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 dominating set 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 degree 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 approximation algorithm 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 graph coloring 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