Approximating the Domatic Numb er

y z x

Uriel Feige Magns M Halldrsson Guy Kortsarz Aravind Srinivasan

Abstract

A set of vertices in a graph is a if every vertex outside the set has a

neighb or in the set The domatic numb er problem is that of partitioning the vertices of a

graph into the maximum numb er of disjoint dominating sets Let n denote the numb er of

vertices the minimum and the maximum degree

We show that every graph has a domatic partition with o ln n dominating

sets and moreover that such a domatic partition can b e found in p olynomial time This

implies a o ln n for domatic numb er since the domatic

numb er is always at most We also show this to b e essentially b est p ossible Namely

extending the approximation hardness of set cover by combining multiprover proto cols with

zeroknowledge techniques we show that for every a ln napproximation implies

O log log n

that NP  DTIME n This makes domatic numb er the rst natural maximiza

tion problem known to the authors that is provably approximable to within p olylogarithmic

factors but no b etter

We also show that every graph has a domatic partition with o ln

dominating sets where the o term go es to zero as increases This can b e turned into

an ecient algorithm that pro duces a domatic partition of ln sets

Intro duction

A dominating set in a graph is a set of vertices such that every vertex in the graph is either in

the set or has a neighb or in the set A domatic partition is a partition of the vertices so that

each part is a dominating set of the graph The domatic number of a graph is the maximum

numb er of dominating sets in a domatic partition of the graph or equivalently the maximum

numb er of disjoint dominating sets

The domatic partition problem is one of the classical NPhard problems It is also one of

the few graph problems in Garey and Johnson whose approximability status on general

graphs has until now b een a blank page with no published upp er or lower b ounds found in a

literature search The purp ose of this pap er is to mend that situation and derive the optimal

approximability within a lower order term

The domatic partition problem arises in various situations of lo cating facilities in a network

Assume that a no de in a network can access only resources lo cated at neighb oring no des or at

Department of Computer Science and Applied Mathematics The Weizmann Institute Rehovot Israel

feigewisdomweizmannacil

y

Department of Computer Science University of Iceland IS Reykjavk Iceland Part of this work was

done while visiting Scho ol of Informatics Kyoto University Japan mmhhiis

z

Department of Computer Science Rutgers University Camden NJ guykcrabrutgersedu

x

Department of Computer Science and Institute for Advanced Computer Studies University of Maryland

College Park MD USA Part of this work was done while at Bell Lab oratories Lucent Technologies

Mountain Avenue Murray Hill NJ USA srincsumdedu

itself Then if there is an essential typ e of resource that must b e accessible from every no de a

hospital a printer a le etc copies of the resource need to b e distributed over a dominating

set of the network If there are several essential typ es of resources each one of them o ccupies a

dominating set If each no de has b ounded capacity there is a limit to the numb er of resources

that can b e supp orted In particular if each no de can only serve a single resource the maximum

numb er of resources supp ortable equals the domatic numb er of the graph We can show

how the general case of larger p ossibly nonuniform capacities can b e reduced to the unit case

We review some elementary facts ab out dominating sets and domatic partitions in light of the

novelty of the problem to many readers Dominating sets satisfy a monotonicity prop erty with

regards to vertex additions if D is a dominating set and D D then D is also a dominating

set This implies that if a graph contains k disjoint dominating sets then its domatic numb er is

at most k those no des not b elonging to any of the k sets can b e added arbitrarily to the sets to

form a prop er partition of the vertex set The domatic numb er can then b e alternatively dened

as the maximum numb er of disjoint dominating sets Every graph G satises D G and

unless G contains an isolated no de D G On the other hand D G where is the

minimum degree the reason b eing that a no de of minimum degree must have some neighb or or

itself in each of the disjoint dominating sets

Fujita has studied several greedy algorithms and shown that their p erformance ratio is

p

n The only other lower b ound on D G given no b etter than for values of up to O

in a recent encyclopdic treatment of domination problems is D G dnn Ge

where n is the numb er of vertices This lower b ound is relevant only in very dense graphs

since it degenerates to D G when G n

A numb er of results are known for sp ecial classes of graphs A graph G is said to b e

domatical ly ful l if D G G the maximum p ossible Determining if a dregular graph

is domatically full is NPcomplete for any d Farb er showed nonconstructively

that strongly chordal graphs are domatically full This class contains the classes of interval

graphs and path graphs Rao and Rangan then gave a linear time algorithm for interval

graphs and Peng and Chang for strongly chordal graphs Farb ers theorem turned out to

b e a sp ecial case of a result of Berge for balanced hyp ergraphs and Kaplan and Shamir

presented a simple algorithm They also showed split graphs and bipartite graphs to b e NP

hard Ecient algorithms are known for partial k trees using generic metho ds Bonucelli

showed that circulararc graphs are NPhard while Marathe et al gave a approximation

algorithm

Let denote the maximum degree of a given graph Our main result is a tight b ound on

the approximability of the domatic numb er problem in general graphs In particular we give

A An algorithm that nds a domatic partition of size o ln n where the o

term go es to zero as n increases

B An algorithm that nds a domatic partition of size at least c ln for some constant c

C A nonconstructive argument showing that the domatic numb er is at least o

ln where the o term go es to zero as increases This shows that the value of

the domatic numb er can b e approximated within a factor of nearly ln

D A b ound on the domatic numb er of random graphs showing that for most graphs the

domatic numb er is at most o ln where the o term go es to zero as

n increases

E A construction showing that for every no p olynomialtime algorithm can approx

imate the domatic numb er problem within a ln n factor unless NP has slightly

log log n

sup erp olynomialtime algorithms NP DTIME n It also yields a o ln

hardness These results hold even for bipartite graphs and split graphs

The o ln napproximation algorithm is a simple randomized assignment though care

is needed not to lose a factor of two in the analysis and is derandomized using the metho d of

conditional probabilities The results B and C ab ove use the Lovsz Lo cal Lemma LLL

as their basic to ol Suitable application of the LLL to our randomized assignment algorithm

ab ove shows that the domatic numb er is at least o ln we then rene this

using a slow partitioning scheme leading to our result that the domatic numb er is at least

o ln The O ln approximation algorithm is a constructive version of the

LLL following an approach of Beck

The hardness construction builds on the pro of of Feige of similar hardness for the set

cover and dominating set problems In fact the construction here generalizes the result of

in that it shows that it is hard to distinguish b etween the following two cases when

the minimum dominating set is large and thus the domatic numb er small or when there

are many small disjoint dominating sets This parallels the situation with the archetypical

minimum partitioning problem where Feige and Kilian showed that it is

hard to distinguish b etween the case when the maximum indep endent set is small and when the

chromatic numb er is small The construction of the current pap er in fact draws additionally

on the zeroknowledge techniques used in

It is instructive to view our results in a larger context that of the study of approximation

algorithms in general It has b een empirically observed and further supp orted by classication of

constraint satisfaction problems that there seem to b e no natural maximization problems

approximable within p olylogarithmic factors but no b etter Our results provide to the b est of

our knowledge the rst maximization problem with such a b ehavior as the domatic numb er is

a maximization problem approximable within logarithmic factors but no b etter

Our algorithmic results give absolute ratios namely b ounds in terms of some basic parame

ters of the graph minimum degree numb er of vertices rather than in terms of the size of the

optimal solution These are in fact the rst nontrivial lower b ounds on the size of an optimal

domatic partition for arbitrary such that ln

D G o

ln

As shown in Section this b ound is b est p ossible up to lower order terms for a large range

of values of n and n

In the past most absolute ratios have b een obtained by fairly simple greedy algorithms

Our algorithms are derandomizations of simple randomized algorithms but their derandomized

versions are not particularly natural and natural greedy algorithms for the problem attain much

worse results It is also interesting that the hardness result gives a gap lo cation at namely

it is equally hard to approximately partition graphs that are domatically full

The rest of the pap er is divided into p ositive results algorithmic and existential on

domatic partitions in Section and hardness results in Section

Approximation algorithms and existential results

This section is devoted to p ositive results for domatic partitions In Section we give an

algorithmic pro of of the b ound D G o ln n and also show that D G

o ln The two usages of o here resp ectively corresp ond to n

and This second result is made algorithmic in Section with a loss in the constant

factor The existential result that D G o ln is then shown in Section

and shown to b e tight on random graphs in Section Section is devoted to a short analysis

of the natural greedy algorithm for domatic partition

Notation Let N v denote the set of neighb ors of a vertex v in the given graph G and let

N v fv g N v Let dv jN v j denote the degree of v and let d v jN v j

dv A partial coloring of G is an arbitrary coloring of an arbitrary subset of the vertices

Given a current partial coloring dene a Bo olean variable A to b e true if there is no vertex

v c

of color c in N v and to b e false otherwise Note that the events A are bad events for

v c

us if A holds for some pair v c then the coloring is not a domatic partition conversely if

v c

none of the events A hold then every vertex v sees every color in N v and we will have

v c

a domatic partition Thus our fo cus will b e on avoiding all of these bad events

Dene to b e the set f g For an event X P X denotes its probability and EX

its exp ectation Finally let e denote the base of the natural logarithm

Logarithmic b ounds

Theorem Any graph admits a polynomialtime constructible domatic partition of size

O log log n log n ln n

Proof Indep endently give each vertex one of lnn ln n colors at random For any

+ +

d v d v

vertexcolor pair v c P A e n ln n Thus summing over all

v c

v c pairs the exp ected total numb er of bad events A is at most ln n Hence the exp ected

v c

numb er of colors that form dominating sets is at least

ln ln n

ln n ln n lnn ln n

The colorclasses that do not form dominating sets can all b e merged into any one color class

that is a dominating set thus we get a domatic partition whose exp ected numb er of sets is at

least as large as the rhs of

This randomized argument can b e derandomized using the metho d of conditional probabil

ities cf Numb er the vertices arbitrarily as v v v and color the vertices in this

n

order never recoloring a vertex as follows Color v arbitrarily Supp ose the rst j vertices

have b een colored with resp ective colors c c c vertex v is colored as follows Let

j j

d v jN v fv v v gj Then the conditional probability of the bad event

j j j n

A is given by

v c

if v j such that v N v and c c

z z z

P A jc c c

v c j

d v

j +1

otherwise

The weight of the current coloring is given by

X X

g c c c P A jc c c

j v c j

c

v V

this is precisely the exp ected numb er of pairs v c pairs for which A will hold after coloring

v c

all vertices given the current coloring c c In each step j we cho ose a color for v

j j

so that the weight of the coloring do es not increase Such a color exists since g c c c is

j

a convex combination of the values fg c c c c c g

j j j

X

g c c c g c c c c

j j j

c

j +1

Then the total numb er of colors that are not dominating sets is at most the weight of the nal

coloring which we ensure is at most the exp ected numb er at the outset or ln n

We now rene this argument using the LLL to get b etter b ounds when n We state

the symmetric simpler version of the LLL

Lemma LLL Let p and let E i k be k events such that P E p for al l i

i i

Suppose there is an integer d such that e p d and each event is independent of al l

but at most d other events More precisely for each E there is a set T of at least k d

i i

other events E such that the conditional probability of E given any Boolean combination of the

j i

V

E events in T equals the unconditional probability of E Then P

i i i

i

Indep endently color each vertex randomly with one of b ln c colors

+

d v

For each v c pair P A We note that each event A is

v c v c

indep endent of all but at most dv dv other such events since vertices of

distance at least from v are completely irrelevant for v More precisely in the notation of

Lemma we can take T to b e the set of all events of the form w c where w is a vertex at

v c

a distance of at least from v and where c is any color from conditioning on any Bo olean

combination of the events in T do es not inuence the colors chosen by vertices in N v The

v c

following lemma now directly follows from the LLL using the fact that and dv

we set d and p in using the LLL

Lemma Any graph admits a domatic partition of size o ln where the o

term tends to zero as

In Section we will rene the ab ove approach by conducting a twostage partitioning that

attains the tight value of o instead of the value o of Lemma However the ab ove

direct approach will help us develop a simple algorithmic version of Lemma in Section It

also motivates the reason for developing the approach of Section

Remark It is interesting to note that the in the b ound of Lemma cannot b e replaced by

d the average degree Consider for example the with vertices nor by

on the left side and vertices on the right side with each vertex on the right side connected

to a particular subset of vertices in the left side The domatic numb er of this graph is two

Indeed say that there are disjoint dominating sets One of these sets S contains at least

vertices on the left side There exists a vertex v on the right side all of whose neighb ors are in

S Hence the two remaining sets must b oth contain the vertex v a contradiction

A parameter that is intermediate b etween average and maximum degree is the inductiveness

G max H It is most notable for giving a tighter upp er b ound on the chromatic

H G

numb er than the maximum degree as G G G The ab ove construction

shows however that cannot replace in the b ound of Lemma on the domatic numb er

O log approximation algorithm

We use an algorithmic version of the LLL due to Beck that derandomizes the probabilistic

argument with some loss in the constants We may assume without loss of generality that G is

connected since we can treat the connected comp onents separately Our algorithm has three

phases assigning colors to successively larger fractions of the vertices After the rst phase each

vertex is either fully satised seeing all colors within its neighb orho o d or has at most onethird

of its neighb ors colored We show that the subgraph induced by no des that are still active

ie either unsatised or not yet colored consists with high probability of only small connected

comp onents of O log n vertices each After the second phase more vertices are colored

with at most twothirds of the neighb ors of yetunsatised vertices b eing colored The connected

comp onents induced by active vertices are now of only O log log n size Then dep ending on

the value of we can either solve each comp onent by exhaustive search or apply Theorem

to obtain a full coloring where each vertex is satised

The algorithm

The rst phase pro ceeds as follows Given a coloring of some of the vertices call a vertex v

dangerous i

at least neighb ors of v have b een colored and

not all the colors app ear in the neighb orho o d of v

Let c ln for a suitably large constant c Order the vertices arbitrarily as v v v

n

and pro cess them in this order When pro cessing v we do the following If v or one of its

i i

neighb ors is dangerous now we freeze v otherwise we assign it one of colors indep endently

i

at random

When the pro cess ends some vertices are colored and some are frozen and some are dan

gerous and some are not The vertices that are not dangerous b elong to one of two categories

Good A go o d vertex sees all colors in its neighb orho o d

Neutral A neutral vertex v do es not see all colors in its neighb orho o d but is not dangerous

This can happ en only if more than of v s neighb ors were frozen

Thus we have two orthogonal partitions coloredfrozen and go o dneutraldangerous

Vertices that are b oth go o d and colored do not need to b e considered further in the later

phases Call the other vertices saved ie those that are dangerous frozen or neutral As in

we are interested in the maximum size of a connected comp onent of the subgraph induced by

saved vertices as this b ounds the size of the indep endent subproblems in the next phase We

show in Section that with probability at least the largest connected comp onent in the

saved graph has size O log n let us assume that this size b ound holds

Phase two of the algorithm is run separately on each connected comp onent induced by the

saved vertices Note that each dangerous or neutral vertex v has at least dv frozen

ie uncolored neighb ors in its connected comp onent Phase two diers from phase one in

that some of the vertices are colored b efore we b egin We leave these colors untouched b ecause

they may b e useful for the vertices that were not saved We only color the frozen vertices and

again dene the notion of phasetwo dangerous frozen and neutral vertices Since the numb er of

vertices we are dealing with now in any connected comp onent is O log n we have essentially

replaced n by O log n Thus analysis similar to that of phase one shows that the connected

comp onents of the newly saved vertices has size at most N O log log log n with

probability at least

In phase three each dangerousneutral vertex has at least dv frozen neighb ors

in its connected comp onent If log log n we have a domatic partition of the frozen vertices

to roughly ln N parts via Theorem As ln N O log this is go o d enough If

log log n then we can nd a domatic partition of size ln whose existence is guaranteed

by Theorem using exhaustive search Since log log n this only takes time

O ln O log log n

N log log n pol y n

This completes the description of the algorithm

Analysis of the algorithm

We now show that with probability at least the largest connected comp onent in the saved

graph in the rst phase has size O log n This will yield a pro of of correctness of our

algorithm

Let X u b e the indicator random variable for vertex u b ecoming dangerous and let q

Lemma Let U fu u u g be any set of vertices with pairwise distance at least

k

k

Then PrX u X u X u q

k

Proof If a vertex v is a neighb or of the set U then it has a unique neighb or in U since the

elements of U have pairwise distance at least Let a a a a b e any sequence of k

k

colors and let S b e the random variable denoting the set of the rst i nonfrozen neighb ors of

i

U Let D a b e the event that for all j k all neighb ors of u in S avoid color

i j i

a Thus D a is the event that even after pro cessing the rst i nonfrozen neighb ors of U

j i

each u in U was missing a particular color a We may assume without loss of generality that

j j

sets S exist for every i up to k b ecause otherwise some vertex u in U do es not have

i

nonfrozen neighb ors and then u cannot b e dangerous Hence D a is dened for all i k

i

We have PrD a Now for every i k PrD a PrD a b ecause the

i i

color of the i st nonfrozen neighb or of U is chosen at random indep endent of the previous

i

colors and indep endent of which vertex it happ ens to b e Hence PrD a so

i

k k k

Pra D a q

k

If all the u are dangerous then there is some a for which D a is true this completes

j

k

the pro of

To prove that the largest connected comp onent in the saved graph is small enough with

reasonable probability we now show that with reasonable probability the maximum numb er

of vertices in a spanning tree of such a comp onent is small This is done as follows By

a standard argument a large connected comp onent contains many vertices with a particular

minimum pairwise distance We rst prove that the numb er of vertices with large pairwise

mutual distance which are all saved is small This indirectly b ounds the maximum numb er of

vertices in a connected comp onent as a function of which is enough for our purp oses

A set of vertices is said to b e separated if it is of mutual distance at least A separated

set of k vertices is said to b e a bad k set if additionally it b ecomes connected if we connect all

vertices of distance exactly As shown next the numb er of such sets in G is at most

k

n

Consider a spanning tree on the set where vertices of distance exactly are connected The

k

numb er of distinct shap es of trees on k vertices is at most Namely such a tree can b e

uniquely represented by ordering the vertices in a lexicographic breadthrst order and attaching

two ag bits to each nonro ot vertex whether it has the same parent as the previous vertex in

the order and whether it has a child or not For each shap e of a tree there are n p ossibilities

of cho osing the ro ot and thereafter at most p ossibilities of cho osing each new vertex since

we already chose its parent in the tree

Let Y u b e the indicator random variable for vertex u b ecoming saved To complete our

argument that no connected comp onent of the saved vertices is large we will show

Lemma For any separated set of vertices v v v

k

k

PrY v Y v Y v q

k

where q is as dened prior to Lemma

Proof The following denition will b e useful for this pro of

P P

X

X w

v N u w N v

Z u X u X v

du

v N u

If vertex u is saved then we have one of three cases i u is dangerous and thus X u

P

X v or iii u is neutral so at least or ii u has a dangerous neighb or and so

v N u

du of its neighb ors are frozen and by the preceding argument it holds for each frozen

P

neighb or v of u that X w Therefore we have the simple but useful observation

w N v

that if Y u then Z u

By Markovs inequality

k k

Y Y

PrY v Y v Y v Pr Z v E Z v

i i

k

i i

Now b ecause Z is linear in the X and using the linearity of exp ectation we can expand

Q

Z v as a linear combination of the terms PrX w X w X w The E

i

k

i

main observation is that since the v have pairwise distance at least the w have pairwise

i i

k

distance at least Thus by Lemma any such term has probability at most q Thus we get

k

Y

du q

k

q Z v q q E

i

du

i

by rst replacing the probability of intersection of events by the pro duct of probabilities and

then unfolding and reversing the ab ove expansion

Consider now a connected comp onent in the subgraph of G of saved vertices If its size is k

or more then it must contain a bad k set which is obtained by rep eating the pro cedure of putting

a vertex in the bad k set and removing all vertices of distance at most Setting the constant c

in the expression c ln large enough we get that q Set k log n log

and recall and Lemma We get that the probability of existence of a connected comp onent

k k

of the saved vertices with cardinality at least k is at most q n

Finally the ab ove Las Vegas algorithm can b e derandomized by the approach of pessimistic

estimators which is a generalization of the metho d of conditional probabilities Briey we

pro ceed as follows As b efore we pro cess the vertices onebyone Supp ose it is currently the

turn of vertex v If v is frozen we skip over it Otherwise we deterministically cho ose a color

for it that minimizes the probability of emergence of a large connected comp onent of the saved

vertices using our b ounds derived ab ove Since k log n log we see from that the

numb er of p ossible bad k sets to b e considered in our analysis ab ove is b ounded by a p olynomial

in n Hence we can write down a p essimistic estimator and cho ose in deterministic p olynomial

time a color for vertex v that minimizes the p essimistic estimator

Getting the right constant

Our next result improves the value o of Lemma to the existentially b estp ossible value

of o

Theorem There is a constant a such that for large enough for every graph G with

c D G b

ln a ln ln

We remark that if the theorem holds by setting a large enough For the sp ecial case

of the value of D G is wellknown via a simple case analysis there is also a lineartime

algorithm for the domatic partition problem if

Proof For the rest of Section any o term will denote a function of alone that go es to

zero as increases We prove the theorem for a b eing any constant greater than for all large

enough this choice of a can b e further improved but we do not attempt this optimization

here

Prepro cessing We prepro cess the graph as follows As long as there is an edge that has b oth

endp oints with degree more than remove such an edge from the graph At the end of this

pro cess the minimum degree remains at and the maximum degree is at most We will

now show a lower b ound on the domatic numb er of this prepro cessed version which clearly will

yield the same lower b ound on the domatic numb er of the given graph This is b ecause the

prepro cessing only removes edges from the given graph The useful prop erty that now holds is

that for each vertex u either du or all neighb ors v of u have dv

There are two cases the rst one b eing simpler

Case I ln We assume that ln a ln ln since the theorem is trivially true

otherwise Dene

ln a ln ln

Color each vertex with a random color from indep endent of all other vertices We will now

V

use the LLL to show that P A in the same way as we did for Lemma For each

uc

uc

u c we have

+

d u

a +

lnln d u a

e ln PrA

uc

Let N u denote the set of vertices at a distance of or from u in G As in our pro of

with v N u so it dep ends on at of Lemma each event A dep ends only on events A

uc

v c

most jN uj other such events Our prepro cessing step ab ove helps b ound jN uj for all u

If du then jN uj If du our prepro cessing ensures that dv

for all neighb ors v of u so jN uj Thus jN uj ln for

all u since ln So each A dep ends on at most O ln other such events Recalling

uc

V

A as required since a the LLL and we see that P

uc

uc

Case I I ln Let ln dene b c and bln bln ln ln c

where b is any constant larger than We will show the existence of a domatic partition of

size ie a coloring of V using colors in such a way that for every vertex u there

is at least one vertex of each color in N u Recall that ln By cho osing b for

instance we can ensure that ln ln ln for all large enough It will b e

convenient to view the colors as elements of We will apply a twostage coloring the

rst coloring determines the rst comp onents of the vertexcolors and the second coloring is for

the second comp onents We can view the rst coloring as a coarse partition which the second

coloring turns into a ne partition The primary purp ose of the rst coloring is to reduce the

dep endencies suciently for our analysis of the second coloring

The rst partitioning is as follows Color each vertex with a random color from inde

p endent of all other vertices For each vertex u and each color c dene X to b e the subset of

uc

N u that receives color c We have EjX j d u Let B b e the bad event that

uc

uc

j d u j d u A Cherno b ound shows that j jX

uc

PrB exp o d u exp o ln

uc

for Once again B is indep endent of any Bo olean combination of events of the form B

uc

v c

vertices v at a distance of or more from u Thus each B dep ends on o other such

uc

V

B events Recalling the LLL shows that Pr

uc

uc

Fix a coloring V which avoids all the events B Cho ose a random color u

uc

for each u indep endent of all other vertices the nal color of u is the pair u u

Let B b e the bad event that there is no vertex of color c c in N u We now use the

uc c

1 2

LLL to show that all these bad events can b e avoided with p ositive probability

fv N u u cg and dene For each vertex u and each c let N

uc

j Since avoids all the events B we have jN d

uc

uc uc

u c d u d d u

uc

Fix an event B First

uc c

1 2

+

d

uc

1

+

d u b b

1 2

PrB e ln O ln

uc c

1 2

the rst inequality here follows from Next which other events do es B dep end on

uc c

1 2

S

Given S V let N S N v Note that B simply says that all elements of

uc c

v S 1 2

N got a value dierent from c Thus we can check that B only dep ends on the

uc c

uc

1 2

1

events in

S u c c fB v N N c g and c

v c c

uc

1

1 2

More precisely we claim that B is indep endent of any Bo olean function of the events lying

uc c

1 2

outside S u c c this can b e veried by seeing that

N N

uc

1

v c

1

B S uc c c v c

1 2

2 v c c 1

1 2

We now b ound jS u c c j in order to apply the LLL once again our prepro cessing step

will b e of help If du then jN N j jN j this is at most O by If

uc uc

1 1

du our prepro cessing ensures that jN N j jN j so from we again get that

uc uc

1 1

jN N j O Thus jS u c c j O O ln We can now apply

uc

1

dep ends on at the LLL using and the facts that i b and ii each bad event B

uc c

1 2

most jS u c c j others This completes the pro of

To see why our twostage coloring helps note that the dep endence jS u c c j in the

o o

second coloring ab ove is only as compared to the dep endence of that we could

get in the directcoloring approach underlying Lemma The constraint v N N in

uc

1

o o

c saves another factor of That saves us a factor of and the constraint c

the rststage coloring eliminates many dep endencies in this fashion is the main idea motivating

this approach

It is an op en question if a domatic partition of the size guaranteed by Theorem can b e

found in p olynomial time

Greedy algorithm

One natural approach to the domatic partition problem is to try to greedily cho ose small dom

inating sets The greedy algorithm iteratively pulls out dominating sets from the graph until

the remainder is no longer dominating The dominating sets are found by a standard O log n

approximate greedy algorithm

Lemma Suppose D G nk Then the greedy algorithm nds a domatic partition of

nk log n sets

Proof We count how many disjoint dominating sets our algorithm nds b efore the set of vertices

in them intersects at least half of the nk vertexdisjoint dominating sets in the graph During

this p erio d there are at least nk disjoint dominating sets thus in each step there exists a

dominating set of size at most k and we nd one of size at most k ln n Hence in each step

a vertex from at most k ln n dierent dominating sets is removed It then requires at least

nk k ln n steps to halve the original numb er of dominating sets in the graph

p

Since D G n the approximation ratio is maximized when k n ln n

p

Corollary The performance ratio of the greedy domatic partition algorithm is O n ln n

Fujita has shown examples where the p erformance of this and some other greedy algo

p

rithms is n

The domatic numb er of random graphs

We now show that the b ound is tight for a large range of values of n by studying

D G for suitable random graphs G Supp ose G is drawn from the random graph mo del Gn p

ie we take n lab eled vertices and put an edge with probability p indep endently b etween each

pair of vertices We will show that with probability o D G o G ln G

Throughout this section the o and notation refers to n getting large

Cho ose any p pn such that np ln n and p o It is easy to check via a

Cherno b ound that with probability o b oth and will lie in the range onp

for our random graph G Fix any constant and let s b lnnppc We will show

that with probability o any dominating set in G will have size more than s Thus with

probability o we will have D G ns completing the pro of For any given subset

of the vertices S with jS j s

s ns

PrS is a dominating set p

psp ns

e

p ns

np

1(p)

nsnp

e

(p)

spnp

e

o(1)

spnp

e

Thus the probability of existence of a dominating set of size s is at most

n o(1) o(1) o(1)

spnp s spnp s s ln npnp

e n s e e s e o

s

o

the b ound s ln n op np follows from the denition of s and from the fact that

np ln n

Hardness of approximating the domatic numb er

We say that a problem is hard to approximate within ratio if having a p olynomial time ran

domized approximation algorithm for it would violate some standard hardness assumption

such as P NP The hardness assumption that we use in this pap er is that NP do es not have

O log log n

randomized algorithms that run in time n for brevity we shall just use the term hard

to approximate

We shall prove the following theorem

Theorem For every xed it is hard to approximate the domatic number within a ratio

of ln jV j

For this purp ose it is helpful to work with a related but dierent problem

Denition A onesided dominating set in a bipartite graph GV V E is a set of vertices

U V such that for every v V there is some u U with u v E Here it is assumed

that the intended bipartition V V is given explicitly as part of the input and that every vertex

in V has some neighbor in V Observe that this problem is merely a reformulation of the wel l

known setcover problem

The onesided domatic numb er of a bipartite graph is the maximum number of mutual ly

disjoint onesided dominating sets that the graph contains

Observe that the dominating set and domatic numb er problems have a relation similar to the one

of the coloring versus maximum indep endent problem A coloring is a packing of indep endent

sets while a domatic partition is a packing of dominating sets

A related problem is the set cover problem where a collection of subsets S of a base set U is

given and we are to nd a minimum cardinality sub collection that contains all elements of U

The set cover packing number is then the maximum numb er of mutually disjoint set covers It

is wellknown that minimum dominating set minimum onesided dominating set and minimum

set cover are strongly related and that ln n is the b est approximation ratio for all of them within

lower order terms details in Section The onesided domatic numb er problem can b e shown

to b e equivalent to the set cover packing problem in terms of optimization We are not aware

of a similar relationship b etween onesided domatic numb er and domatic numb er However we

can give a reduction that yields the necessary result

Prop osition Let c and consider bipartite graphs GV V E with jV j large enough

c c

eg jV j jV j jV j and the fol lowing promise for some c and for r

and q satisfying r q jV j ln jV j r and q may depend on the size of G either

The size of the smal lest onesided dominating set in G is at least r or

The onesided domatic number of G is at least q

Note that the two cases for the promise cannot both hold If it is hard to distinguish which

of the two cases holds then it is hard to approximate the domatic number within a ratio of

c ln jV j

The pro of of Prop osition is given in Section The main result of the section is the

following theorem proved in Section

Theorem For every and every integer c the two cases of Proposition cannot

be distinguished in random polynomial time unless NP has randomized algorithms that run

O log log n

in time n

Theorem now follows from Theorem and Prop osition

Remark Theorem implies a ln hardness of approximation result for n for some

which is close to To obtain ln hardness of approximation results when is much

smaller compared to n simply make many disjoint copies of the graph increasing n without

changing or the domatic numb er

Overview and intuition

Before presenting the pro of of Theorem let us provide some background on proving hardness

of approximation results in general and how hardness of approximation results were proved for

problems related to domatic numb er

A convenient starting p oint for proving hardness of approximation results is the problem of

Max SAT The input to this problem is a CNF formula and the desired output is an assignment

to the variables that satises as many clauses as p ossible The well known PCP theorem of

implies or in fact is equivalent to the following gap for some it is NPhard to distinguish

b etween CNF formulas that are satisable which we call yes instances and CNF formulas

in which every assignment satises at most a fraction of the clauses which we call no

instances This hardness result can b e extended to a restricted version of Max SAT in which

the input CNF formula has the prop erty that each variable app ears in exactly clauses The

choice of is arbitrary here Any other constant greater than would do as well We call this

restricted version Max SAT

As noted ab ove the set cover problem is strongly related to the dominating set problem

which in turn is related to the domatic numb er problem Moreover the onesided domatic

numb er problem is equivalent to the set cover packing problem Hence our plan for proving

Theorem is to take known results regarding the hardness of approximation of set cover and

mo dify their pro of so that it shows hardness of approximation for the set cover packing problem

as well and hence also for onesided domatic numb er To see more explicitly what needs to b e

done let us rst review at a very high level the known result that set cover and onesided

dominating set is hard to approximate within a factor of ln n

The pro of in reduces instances of Max SAT to instances of onesided dominating set

The reduction is slightly sup er p olynomial instances of size n are mapp ed to instances of size

O log log n

n and is a gap reduction in the following sense yes instances give bipartite graphs

that have small onesided dominating sets and no instances give graphs all of whose onesided

dominating sets are much larger To prove hardness of approximation for onesided domatic

numb er the requirement for no instances do es not change as it already implies a small one

sided domatic numb er However we would like yes instances to give bipartite graphs that have

not just one small onesided dominating set but many disjoint small onesided dominating sets

and hence a large onesided domatic numb er

To achieve this extra prop erty we invoke an idea used in when proving hardness of

approximation for chromatic numb er In our context it suces to change the starting p oint

of the reduction from the problem Max SAT to the problem Maxcolorability This

is the problem of coloring a regular graph with colors so as to maximize the numb er of

edges legally colored see Section It also has a gap similar to that of Max SAT The

imp ortant prop erty of Max colorability is that yes instances of it necessarily have many

disjoint solutions When these instances are reduced to instances of onesided dominating set

the resulting bipartite graph has many disjoint small onesided dominating sets This implies a

large onesided domatic numb er as required by Theorem

Hence to complete the pro of of Theorem we need to accomplish three things

Intro duce the problem of Max colorability and its prop erties This is done in Sec

tion

Give the reduction from Max colorability to onesided dominating set This reduction

closely follows the reduction of from Max SAT to set cover except for one small

extra step vertices on one side of the bipartite graph are duplicated many times an

op eration that was not p erformed in the reduction to set cover This reduction is fairly

complicated but essentially all complications come from the reduction in

Prove the prop erties of the reduction This has two parts One is to show that yes in

stances of Max colorability are reduced to bipartite graphs with high onesided domatic

numb er which is done in Lemma The other is to show that no instances are reduced

to bipartite graphs with only large onesided dominating sets This part is not proved

in this pap er b ecause the pro of in that no instances of Max SAT give instances

with a large set cover extends virtually without change to our adaptation of the reduction

of

App endix B reviews some of the ingredients of the reduction used in from Max SAT

to set cover and explains some mo dications used in our context Some of these ingredients are

only used in the analysis of what happ ens on no instances and hence do not come into any of

the pro ofs in this pap er They are included in the overview only so as to give some indication

on what led to the construction of the reduction For more details see

Domination and onesided domination

The three problems minimum dominating set onesided dominating set and set cover are

equivalent in the following sense see eg

Prop osition There is a polynomialtime reduction between any of the three problems

dominating set onesided dominating set and set cover that preserves the value of the minimum

solution

Proof To reduce dominating set to set cover let V the set of vertices b ecome U the ground

set and let the collection S include the sets N v for every v V To reduce set cover to

onesided dominating set construct a bipartite graph GV V E in which V is the ground set

U and the vertices of V each represent a set in S Put an edge u v E if v V corresp onds

to an item that is contained in the set that corresp onds to the vertex u V Finally to

reduce onesided dominating set to dominating set make a out of all vertices of V and

S

V it is not hard to see that these reductions reserve feasibility of solutions and let V V

in addition it is not hard to see that the size of the minimum dominating set is preserved

For set cover let n jU j It is known that minimum set cover can b e approximated within

a ratio of ln n and that for every xed it is hard to approximate it within a

ratio of ln n By prop osition this implies a similar result for onesided dominating

set with n jV j For dominating set it is natural to take n jV j Then the approximation

ratio of ln n trivially applies but in order to transfer the ln n hardness result from one

sided dominating set to dominating set using the reduction of Prop osition we also need that

ln jV j ln jV j which holds whenever jV j jV j It turns out that this is indeed the case in

the construction of Hence it is known that up to low order terms ln n is the b est p ossible

approximation ratio for all three problems

A pro of similar to that of Prop osition shows that there is a p olynomialtime reduction

from domatic numb er whether onesided or not to set cover packing numb er that preserves

the value of the maximum solution Likewise there is a p olynomialtime reduction from set

cover packing numb er to onesided domatic numb er that preserves the value of the maximum

solution However we are not aware of such a reduction from onesided domatic numb er to

domatic numb er Instead we use the following prop osition

Prop osition For every integer k where k may be an arbitrary function bounded by a

polynomial in the size of the input there is a polynomial time transformation mapping a bipartite

graph V V E to a graph V E with the fol lowing properties

jV j jV j k jV j and

For the original graph let q and r respectively denote the onesided domatic number and the

minimum cardinality of a onesided dominating set hence q jV jr Let p denote the domatic

number of the new graph Then k q p minjV jr k jV jr

Proof Let G V V E b e a bipartite graph for which we are interested in computing the

S

k

i

where for V onesided domatic numb er Construct a graph G V E as follows V

i

S

i

V which gives jV j jV j k jV j For every v V is a copy of V V V every i V

i

place an edge u v E i there is an edge u v E In addition all i k and u V

form a clique vertices of V

Observe that every onesided dominating set in G corresp onds in a natural way to k mutually

i

Hence the optimal solution for G can disjoint dominating sets in G one on each copy of V

b e copied k times on G giving p k q If in addition V has no isolated vertices then V can

serve as one more dominating set disjoint from all the others giving p k q

To upp er b ound p let D D b e a maximum cardinality collection of disjoint dominating

p

sets in G Similar to the pro of of Prop osition we can see that we can change maximum

As of no larger size fully contained in V dominating set D to maximum dominating set D

j

j

j r for all j Hence p jV jr must dominate V we get that jD j jD D

j

j j

We now turn to proving the second part of the upp er b ound namely p k jV jr

S T

is a dominating set This gives a tighter b ound when jV j k jV j Observe that D D V

i j

So supp ose we pair up any bpc of the D s into bpc pairs each of which contained in V

i

forms a dominating set of size at least r Then r bpc k jV j so p k jV jr

We are now ready to prove Prop osition

Proof Perform the reduction of Prop osition with k jV j If the rst case holds then the

domatic numb er of G is at most jV jr If the second case holds then the domatic numb er

is at least q jV j The ratio b etween these two values is q jV jjV jr q r jV jjV j We use

jV jjV j jV j and q r jV j ln jV j to obtain that the ratio is at least

c c

jV j ln jV j Using jV j jV j and jV j we obtain that ln jV j c ln jV j

Now Prop osition follows assuming that jV j c

Remark The graphs G formed in the ab ove reduction are split graphs which are graphs

whose vertexset can b e partitioned into a clique and an indep endent set These graphs are

b oth chordal and complements of chordal graphs where a graph is chordal if it contains no cycle

with four or more vertices as an induced subgraph Thus a o ln n hardness for domatic

numb er holds also for split and thus chordal and cochordal graphs Furthermore one can get

a similar result for bipartite graphs by the following mo dication Instead of making a clique

out of V we add vertices and connect them to every no de in V This aects the domatic

numb er by at most and the approximability hardness follows

The problem Max colorability

We now intro duce the NPlanguage that serves as the basis to our reduction

Denition Max colorability is the problem of coloring the vertices of a graph with three

colors so as to maximize the number of legal ly colored edges edges whose endpoints are colored

dierently

An NPwitness for colorability is an assignment of colors to the vertices such that each

edge is legally colored The witness can b e checked in a probabilistic sense by sampling an edge

at random and checking whether the colors of its two endp oints disagree The actual names of

the colors of the vertices play no role b ecause the names of the three colors can b e arbitrarily

p ermuted without changing the legality of the coloring Hence every NPwitness gives rise to

six dierent witnesses dep ending on the p ermutation used for the names of the colors and

cycling over the six witnesses every edge gets its six legal colorings In our reductions to one

sided dominating set we shall use this structure of the set of witnesses for colorability in order

to show that if the resulting bipartite graph has a small onesided dominating set then in fact it

has many disjoint onesided dominating sets We note that the same structure was used in

to construct a zero knowledge pro of system for NP

In order to prove a hardness result for Max colorability we rely on the hardness of Max

colorability and the use of expanders Call a graph H with h vertices a expander if for

any subset S of the vertices of H with jS j h the numb er of edges leaving S is at least jS j

For some constant and for any there is a h such that for all h h there is an

explicitly constructible expander with maximum degree at most and with the numb er

of vertices lying in the range h h Note that a expander is necessarily also

a expander In particular this implies that for any given integer h we can construct

a expander with maximum degree at most and with the numb er of vertices b eing

some F h that satises h F h h for some absolute constant We can pro ceed

as follows If h h where h is a suciently large constant just construct any connected

hvertex graph of maximum degree If h h construct a expander with maximum

degree at most and with the numb er of vertices lying in the range h h using

We will also need the following theorem of

Theorem For some explicit constant it is NPhard to distinguish between

graphs that have a legal coloring that colors al l edges legal ly and graphs for which every

coloring legal ly colors at most a fraction of the edges

Prop osition For some explicit constant it is NPhard to distinguish between

regular graphs that have a legal coloring and regular graphs for which every coloring legal ly

colors at most a fraction of the edges

Proof Given a graph GV E we show that it can b e mo died in p olynomial time to a graph

G V E such that i G is regular ii jE j O jE j iii G is legally colorable i G is

and iv For some constant c and every k jE j every coloring of G that leaves k

edges illegally colored can b e transformed in p olynomial time to a coloring of G that leaves at

most ck illegally colored edges We can then invoke Theorem

First we may assume that every vertex in G has degree at least three Indeed supp ose we

rep eatedly remove any vertex of degree at most until the remaining graph G has minimum

degree at least It is easy to see that given any coloring of the vertices of G we can add

back the deleted vertices of G and color these addedback vertices in such a way that all the

edges incident with them are legally colored So we assume that G has minimum degree at least

Assume rst that all vertices in G have degree at most Then we replace each vertex v

1

by a cluster of dv vertices where each vertex handles one outgoing edge That is if v u

1

Recall that d(v ) denotes the degree of vertex v v1

v2

Figure Equality gadget

is an edge in G then the vertex representing u in v s cluster is made adjacent to the vertex

representing v in us cluster These dv vertices are connected in a cycle by equality gadgets

shown in Fig These equality gadgets are subgraphs that contain twelve vertices Two of the

vertices are sp ecial and have degree two and all the rest have degree ve The equality gadget

has the prop erty that it is legally colorable i the sp ecial vertices receive the same color The

two cluster vertices serve as the sp ecial vertices As each cluster vertex participates in two

gadgets and also has one outgoing edge its degree is Hence G is regular The numb er of

P

dv jE j and hence jE j jE j Every legal coloring of G edges added is at most

v

colors all cluster vertices with the same color and hence naturally gives a legal coloring of G

Similarly a legal coloring of G can b e extended to a legal coloring of G Any coloring of G

in which cluster vertices receive dierent colors causes at least two edges of G to b e miscolored

Coloring the vertex corresp onding to the cluster with an arbitrary color in G causes at most

dv edges to b e miscolored Hence the numb er of illegally colored edges in G is smaller

than that of G by a factor of at most

If G has a vertex of degree more than we create a new graph G as follows Let F

and b e as in our discussion on expanders ab ove Similarly as G is the same as the

G constructed in the previous paragraph except that for each vertex v we create a

expander with F dv vertices and maximum degree at most instead of a cycle with dv

vertices and do the ab oveseen op eration of replacing the expander edges by equality gadgets

Of the F dv vertices dv are real and represent the neighb ors of v and F dv dv

are dummy vertices As in the previous paragraph if v u is an edge in G then the vertex

representing u in v s cluster is made adjacent to the vertex representing v in us cluster Note

P

dv O jE j edges Once that G has maximum degree at most and has O

v

again a coloring of G can b e eciently transformed into one for G in which the numb er of

miscolored edges remains the same Conversely supp ose we have a coloring of G Let

X b e the numb er of original edges of G that are miscolored by and let Y b e the numb er

of edges in the equality gadgets miscolored by We will pro duce the following coloring of

G and then analyze For each vertex v cho ose a largest sub cluster C v of vertices from its

cluster of F dv vertices that receive the same color in and dene v to b e the color

assigned to the vertices in C v by Call an edge u v of G bad if it was prop erly colored

by in G and was miscolored by in G Let X b e the numb er of bad edges Then the

numb er of edges miscolored by is at most X X An edge u v is bad only if u C v or

P

xv v C u holds Thus letting xv F dv jC v j F dv we see that X

v

Since xv F dv and recalling the prop erty of expansion we can check that

the numb er of edges in all the equality gadgets of v that are miscolored in is xv Hence

Y X Armed with this prop erty and the facts that G has O jE j edges and maximum

degree we transform G into G as describ ed in the previous paragraph

Preliminaries

Our reduction closely follows that of The main dierences are as follows i The outcome

of the reduction is onesided dominating set rather than set cover which is the same thing

termed dierently ii The starting p oint of the reduction is Max colorability rather than

Max SAT As mentioned earlier the purp ose of this change is to have the reduction apply

to onesided domatic numb er rather than just onesided dominating set iii Every vertex in

l

the V side of the bipartite graph will b e duplicated times where l is a parameter of the

reduction This is a technical condition that seems to b e required when we reason ab out the

onesided domatic numb er

To describ e the reduction we recall two notions used in

Denition A k l Hadamard co de is a set of k binary words of length l where every code

word has Hamming weight l and the Hamming distance between every two codewords is l

There is a simple construction of Hadamard co des when l is a p ower of and k l see

eg

Denition A partition system B m L k d has the fol lowing properties

There is a ground set B of m points

There is a col lection of L distinct partitions p p

L

For i L partition p is a col lection of k disjoint subsets whose union is B

i

Any cover of the m points by subsets that appear in pairwise dierent partitions requires at

least d subsets

The following lemma is proved in

Lemma For every c and m suciently large there is a partition system B m L k d

whose parameters satisfy the fol lowing

c

L log m

ln m

k can be chosen arbitrarily as long as k

ln ln m

d f k k ln m where f k as k

Moreover a random col lection of L partitions into k subsets with parameters chosen as above

gives with high probability a partition system with f k k

The randomized construction can b e replaced by a deterministic construction with a some

what larger value for f k using techniques develop ed in

The construction

The input to the reduction is a regular graph GV E for which we want to determine whether

it is legally colorable or whether every coloring of its vertices legally colors at most jE j

edges As noted in Prop osition for some explicit this problem is NPhard The

reduction uses the following parameters which are chosen so that k l Hadamard co des exist

and Lemma holds

k l c log log jV j for some suciently large constant c We assume that l is a p ower

of

l l

L and m jV j

The output of the reduction is a bipartite graph G V V E when we say color b elow

l

we refer to a colorset of colors The left hand side vertex set V is comp osed of jE j clusters

of vertices where each cluster contains m vertices Each left hand side cluster is lab eled by a

sequence of l edges in G and a sequence of l bits for each i the ith bit in the bitsequence denotes

one endp oint vertex of the ith edge in the edgesequence Hence this sequence of bits can also

b e viewed as a sequence of vertices The right hand side vertex set V is comp osed of k disjoint

rays and each ray is lab eled by a co deword of the k l Hadamard co de Each ray is comp osed

l l l

of jV j jE j clusters where each cluster contains vertices Each right hand side cluster

is lab eled by a sequence of l vertices in G and a sequence of l edges in G Equivalently

we may merge these two sequences to one sequence of length l where the co deword of the ray

containing the cluster is used as a selector function sp ecifying the order in which vertices and

edges are merged a vertex when the corresp onding bit in the co deword is and an edge when

the corresp onding bit in the co deword is This is called the merged lab el of a right hand side

cluster Individual vertices of right hand side clusters are further lab eled by a sequence of l

colors ie a ternary sequence a sequence of l pairs of distinct colors ie a sequence in

l

base and a numb er b etween and Simple counting shows that for each of the lab eling

schemes that we dened the numb er of available lab els is exactly equal to the numb er of ob jects

that need to b e lab eled

We say that a left hand side cluster and a right hand side cluster are compatible if their

lab els agree co ordinatewise in the following sense for co ordinate i if the merged lab el of the

right hand side cluster has an edge then this is the ith edge in the sequence of edges lab eling

the left hand side cluster and if the merged lab el has a vertex then this is the ith vertex in

the sequence of vertices lab eling the left hand side cluster Edges in G only connect compatible

clusters compatibility is necessary but not sucient as will b e seen shortly Note that each

left hand side cluster is compatible with exactly one cluster in each ray Each right hand side

l l

cluster is compatible with left hand side clusters the term here arises from the fact

that G is regular and the term follows from the fact that every edge has two endp oints

In order to describ e the edge set E we use the notion of a partition system For each left

l

hand side cluster C we have L partitions with prop erties as in Denition Recall that

C has m elements as required by Denition Each partition is lab eled by a sequence of l

colors This sequence of colors is interpreted as a sequence of colors for the sequence of vertices

that lab el C Note that the same vertex of G may app ear several times in the sequence of

vertices that lab els C we do not require the colors given to this vertex to b e the same

Consider an arbitrary vertex v in a cluster C that b elongs to ray i We now describ e the

r

set of neighb ors that it has in a compatible left hand side cluster C Recall that v is lab eled

by a sequence of l colors and a sequence of l pairs of distinct colors These colors give in

a natural way a coloring for the merged sequence of vertices and edges lab eling C The vertex

r

l

v was also lab eled by a numb er b etween and this lab el is ignored when determining the

l

set of neighb ors of v that is C has identical copies of v

r

The coloring of the merged sequence of C induces in a natural way a coloring for the sequence

r

of vertices lab eling C This coloring lab els one particular partition p Vertex v is connected to

all vertices p oints of the ith part of partition p recall that i is the ray to which C b elongs

r

This completes the description of E

l

Lemma If G is legal ly colorable then the onesided domatic number of G is

Proof We rst show that G has a onesided dominating set that includes exactly one vertex

from every right hand side cluster Consider a sequence of l arbitrary legal colorings of G the

same legal coloring may app ear multiple times in the sequence Consider now an arbitrary

right hand side cluster It is lab eled by a length l merged sequence of vertices and edges The

sequence of legal colorings induces a coloring on this merged sequence The cluster contains

l

exactly vertices whose lab el induces the same coloring Select one of them arbitrarily to b e

included in the onesided dominating set

To show that indeed we have a onesided dominating set we need to show that every left

hand side vertex u is covered Consider the cluster C to which u b elongs It is lab eled by a

sequence of l edges and a sequence of l vertices The sequence of legal colorings induces a

coloring for the sequence of vertices This coloring agrees with the name of one partition p Let

i b e the part of partition p to which u b elongs Consider the right hand side cluster C that is

r

compatible with C and b elongs to ray i The vertex selected from C necessarily covers u

r

l l

We now show that the onesided domatic numb er is at least in fact it is exactly

l

Consider an arbitrary legal coloring of G From it we can derive distinct length l sequences

of legal colorings where in each of the l co ordinates we put one of the six p ermutations of the

legal coloring Each of these sequences gives a onesided dominating set as describ ed ab ove

We show that these onesided dominating sets can b e chosen to b e distinct This follows from

the fact that for each sequence of legal colorings and every right hand side cluster we can have

l l l

equivalence classes of vertices who dier only in the numb er they are given in the

l l l

third lab el and equivalence classes of sequences of colorings who dier only in the

way they color vertices not in the merged sequence of the right hand side cluster Preserving

the structure of the equivalent classes we can match the sequences of legal colorings with the

vertices of a right hand side cluster

Lemma If every coloring of G legal ly colors at most jE j edges than the smal lest one

l

sided dominating set in G is of cardinality at least ojV j ln m

Proof The pro of is essentially identical to that of Lemma in and is omitted

By making m suciently large we can have jV j jV j and ln m ln jV j and the pro of of

Theorem follows

Acknowledgments

This work app ears in preliminary form in and

We would like to thank Satoshi Fujita and Mario Szegedy for helpful discussions and the

two referees for their helpful comments The rst author is the Incumb ent of the Joseph and

Celia Reskin Career Development Chair The research of the rst author was supp orted in part

by a Minerva grant The second author would like to thank Kazuo Iwama at Kyoto University

for his hospitality The last author was supp orted in part by NSF Award CCR

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A Multicoloring version

In the domatic multipartition problem we are additionally given an integral weight x V N

indicating in how many dominating sets each vertex can app ear The domatic numb er problem

has xv for each v the r Conf problem of has xv r for each v

We can reduce the multipartition problem to the ordinary partition problem Given a graph

G and weight vector x form a graph G as follows G has xv copies of each vertex v connected

as a clique For each edge uv in G the copies of u and v form a in

G Any minimal dominating set in G is also a dominating set in G and taking one copy of

each vertex of a dominating set in G also gives a dominating set in G Further a domatic

partition of G is in onetoone corresp ondence with a multipartition of G within the weight

constraints Thus the results obtained in this pap er for the domatic numb er problem carry

P

xv over to the domatic multipartition problem replacing by min dv xv and n by

v

v

B Overview of some ingredients from

Parallel rep etition of twoprover pro of systems There is a straightforward oneround

twoprover pro of system for Max colorability that has the following prop erties on yes in

stances the verier always accepts and on no instances when at most a fraction of the

edges can b e legally colored simultaneously the verier accepts with probability at most

Give a regular graph G the pro of system pro ceeds as follows The verier sends to the rst

prover a random edge in G and to the second prover a random vertex from that edge We

call this vertex the common vertex It is imp ortant that the rst prover do es not know which

endp oint of the edge is the common vertex and that the second prover do es not know which

edge was received by the rst prover The rst prover replies with two dierent colors out of

the three allowable colors for the two vertices that are the endp oints of the edge The second

prover replies with a color for the common vertex The verier accepts only if the two provers

give the same color to the common vertex For a yes instance G the two provers can answer

according to the global legal coloring and ensure that the verier accepts with probability

For a no instance G regardless of the strategy of each prover where a strategy is a function

from questions to answers the acceptance probability is at most where probability is

computed over the random choices of the verier

The l fold parallel rep etition of this oneround twoprover pro of system is as follows The

verier sends to the rst prover a tuple of l random edges and to the second prover a tuple

of l random vertices one from each of these edges Each prover replies with colors to all the

vertices that it receives The verier accepts if on every one of the l common vertices the two

provers agree on the color that they give It is not hard to see that on yes instances the provers

still have a strategy that makes the verier accept with probability The parallel rep etition

cl

theorem shows that on no instances the verier accepts with probability at most

where c is a constant that dep ends on

Separating co des and k provers To eventually get ln n hardness of approximation re

sults twoprover pro of systems are extended to k prover systems in a sp ecial way As in the

twoprover pro of system the verier selects l edges at random and a random vertex within each

l

edge This gives a total of jE j p ossible queries Each of the k provers is sent some subset

of the l edges and l vertices using the following rule A k l Hadamard co de is a collection of

k binary words of length l each for which the Hamming distance b etween any pair of words

is large sp ecically l With each prover we asso ciate one co deword from the Hadamard

co de The verier sends to each prover l edges and l vertices selected according to the s

and s in the co deword of the resp ective prover Namely if the ith bit in the co de is the

next entry in the query is the resp ective ith edge among the l edges in the tuple If the ith

entry is the ith vertex in the vertex tuple is sent Considering separately two of the provers

the fact that their co dewords have a large Hamming distance implies that in many co ordinates

one prover will receive an edge while the other prover will receive a vertex on this edge This

is similar to the scenario in the twoprover pro of system Intuitively we may view the k prover

k

pro of system as correlated twoprover pro of systems going on in parallel It is natural to

k

have the verier accept if all pro of systems are accepting The denition of acceptance used

in is more subtle and will not b e discussed in this overview

Next a bipartite graph G V V E is built based on this scenario Sp ecically the vertices

in V describ e the proversqueries schema The set V corresp onds to the provers and is parti

j

k

are further partitioned tioned into k rays V j k one ray for each prover The sets V

j

l l

into disjoint subsets V Q where Q ranges over all the jV j jE j p ossible questions that a

prover can receive On each p ossible question the prover must answer with a coloring of the

j

l edges and l vertices received Thus the V Q sets are further divided into all p ossible

j

l l

answers Therefore there are p oints inside V Q in this nal division a p oint for each

p ossible coloring of the l vertices and l edges In addition for technical reasons each p oint

l

p ossible answer of each prover is duplicated times

Ground sets and random partitions To determine the structure of V some ideas essen

tially due to and extended in are used

l

The set V is partitioned into jE j ground sets C one ground set p er each p ossible query

l

Each ground set contains m p oints for m jV j It is instructive to note the following

asymmetry Given a query namely a sequence of l edges and l corresp onding endp oints for the

j

edges consider its corresp onding ground set in V There are exactly k sets V Q that are

compatible with this sequence one p er prover since the question to the provers is completely

j

determined by the Hadamard co de On the other hand given a question cluster V Q many

p ossible querybase ground sets in V could have caused this question Indeed since the question

l l

contains only l edges and l vertices this partial information can b e completed in

ways to give compatible l edges and l vertices Each vertex sent to this prover has neighb ors

and thus ways of completing this vertex into a compatible edge For each edge sent to this

prover there are p ossibilities for which of the two vertices to put in the resulting query

l

We use L random partitions of C each partitioning C into k parts Consider the

l tuple of vertices in a p ossible query namely the part of the query corresp onding to the chosen

l

vertices one p er each edge Note that the numb er L of partitions of each ground set

l

exactly equals the numb er of p ossible colorings of the vertices in that query We can thus

l

form a corresp ondence b etween the partitionings and the coloring not necessarily a

consistent one of the l vertices in the query base Each partition is now regarded as a coloring

and viceversa Recall that a p oint in the nal subpartition cluster of V corresp onds to an

answer to the coloring of the query of l edgevertex pairs This corresp onds to a coloring of

the l vertices of the cluster C hence gives a vertex coloring and thus a partition of C The

edges b etween the vertices of V and V are dened as follows Each vertex namely coloring or

j

answer v V Q is connected to the j th part of the partition corresp onding to v as v is a

coloring a partition is also asso ciated with v

Now consider the graph G resulting from a colorable graph G We can cho ose one p oint

answer p er each questioncluster in V so as to cover each C Thus all the C are covered

l

using at most k E vertices This will happ en if the provers use strategies which are all

part of a global coloring and this gives a small dominating set On the other hand consider

the resulting graph G when only of the edges of G can b e simultaneously legally colored

Because of the parallel rep etition which greatly increases the gap the provers essentially cannot

j q

use a joint strategy This means that each pair of answers v V Q v V Q chosen

j q

corresp onds to two dierent colorings Hence a ground set C is essentially covered via random

sets each containing mk random elements of C Thus the numb er of elements needed in order

l

to cover C is roughly k ln m This gives a total of roughly k E ln jV j minimum onesided

dominating set for a logarithmic gap

Zero knowledge and the domatic numb er Given the fact that a single small onesided

dominating set exists we can p ermute the colors as done in zeroknowledge proto cols in order

to hide information to show the existence of many disjoint dominating sets hence a large

domatic numb er Namely once a coloring of G exists one gets legal colorings for G This

l

implies a total of colorings which can b e completed into global co ordinatewisecompatible

colorings of any l tuple of vertices The p ossibility for color p ermutation was the main reason

for reducing from a coloring and not a satisability problem Hence we either get many small

dominating sets which implies a large onesided domatic numb er or any dominating set is large

which gives a a small onesided domatic numb er