possible and such a vertex

w

(with 4-chromaticNc

(w)

) does not exist. Sincexyis a dominating edge, we can partition the uncolored vertices ofGinto the following sets: setsUi,jconsisting of vertices adjacent to vertices with coloriandjfor

(

i

,

j

)

∈ {

(

1

,

2

), (

1

,

3

), (

1

,

4

), (

2

,

3

), (

2

,

4

)

}

and

setsUiconsisting of verticesonlyadjacent to colorifori

=

1

,

2. Then we extend the precoloring ofFby assigning color 1 to

the vertices inU1,2

∪

U1,4

∪

U2,4

∪

U1

∪

U2and color 2 to the vertices inU1,3

∪

U2,3. This proves case (i).

(ii) SupposeGdoes not have diameter 2. If diam

(

G

)

=

1, thenGis a complete graph and does not have a diagonal coloring. Let diam

(

G

)

≥

3. Then there exist verticesu

, v

inGwithdG

(

u

, v)

=

diam

(

G

)

≥

3. We can find such a pairu

, v

in polynomial

time. We assign color 1 tou, color 2 to all neighbors ofu, color 3 to all vertices of distance 2 fromu, and color 4 to all remaining vertices inG. As this coloring is diagonal, we have shown case (ii).

(iii) SupposeGhas maximum degreedfor some fixed integerd. By (ii) we may assume thatGhas diameter 2. This means thatVhas at mostd2

₊

_{1 vertices, which proves case (iii).}

(iv) SupposeGis triangle-free. By (ii) we may assume that diam

(

G

)

=

2. IfGhas adominating vertex u(i.e.,N

(

u

)

=

V

\{

u

}

) thenGdoes not have a diagonal coloring (sinceuwould become adjacent to a forbidden color). SupposeGdoes not have a dominating vertex. We claim thatGhas a diagonal coloring if and only if

|

V

| ≥

4.

SupposeGhas a diagonal coloringc. As

|

c

(

V

)

| =

4, we obtain

|

V

| ≥

4. To prove the reverse implication, suppose

|

V

| ≥

4. Letu

∈

Vbe a vertex with degree at least two. We coloruby 1, one of its neighbors by 2, its remaining neighbors by 4 and all the other vertices by 3 (asuis not dominating,Ghas at least one vertex not adjacent tou). SinceGis triangle-free,

N

(

u

)

is an independent set, and we have obtained a diagonal coloring ofG. This proves case (iv) and completes the proof of

Theorem 8.

Corollary 9. Biclique Vertex-Coverfor fixed k

=

2can be solved in polynomial time for the class of graphs that contain a pair of nonadjacent vertices with no common neighbor. In particular,Biclique Vertex-Coverfor fixed k

=

2can be solved in polynomial time for bipartite graphs.

Proof. The first statement immediately follows fromObservation 7andTheorem 8. So, letGbe a bipartite graph with bipartition classesA

,

B. IfGhas two nonadjacent verticesx

∈

Aandy

∈

B, then we are done by the first statement. In the other caseGis a biclique. This provesCorollary 9.

Remark 10. A homomorphismffrom a graphGto a graphHis callededge-surjectiveor acompactionif for eachxy

∈

E

(

H

)

withx

6=

ythere is someu

v

∈

E

(

G

)

withf

(

u

)

f

(v)

=

xy. The problem that asks whether there exists a compaction from a given graph to the reflexive closure ofC4is known to be NP-complete [23]. For a graphGwith diameter 2, it is equivalent to

asking ifGallows a vertex-surjective homomorphism to the reflexive closure ofC4. This can be seen as follows.

We first note that any compaction, which is edge-surjective, is also vertex-surjective. To show the remaining implication, supposefis a vertex-surjective homomorphism fromGto the reflexive closure ofC4. Supposef

(

u

)

=

c1andf

(v)

=

c2. If

u

v

∈

E, thenc1andc2are images of the end vertices of an edge. Otherwise, sinceGhas diameter two, there exists a vertex

sadjacent touand

v

. Thenf

(

s

)

=

c1orf

(

s

)

=

c2. In the first cases

v

and in the second casesuis the desired edge. We use

So far, we could only show that the problem that asks ifGallows a compaction to the reflexive closure ofC4stays

NP-complete if we restrict the input graphs to graphs with diameter 3. We do this by slightly modifying the NP-NP-completeness reduction given in [23]. As this is beyond the scope of this paper, we leave out the proof details.

Remark 11. Of related interest is the concept ofH-partitionsas studied by Dantas et al. [4]. LetHbe a fixed graph with four verticesh1

, . . . ,

h4. AnH-partition of a graphG

=

(

V

,

E

)

is a partition ofVinto fournonemptysetsX1

, . . . ,

X4such

that wheneverhihjis an edge of H, thenGcontains the bicliqueK

=

((

Xi

,

Xj

),

Ek

)

.H-partitiondenotes the problem

of deciding whether a given graph admits anH-partition. Evidently,Biclique Vertex-Coverfork

=

2 is equivalent to

the problem 2K2-partitionwhere 2K2denotes the graph on four vertices with two independent edges.H

=

2K2is the

only case for which the complexity ofH-partitionis not known. All other cases are known to be solvable in polynomial time.

Remark 12. TheBiclique Vertex-Coverproblem fork

=

2 is also equivalent to asking ifGhas adisconnected cut, i.e., a set

U

⊆

Vsuch that bothG

[

U

]

andG

[

V

\

U

]

are disconnected. We are not aware of any previous work on the problem expressed this way.

4.3. Bounding one side of the bicliques

In the following, we study the question of whetherBiclique Vertex-Coverbecomes easier when the number of vertices

in one of the two classes of the vertex bipartition of bicliques is bounded. For a complete bipartite graphK

=

((

U1

,

U2

),

E

)

we define

β(

K

)

=

min

{|

U1

|

,

|

U2

|}

. Clearly,

β(

K

)

=

1 if and only ifKis a star. Ab-bounded bicliqueis a bicliqueKsuch that

β(

K

)

≤

b. Ab-biclique vertex-coverof a graphGis a set ofb-bounded bicliques ofGsuch that each vertex ofGis contained in one of these bicliques.

Letbbe a fixed positive integer. We consider the following parameterized problem.

b-Biclique Vertex-Cover

Instance:A graphGand a positive integerk Parameter:.

Question:The integerk.

Does there exist ab-biclique vertex-coverSofGsuch that

|

S

| ≤

k?

It is not difficult to see thatb-Biclique Vertex-Coveris in XP as follows. LetG

=

(

V

,

E

)

be a graph withnvertices and k

>

0. We assume w.l.o.g. thatGdoes not contain isolated vertices. We choose, independently, subsetsX1

, . . . ,

Xk

⊆

Vof

size at mostb, and there areO

(

nkb

)

possibilities. For each choiceX1

, . . . ,

Xkwe defineY1

, . . . ,

YkwhereYi

=

T

x∈XiN

(

x

)

.

Then we check in polynomial time if every vertex ofGis in at least one setXiorYi, and if allYiare non-empty (note that

Yi

= ∅

implies

|

Xi

| ≥

2 because we assumeGdoes not have isolated vertices). If both conditions are satisfied, then we have

found ab-biclique vertex-cover of size at mostk. Furthermore, if there exists ab-biclique cover of size at mostkthen one of the guesses will succeed.

Next we will identify the exact parameterized complexity ofb-Biclique Vertex-Cover. InLemma 14we show that theb

-Biclique Vertex-Coverproblem is in W[2]. InLemma 15we show that theb-Biclique Vertex-Coverproblem is W[2]-hard.

These two lemmas together imply the following result.

Theorem 13.The b-Biclique Vertex-Coverproblem is W[2]-complete for every b

≥

1. This also holds if only bipartite graphs are considered.

To show W[2]-membership we use the following parameterized problem known to be W[2]-complete [6].

Dominating Set

Instance:A graphGand a positive integerk.

Parameter:The integerk.

Question:Does there exist a dominating set ofGof size at mostk?

Lemma 14. There is an fpt-reduction from b-Biclique Vertex-CovertoDominating Set.

Proof. Consider an instance

(

G

,

k

)

ofb-Biclique Vertex-Cover. For a setS

⊆

V

(

G

)

letS0

⊆

V

(

G

)

denote the set of common

neighbors of vertices inS, i.e.,S0

=

T

v∈SN

(v)

. Furthermore, letT denote the set of subsetsS

⊆

V

(

G

)

with 1

≤ |

S

| ≤

band

S0

_{6= ∅}

_.

We construct a graphH

=

(

V0

_,

_E0

₎

_{as follows. We letV}0_{consist ofV}

₍

_G

₎

_{, two new verticesz}

_,

_z0_{and a new vertex}

_v

Sfor

everyS

∈

T. We letE0_{consist ofE}

₍

_G

₎

_{together with the edgezz}0_{and all edges}

_v

S

w

for

w

∈

S

∪

S0

∪ {

z

}

,S

∈

T. Note thatH

can be constructed in polynomial time as

|

T

| =

O

(

bnb

)

wheren

= |

V

(

G

)

|

. We show thatGhas ab-biclique vertex-cover of size at mostkif and only ifHhas a dominating set of size at mostk

+

1.

LetSbe ab-biclique vertex-cover ofGand

|

S

| ≤

k. Note thatS

⊆

T. For everyK

∈

Swe choose a vertexxK

∈

V

(

H

)

as

follows. IfKis trivial (i.e.,V

(

K

)

= {

v

}

) then we putxK

=

v

. Otherwise we putxK

=

v

S. EvidentlyD

= {

xK

:

K

∈

S

} ∪ {

z

}

is

Fig. 2.The graphG0

for the instance((Q,C),3)andb=1, whereQ= {q1, . . . ,q5}andC= {C1, . . . ,C6}withC1= {q1},C2= {q2,q4},C3= {q2,q3},

C4= {q1,q5},C5= {q3,q5}andC6= {q4,q5}.

Conversely, letDbe a dominating set ofHwith

|

D

| ≤

k

+

1. We may assume, w.l.o.g., thatz

∈

D(otherwisez0

_∈

_Dand

we can replacez0byz). For everyx

∈

D

\{

z

}

we identify a bicliqueKxofGas follows. Ifx

=

v

Sfor some setS

⊆

V

(

G

)

then we

letKx

=

((

S

,

S0

),

EKx). Otherwise, ifx

∈

V

(

G

)

, then we defineKx

=

((

{

x

}

,

N

(

x

)),

EKx). We letS

= {

Kx

:

x

∈

D

\{

z

} }

. Again it

is easy to verify thatSis ab-biclique vertex cover ofG, and clearly

|

S

| ≤ |

D

| −

1

=

k.

To show W[2]-hardness we use the following parameterized problem known to be W[2]-complete [6].

Hitting Set

Instance:A pair

(

Q

,

C

)

, whereQ

= {

q1

, . . . ,

qm

}

andC

= {

C1

, . . . ,

Cn

}

withCi

⊆

Sfori

=

1

, . . . ,

n, and a positive

integerk.

Parameter:The integerk.

Question:Does there exist a subsetH

⊆

Q with

|

H

| ≤

k, such thatH

∩

Ci

6= ∅

fori

=

1

, . . . ,

n?

Lemma 15. There is an fpt-reduction fromHitting Setto b-Biclique Vertex-Coverfor bipartite graphs.

Proof. LetI

=

((

Q

,

C

),

k

)

be an instance ofHitting Set, whereQ

= {

q1

, . . . ,

qm

}

andC

= {

C1

, . . . ,

Cn

}

. We transformIinto

an instance ofb-Biclique Vertex-Coveras follows. First construct a bipartite graphG

=

((

Q

,

C

),

E

)

by lettingqiCj

∈

E

(

G

)

if and only ifqi

∈

Cj. Now add two new verticeszandz0toG, such thatzis adjacent to everyqiandz0is adjacent tozonly.

Finally, for each vertexCjaddbknew vertices

v

j1

, . . . , v

jbkand add edges such thatN

(v

jd

)

:=

N

(

Cj

)

,d

=

1

, . . . ,

bk. Call the

resulting graphG0_{. An example of a graphG}0_{is given inFig. 2. Clearly,G}0_{is bipartite. LetU}0

_,

_V0_{be the bipartition classes of} V

(

G0

₎

_{, such thatz}

_∈

_V0_{, and consequently,U}0

₌

_Q

_{∪ {}

_z0

_}

_{. We show that}

₍

_Q

_,

_C

₎

_{has a hitting set of size at mostkif and only}

ifG0has ab-biclique vertex-cover of size at mostk

+

1.

LetHbe a hitting set of

(

Q

,

C

)

with

|

H

| ≤

k. DefineKq

:=

NG0

(

q

)

∪ {

q

}

for allq

∈

H. These

|

H

|

stars, together with the

star that consists ofz

,

z0_{and the elements ofQ}

_\

_{H, form ab-biclique vertex-cover ofG}0_{with size}

_|

_H

_{| +}

₁

_≤

_k

₊

_1.

Conversely, suppose thatG0_{has ab-biclique vertex-coverSof size at mostk}

₊

_{1. We may assume, w.l.o.g., thatScontains}

a starK0centered at the vertexz. LetS0

:=

S

\{

K0

}

. For a bicliqueK

=

((

X

,

Y

),

EK

)

∈

S0we may assume, w.l.o.g., that

X

⊆

Q andY

⊆

V0. Then

|

X

| ≤

bor

|

Y

| ≤

b. LetQ0be the union of all vertices that are in a setXof at least one bicliqueK

=

((

X

,

Y

),

EK

)

∈

S0with

|

X

| ≤

b. We claim thatC

=

NG

(

Q0

)

. Suppose to the contrary that there is a vertex

Cj

∈

C

\

NG

(

Q0

)

. Consider the setVj

= {

Cj

, v

j1

, . . . , v

jbk

}

. SinceCj

∈

/

NG

(

Q

0

₎

_{, we haveV}

j

∩

NG0

(

Q0

)

= ∅

by construction of

G0. Thus, for each bicliqueK

=

((

X0

,

Y0

),

EK

)

∈

S0containing an element

v

∈

Vj, it follows that

|

X0

|

>

band

|

Y0

| ≤

b.

However, then

|

S0

_{| ≥}

_k

₊

_{1, since}

_|

_V

j

|

>

bk. This means that

|

S

| = |

S0

| +

1

≥

k

+

2. This is a contradiction. Therefore, we

obtain a setH

⊆

Q that is a hitting set of

(

Q

,

C

)

of size at mostkby including inHprecisely one vertex inQ0

_∩

_{Xfor each} K

=

((

X

,

Y

),

EK

)

∈

S0.

Remark 16. Since the non-parameterizedHitting Setproblem, wherekis just part of the input and not a parameter, is well

known to be NP-hard, and since the reduction in the proof ofLemma 15is in fact a polynomial-time reduction, it follows that the non-parameterizedb-Biclique Vertex-Coverproblem is NP-hard.

5. Final remarks

We have classified the parameterized complexity of the problemsBiclique Cover,Biclique Partition,Biclique Vertex-Cover, andBiclique Vertex-Partition: the first two are fixed-parameter tractable, the latter two are equivalent and not fixed-parameter tractable unless P

=

NP. It would be interesting to improve our algorithm forBiclique CoverandBiclique Partition. In particular, it would be interesting to improve on the 3k_{kernel or to show that, under plausible complexity}

theoretic assumptions, a kernelization to a kernel of size polynomial inkis not possible. Our results for theBiclique Vertex-Coverproblem are negative. It would be interesting to identify special graph classes for which the problem becomes

fixed-parameter tractable, and to determine the complexity ofBiclique Vertex-Coverfor fixedk

=

2.

Acknowledgment

The authors thank Mike Fellows for helpful discussions. The second author’s research was supported by International Science Programme (ISP) of Sweden, under the project ‘‘The Eastern African Universities Mathematics Programme (EAUMP).’’ The third author’s research supported by the EPSRC, project EP/D053633/1. The fourth author’s research supported by the EPSRC, project EP/E001394/1.

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