Counting the Number of Minimum Roman Dominating Functions of a Graph

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1 Coutig the Number of Miimum Roma Domiatig Fuctios of a Graph SHI ZHENG ad KOH KHEE MENG, Natioal Uiversity of Sigapore We provide two algorithms coutig the umber of miimum Roma domiatig fuctios of a graph o vertices i ( ) time ad polyomial space. We also show that the time complexity ca be reduced to ( ) if expoetial space is used. Our result is obtaied by trasformig the Roma domiatio problem ito other combiatorial problems o graphs for which exact algorithms already exist. Categories ad Subject Descriptors: G.2.2 [Discrete Mathematics]: Graph Theory Graph Algorithms Geeral Terms: Algorithms, Theory Additioal Key Words ad Phrases: Roma domiatio, exact algorithm, set cover, partial domiatig set, red-blue domiatig set 1. INTRODUCTION I algorithmic graph theory, exact expoetial-time algorithms are those usig expoetial executive time to give exact solutios to the problem. This field has grow tremedously recetly ad may ew techiques have bee developed, for istace, measure ad coquer by Fomi et al. [2005] which is a techique for estimatig bouds o the ruig time. There are also may kow results itroduced to the field, for istace, iclusio/exclusio [Bjӧrklud ad Husfeldt. 2006] ad dyamic programmig o graph decompositios [Fomi ad Stepaov. 2007; va Rooij et al. 2009]. Exact expoetial-time algorithms o may problems i domiatio ad its variats have bee desiged ad the bouds o ruig time cotiue to be improved by ewly developed or itroduced techiques. Roma domiatio is a variat of domiatio raised by Cockaye et al. [2004] i which the algorithmic ad complexity properties of Roma domiatio was metioed as a ope problem. May theoretical results i Roma domiatio have bee foud ad there exist polyomial time algorithms for computig the Roma domiatio umbers of some special classes of graphs [Cockaye et al. 2004; Liedloff et al. 2005]. However, the Roma domiatio problem for geeral graphs is NPcomplete as stated i Cockaye et al. [2004], ad thus we oly expect to costruct exact expoetial time algorithm o the Roma domiatio problem. Util ow, o exact algorithm has bee costructed o the Roma domiatio problem, ad the (1) of the trivial algorithm by listig ad best result o time complexity is (3 ) checkig. Our Results. I this paper, we are goig to itroduce exact expoetial-time algorithms coutig the umber of miimum Roma domiatig fuctios by kow results i exact algorithms o domiatio ad its variats. Two algorithms usig ( ) time ad polyomial space will be itroduced. It is oted that the approaches to costruct these two algorithms are differet. The first algorithm is ispired by usig a weight set cover istace to model the Roma domiatio problem, while the secod is costructed by relatig Roma domiatio to partial domiatio. I additio, a algorithm usig ( ) time ad expoetial space ca be derived from the secod algorithm. This paper falls ito three sectios. I Sectio 2, we provide some basic defiitios ad results. I Sectio 3, the algorithms coutig the umber of miimum Roma domiatig fuctios are give. 1

2 2. DEFINITIONS AND BASIC RESULTS 2.1 Domiatio i graphs Let G ( V, E) be a simple graph with vertex set V ad edge set E. The ope eighborhood of a vertex v, deoted as Nv, () is the set of vertices i G that are adjacet to v, ad N[ v] { v} N( v) is the closed eighborhood of v. A subset D V is called a domiatig set of G if for every v V, v either belogs to D or is adjacet to some vertex i D. A domiatig set is said to be miimal if o proper subset of it is a domiatig set of G. The miimum cardiality of a domiatig set is called the domiatio umber of G, deoted as ( G), ad ay domiatig set of cardiality ( G) is called a miimum domiatig set or -set. If there is a weight fuctio w: V, the D V is called a miimum weight domiatig set if D is a domiatig set of the miimum weight. I the case w: V {1}, the miimum weight domiatig set problem is reduced to the miimum domiatig set problem. A subset D V is called a partial domiatig set if there are at most l vertices that are either i D or adjacet to ay vertex i D. I other words, D domiates at least l vertices. I a bipartite graph G ( R B, E), D R is called a red-blue domiatig set if B N( D) ad D is said to be miimum if it has the miimum cardiality. For a fuctio f : V {0,1, 2}, f is called a Roma domiatig fuctio, or abbreviated as RDF, if for every u V0, V2 N( u ) is ot empty. For a subset V ' V, defie f ( V ') v V' f ( v) ad the weight of f is f ( V ) v V f ( v). The miimum weight of a RDF of G is the Roma domiatio umber of G, deoted as R ( G), ad a RDF f is called a miimum Roma domiatig fuctio or R -fuctio if w( f ) R ( G). The triple { V0, V1, V 2} is a vertex partitio, where Vi { v f ( v) i} for i = 0, 1, 2, ad sice there is a 1-1 correspodece betwee fuctios f : V {0,1, 2} ad vertex partitios { V0, V1, V 2}, we may write f { V0, V1, V2} for simplicity. 2.2 Set cover A set cover istace is defied by a uiverse ad a collectio of subsets such that R R. A set cover of a set cover istace (, ) is a sub-collectio ' such that R ' R. The cardiality of a set cover is the umber of subsets i the set cover. A set cover is said to be miimal if oe of its proper sub-collectio is a set cover. A miimum set cover is the oe of the miimum cardiality. If there is a weight fuctio w :, the ' is called a miimum weight set cover if ' is a set cover of the miimum weight. A set cover istace (, ) ca also be formulated as the red-blue domiatig set problem o G ( R B, E) by takig R { S S }, B { e e } ad ( e, S) E if ad oly if e S for ay e ad S. G ( R B, E) is called the icidece graph of the set cover istace (, ). Sometimes the red-blue domiatig set problem is cosidered istead of the set cover problem. For a graph G ( V, E), if we take V ad { N[ v] v V}, the D V is a miimal or miimum domiatig set if ad oly if ' { N[ v] v D} is a miimal or miimum, respectively, set cover of the set cover istace (, ). Hece (, ) defied above is ofte used as the set cover modellig of the domiatig set problem o G. Almost all expoetial-time exact algorithms after Gradoi [2006] apply algorithms desiged o the set cover problems o the set cover modellig of the domiatig set problem. 2

3 3. ALGORITHMS I this sectio, we are goig to itroduce two exact algorithms o Roma domiatio. Both of the algorithms cout the umber of R -fuctios of a graph G of vertices usig ( ) time ad polyomial space, however they are costructed by differet approaches. The first algorithm is desiged by usig a weight set cover modellig of the Roma domiatio problem ad the secod is desiged by relatig the Roma domiatio problem to the partial domiatig set problem i graphs. There exist exact algorithms o both weight set cover problems ad partial domiatig set problems ad therefore we will directly iclude them as subrouties i our algorithms o Roma domiatio. I additio, by the secod approach, we ca also itroduce a algorithm usig ( ) time ad expoetial space to cout the umber of R -fuctios of a graph G. 3.1 A algorithm coutig the umber of R -fuctios For ay simple graph G ( V, E) o vertices, we itroduce a weight set cover istace ( R, R ), here R V ad R { v v V} { N[ v] v V} with the weight fuctio w({ v}) 1 ad wnv ( [ ]) 2 for each v V. We otice that for ay RDF f { V0, V1, V2} of G, { v v V1} { N[ v] v V2} is a set cover of ( R, R ). Ad for a set cover { v v V1} { N[ v] v V2} such that V1 V 2 is empty, the fuctio f { V \{ V1 V2}, V1, V2} is a RDF. It is ot hard to see that for ay miimum weight set cover { v v V1} { N[ v] v V2}, V1 V2 ; otherwise, if v V1 V2, the \ { v } is a set cover of weight smaller tha the weight of, which is a cotradictio. Hece, solvig the Roma domiatio problem o G is equivalet to solvig the miimum weight set cover problem o ( R, R ) as defied above. Va Rooij [2010] showed that the miimum weight domiatig set problem ca be solved i ( ) time ad polyomial space if the set of possible weight sums is polyomially bouded. I his work, a algorithm listig the umber of set covers of each weight was desiged ad applied o a weight set cover istace reduced from the miimum weight domiatig set problem. Here we ca adopt this algorithm to solve our miimum weight set cover problem o ( R, R ) sice the possible weight of a subset is either 1 or 2. ALGORITHM 1. A Algorithm Computig the Number of R -fuctios Iput: A simple graph G ( V, E) Output: The umber of R -fuctios(# R -fuctios) 1. Geerate a weight set cover istace ( R, R ) with weight fuctio w : R V, R { v v V} { N[ v] v V} ad w({ v}) 1 ad wnv ( [ ]) 2 for each v V 2. Geerate the icidece graph I( R R, E ) of ( R, R ) : ( e, S) E if ad oly if e S for ay e R ad S R 3. Compute the umber of set covers of each possible weight: CWSC ( I ( R R, E), ) 4. Retur the umber of R -fuctios: # R -fuctios is the umber of set covers of the miimum weight SUBROUTINE CWSC. A Algorithm Coutig the Number of Set Covers of Each Weight Iput: The icidece graph I(, E) ad a set of aotated vertices A Output: A list cotaiig the umber of set covers of (, ) of each weight 3

4 Table I. The weights vi () ad wi () i complexity aalysis of Algorithm 1 i >6 vi () wi () Source: [Va Rooij 2010] The subroutie CWSC will call itself recursively ad the iput aotated vertices A is set to be empty iitially. For more details of CWSC see va Rooij [2010]. THEOREM 3.1. The Roma domiatio problem ca be solved i ( ) ad polyomial space. time PROOF. We cosider usig Algorithm 1 to solve the Roma domiatio problem. The mai part of Algorithm 1 is obviously i lie 3 which is applyig CWSC o the weight set cover istace ( R, R ), ad the time complexity of Algorithm 1 ca be bouded by that of CWSC ( I( R R, E), ). By usig measure ad coquer techique, va Rooij [2010] showed that the time complexity of CWSC ca be bouded by k( ) ( ), here k ( ) v ( d ( e )) w ( d ( S )) e ( )\ A ( )\ A S dx ( u) is the degree of u X i the iduced graph GX [ ] ad vi () ad wi () are weights set as i Table I. Hece, for the set cover istace ( R, R ), k( ) ( w(1) w( 6) v( 6)) R ad therefore the time complexity of Algorithm 1 ca be bouded from above by ( ) ( ). Actually this is also the complexity of applyig CWSC o the set cover modellig of the miimum domiatig set problem sice vertices represetig subsets of degree 1 i G[( ) \ A ] do ot cotribute to the complexity. However, there is a faster versio of CWSC that uses expoetial space i which the vertices represetig subsets of degree 1 will cotribute to the complexity ad the cost of applyig it o ( R, R ) will be expoetial space ad greater tha ( ) time. Hece we are ot goig to itroduce expoetial space exact algorithms by usig the weight set cover modellig here. 3.2 Aother algorithm coutig the umber of R -fuctios Give a simple graph G ( V, E) ad a R -fuctio f { V0, V1, V2} of G, for every vertex u V0, u is adjacet to at least oe vertex v V2. Moreover, there is o edge joiig V1 ad V 2. Hece, V 2 is a partial domiatig set that domiates exactly V 1 vertices ad the umber of RDF such that V1 k ad V2 j ad there is o edge joiig V1 ad V2 is the umber of partial domiatig sets of cardiality j that domiates exactly k vertices. 4

5 Recall that for a graph G ( V, E), the set cover modellig of the miimum domiatig set problem o G is a set cover istace (, ), where V ad { N[ v] v V} ad the icidece graph I(, E ) of (, ) is a bipartite graph such that ( e, S) E if ad oly if e S for ay e ad S. The D is a partial domiatig set of cardiality j that domiates exactly k vertices of G if ad oly if D is a partial red-blue domiatig set of j vertices i that domiates exactly k vertices i of the icidece graph I. Nederlof ad va Rooij [2010] showed that the partial domiatig set problem ca be solved i ( ) time ad polyomial space by a algorithm that computes a matrix P cotaiig the umber of partial red-blue domiatig sets of j red vertices domiatig exactly l blue vertices for each give 0 j R ad 0 l B for ay bipartite graph G ( R B, E). Here we will iclude this algorithm as a subroutie. ALGORITHM 2. A Algorithm Computig the Number of R -fuctios Iput: A simple graph G ( V, E) Output: The umber of R -fuctios(# R -fuctios) 1. Geerate a set cover istace (, ) : V ad { N[ v] v V} 2. Geerate the icidece graph I(, E ) of (, ) : ( e, S) E if ad oly if e S for ay e ad S 3. Compute a matrix P cotaiig the umber of partial red-blue domiatig sets of cardiality j that domiates exactly l vertices: CPSC ( I(, E), ) 4. Compute a matrix A cotaiig the umber of RDF such that V1 k ad V2 j ad there is o edge joiig V1 ad V 2 : ajk p, j k 5. Compute the Roma domiatio umber: R mi {2 j k} a jk 0 6. Compute the umber of R -fuctios: # R -fuctios = 2 j k R a jk SUBROUTINE CPSC. A Algorithm Computig the Number of Partial Red-Blue Domiatig Sets Iput: A bipartite graph G ( R B, E) ad a set of aotated vertices A Output: A matrix cotaiig the umber of partial red-blue domiatig sets of cardiality j that domiates exactly l vertices for each give 0 j R ad 0 l B The subroutie CPSC is very similar to CWSC as the iputs are both a bipartite graph ad a set of aotated vertices that is empty iitially ad the outputs ca be geerated by similar rules. This similarity leads to that the complexity aalysis of CPSC is the same as that of CWSC. AN ALTERNATE PROOF FOR THEOREM 3.1. We cosider usig Algorithm 2 to solve the Roma domiatio problem. The mai part of Algorithm 2 is lie 3 which applies CPSC o the weight set cover istace (, ) ad the time complexity of Algorithm 2 ca be bouded by that of CPSC ( I(, E), ). The complexity was aalyzed to be ( ) time ad polyomial space [Nederlof ad va Rooij. 2010]. There is a expoetial-space versio of CPSC usig ( ) time solvig the partial domiatig set problem. For more details see Nederlof ad va Rooij [2010]. THEOREM 3.2. The Roma domiatio problem ca be solved i ( ) ad expoetial space. time 5

6 PROOF. We cosider substitutig CPSC i Algorithm 2 with its expoetial-space versio ad use it to solve the Roma domiatio problem. The time complexity ca be estimated i the way similar to the alterate proof for Theorem 3.1 to be ( ). REFERENCES Bjӧrklud, A. ad Husfeldt, T Iclusio--Exclusio Algorithms for Coutig Set Partitios. I Foudatios of Computer Sciece, FOCS '06. 47th Aual IEEE Symposium o, Cockaye, E.J., Dreyer, P.A., Hedetiemi, S.M. ad Hedetiemi, S.T Roma domiatio i graphs. Discrete Mathematics 278, Fomi, F., Gradoi, F. ad Kratsch, D Measure ad Coquer: Domiatio A Case Study. I Automata, Laguages ad Programmig, L. CAIRES, G. ITALIANO, L. MONTEIRO, C. PALAMIDESSI AND M. YUNG Eds. Spriger Berli Heidelberg, Fomi, F. ad Stepaov, A Coutig Miimum Weighted Domiatig Sets. I Computig ad Combiatorics, G. LIN Ed. Spriger Berli Heidelberg, Gradoi, F A ote o the complexity of miimum domiatig set. Joural of Discrete Algorithms 4, Liedloff, M., Kloks, T., Liu, J. ad Peg, S.-L Roma Domiatio over Some Graph Classes. I Graph-Theoretic Cocepts i Computer Sciece, D. KRATSCH Ed. Spriger Berli Heidelberg, Nederlof, J. ad va Rooij, J.M.M.V Iclusio/Exclusio Brachig for Partial Domiatig Set ad Set Splittig. Va Rooij, J.M Polyomial Space Algorithms for Coutig Domiatig Sets ad the Domatic Number. I Algorithms ad Complexity, T. CALAMONERI AND J. DIAZ Eds. Spriger Berli Heidelberg, Va Rooij, J.M., Bodlaeder, H. ad Rossmaith, P Dyamic Programmig o Tree Decompositios Usig Geeralised Fast Subset Covolutio. I Algorithms - ESA 2009, A. FIAT AND P. SANDERS Eds. Spriger Berli Heidelberg,