Approximating the Chromatic Polynomial of a Graph NaiWei Lin Department of Computer Science University of Arizona The Tucson AZ naiweicsarizonaedu Abstract The problem of computing the chromatic p olynomial of a graph is Phard This pap er presents an approximation algorithm for computing the chromatic p olynomial of a graph 2 2 n log n nm for a graph with n vertices and m This algorithm has time complexity O edges This pap er also shows that the problem of computing the chromatic p olynomial of a chordal graph can b e solved in p olynomial time Kno wledge ab out the chromatic p olynomial of graphs can b e employed to improve the p erformance of logic programs and deductive databases Intro duction This pap er considers nite graphs without lo ops ie edges joining a vertex to itself and multiple edges b etween any pair of vertices The chromatic number of a graph G written as G is the minimum numb er of colors necessary to color G such that no adjacent vertices have the same color The chromatic p olynomial of a graph G denoted by C G x is a p olynomial in x representing the numb er of dierent ways in which G can b e colored by using at most x colors The problem of k colorability of a graph G is the one of deciding whether the value C G k Since the problem of deciding whether the value C G k is NPcomplete the problem of computing the value C G k is Pcomplete The more general problem of computing the chromatic p olynomial of a graph is therefore Phard This pap er presents an approximation algorithm for computing the chromatic p olynomial of a graph This algorithm is based on the gr eedy metho d We rst determine an ordering According to the determined ordering we on the vertices of the graph using some heuristics This work was supp orted in part by the National Science Foundation under grant numb er CCR next derive an upp er b ound and a lower b ound on how many dierent ways each vertex can b e colored The pro duct of the upp er b ounds and the pro duct of the lower b ounds for all the vertices in the graph then give resp ectively an upp er b ound and a lower b ound on the total the numb er of available colors is numb er of dierent ways the entire graph can b e colored If given as a symb olic variable then the two pro ducts are p olynomials in this variable Finally we take a mean of these two p olynomials as an approximation of the chromatic p olynomial of 2 2 for a graph with n vertices n log n nm the graph This algorithm has time complexity O and m edges A graph is called chordal if every cycle of length greater than has an edge joining two nonconsecutive vertices of the cycle Chordal graphs arise in many contexts and contain the following families of graphs interval graphs cactus graphs adjoint graphs of cactus graphs and so on Gavril has shown that problems of nding a minimum coloring a minimum a maximum clique and a maximum indep endent set of a chordal graph covering by cliques can b e solved in p olynomial time In this pap er we show that the problem of computing the chromatic p olynomial of a chordal graph can also b e solved in p olynomial time Man y problems in areas such as op erations research and articial intelligence require enu merating all the solutions that satisfy a set of binary equality or disequality constraints on y or variables ranging over a nite domain of values The constraints are of the form x x y These constraint satisfaction problems can b e reduced to the graph coloring problem as h vertex in the graph corresp onds to a variable in the constraints each edge in the follows Eac graph corresp onds to a disequality constraint and the edges of two vertices are merged if there each coloring of the is an equality constraint b etween the corresp onding two variables Hence graph corresp onds to a solution to the original constraint satisfaction problem Accordingly the asso ciated counting problem for each such constraint satisfaction problem reduces to the computation of the chromatic p olynomial of its corresp onding graph in the settings such as This class of constraint satisfaction problems is frequently realized logic programs and deductive databases where it is easy to generate multiple solutions of a problem if required The ability to estimate the chromatic p olynomial of a graph allows us to estimate the numb er of solutions generated by the pro cedures realizing these constraint satisfaction problems Information ab out the numb er of solutions generated by pro cedures has b een used to estimate the cost of executing pro cedures and the cost information has b een further used to optimize programs and queries in these settings For example this information has b een employed for query optimization in deductive databases by appropriately rearranging the evaluation order of subgoals and for pro cess granularity control in parallel logic programming systems by prop erly preventing smallgrain pro cesses from spawning The remainder of this pap er is organized as follows Section derives a numb er of upp er and lower b ounds on the chromatic p olynomial of a graph Section discusses how the orderings presents on the vertices of a graph aect the b ounds on its chromatic p olynomial Section s H A H H A H s s A A s s Figure An example an approximation algorithm for computing the chromatic p olynomial of a graph based on an ordering prop osed in Section Section shows some exp erimental results on the p erformance b ehavior of the algorithm Finally Section concludes this pap er Bounds on Chromatic Polynomials Since the value of the chromatic p olynomial C G x is always nonnegative for any graph G x really means the function and natural numb er x we shall assume that any p olynomial P maxP x Furthermore for any two p olynomials P x and P x we dene P x P x 1 2 1 2 if and only if maxP x maxP x for all natural numb ers x We shall derive b ounds 1 2 on the chromatic p olynomial of a graph based on the greedy metho d Let G V E b e a graph The order of G denoted by jGj is the numb er of vertices in G Let U b e a subset of V The subgraph of G induced by U is a graph H U F such that F consists of all the edges in E b oth of whose vertices b elong to U The neighbors v fw V jv w E g of a vertex v in G is the set of vertices adjacent to v The N G v of v is the subgraph of G induced by N v adjacency graph Adj G G Let G V E b e a graph of order n and v v b e an ordering on V We 1 n dene two sequences of subgraphs of G according to The rst is a sequence of subgraphs G G called accumulating subgraphs where G is the subgraph induced by 1 n i 0 0 G V fv v g for i n The second is a sequence of subgraphs G i 1 i 2 n 0 called interfacing subgraphs where G is the adjacency graph Adj v for i n i ( ) G i 1 i For example consider the graph shown in Figure The imp osed ordering is denoted by the lab els of vertices The corresp onding accumulating subgraphs and interfacing subgraphs are shown in Figure The following prop osition gives an upp er b ound and a lower b ound on the chromatic p oly nomial of a graph in terms of resp ectively the chromatic numb er and the order of interfacing subgraphs i s s s s accumulating H s s A s A A H s subgraphs s s s As s As A s G i s s s interfacing s subgraphs s s s 0 G i Figure The accumulating and interfacing subgraphs of the graph in Figure Prop osition Let G V E be a graph of order n and be the set of al l possible order ings on V Suppose the interfacing subgraphs of G corresponding to an ordering are 0 0 G G Then n 2 n n Y Y 0 0 x G g max fx x jG jg C G x min fx 2 2 i i i=2 i=2 Pro of Supp ose G G are the accumulating subgraphs of G corresp onding to an 1 n ordering The pro of is by induction on G for j n In base case G is a graph j 1 Q j 0 consisting of one vertex v so C G x x Supp ose x j C G x x jG 1 1 j i i=2 Q j 0 and for some j j n Then consider adding the vertex v x G x j +1 i i=2 0 asso ciated edges into G to form G For the lower b ound since jG j is the j j +1 j +1 0 degree of v in G we have x jG j C G x C G x From the j +1 j +1 j j +1 j +1 hyp othsis we obtain j +1 Y 0 0 x x jG j x jG j C G x C G x j j +1 i j +1 i=2 0 For the upp er b ound since G is the minimum numb er of colors necessary for coloring j +1 0 0 G we have C G x x G C G x From the hyp othsis we j +1 j j +1 j +1 obtain j +1 Y 0 0 C G x x G x G C G x x j +1 j j +1 i i=2 Since G G we have the following formula n n n Y Y 0 0 x G x jG j C G x x x i i i=2 i=2 Finally b ecause Formula holds for any ordering Formula follows 2 The upp er b ound in Formula is in terms of the chromatic numb er of interfacing sub graphs Unfortunately the computation of the chromatic numb er of a graph is NPcomplete Nevertheless it turns out that if each of the chromatic numb ers in Formula is replaced by any of its lower b ounds the resultant expression is still an upp er b ound on the chromatic gives a lower b ound on the chromatic numb er p olynomial The following theorem by Bondy of a graph that can b e computed eciently Theorem Let G be a graph of order n d dn be the degrees of nodes in G be dene and d recursively by j 1 j X n d j i i=1 Suppose k n is some integer satisfying 1 k X n j j =1 Then G k 2 We dene G called the Bondy number of a graph G to b e the largest integer k n satisfying Formula Then an upp er b ound on the chromatic p olynomial of