An Algebraic Characterization of Rainbow Connectivity
The use of algebraic techniques to solve combinatorial problems is studied in this paper. We formulate the rainbow connectivity problem as a system of polynomial equations. We first consider the case of two colors for which the problem is known to be…
Authors: Prabhanjan Ananth, Ambedkar Dukkipati
AN ALGEBRAIC CHARA CTERIZA TION OF RAINBO W CONNECT I VITY PRABHANJAN ANANTH AND AMBEDKAR DUKKIP A TI Abstra ct. The use of alg ebraic techniques to solv e com binatorial prob- lems is studied in this pap er. W e formulate the rainbow connectivit y problem as a system of p olynomial equations. W e first consider t he case of t w o colors for whic h the problem is kn o wn to b e hard and we then extend the approac h to the general case. W e also give a form ulation of the rain b o w connectivity problem as an idea l membership problem. 1. Introduction The use of algebraic concepts to solv e combinatoria l optimization p rob- lems has b een a fascinating fi eld of study exp lored by man y researc hers in theoretical computer science. The com binatorial metho d in tro duced b y Noga Alon [1] offered a new direction in obtaining structural results in graph theory . Lo v´ asz [9], De Lo era [5] and others formulat ed p opular grap h p r ob- lems lik e verte x coloring, indep endent set as a system of p olynomial equa- tions in su c h a wa y that solving the system of equatio ns is equiv alen t to solving the combinato rial p roblem. This formulation ensu red the fact that the system has a solution if and only if the corresp onding instance has a “y es” answ er. Solving s ystem of p olynomial equations is a well stu died problem with a we alth of literature on this topic. It is w ell kno wn that solving s y s tem of equ ations is a notoriously hard p r oblem. De Loera et al. [6] prop osed the NulLA approac h (Nullstellensatz Lin ear Algebra) w h ic h used Hilb ert’s Nullstellensatz to determine the feasibilit y among a sy s tem of equations. This app roac h was furth er used to charac terize some classes of graphs based on degrees of the Nullstellensatz certificate. In Section 2, we review th e basics of enco d in g of com binatorial problems as systems of p olynomial equations. F urther, w e describ e NulLA along with th e preliminaries of rain b o w connectivit y . In Section 3, we prop ose a formulation of the r ain b o w connectivit y pr oblem as an ideal membersh ip prob lem. W e then present enco dings of the rain b ow connectivit y problem as a system of p olynomial equations in Section 4. 2. Back ground and Preliminaries The encod ing of w ell kno wn com binatorial p roblems as system of p oly- nomial equations is describ ed in this section. The encod ing sc hemes of the vertex coloring and th e indep en den t set p roblem is p r esen ted. Encod in g sc hemes of well kn own problems lik e Hamiltonian cycle problem, MAXCUT, SA T and others can b e fou n d in [10]. The term enco ding is formally defined as follo ws: 1 2 P . ANANTH AND A. DUK KIP A TI Definition 1. Give n a language L , if ther e exists a p o lynomial-time algo - rithm A that takes an input string I , and pr o duc es as output a system of p olynomial e quations such that the system has a solution if and only if I ∈ L , then we say that the system of p olynomial e q uations enc o des I . It is a n ecessit y that th e algorithm that transform s an instance in to a system of p olynomial equations has a p olynomial ru nning time in the size of the instance I . Else, the pr oblem can b e solv ed b y brute force and trivial equations 0 = 0 (“y es” instance) or 1 = 0 (“no” instance) can b e output. F urther sin ce the algorithm run s in p olynomial time, th e size of the output system of p olynomial equations is b ou n ded ab o v e b y a p olynomial in the size of I . The enco din gs of vertex coloring and stable set problems are p resen ted next. W e use th e follo wing notati on throughout th is pap er. Unless otherwise men tioned all the graph s G = ( V , E ) ha ve th e v ertex set V = { v 1 , . . . , v n } and the edge set E = { e 1 , . . . , e m } . The notation v i 1 − v i 2 − · · · − v i s is u sed to denote a path P in G , wh ere e i 1 = ( v i 1 , v i 2 ) , . . . , e i s − 1 = ( v i s − 1 , v i s ) ∈ E . The path P is also denoted b y v i 1 − e i 1 − · · · − e i s − 1 − v i s and v i 1 − P − v i s . 2.1. k -v ertex coloring and stable set problem. The ve rtex coloring problem is one of the most p opular pr ob lems in graph theory . The minimum n umb er of colors required to color the v er tices of the graph su c h that no t w o adjacen t v ertices get the same color is termed as the ve rtex coloring problem. W e consider the decision version of the v er tex coloring p roblem. The k -vertex coloring problem is defin ed as follo ws: Giv en a graph G , d o es there exist a ve rtex coloring of G with k colors suc h th at no t wo adjacen t v ertices get the same color. Th ere are a quite a few enco dings kno wn for the k -vertex colorabilit y problem. W e presen t on e suc h enco din g giv en b y Ba yer [3 ]. The p olynomial ring u nder consideration is k [ x 1 , . . . , x n ]. Theorem 2.1. A gr aph G = ( V , E ) is k -c olor able if and only if the fol lowing zer o-dimensional system of e quations has a solution: x k i − 1 = 0 , ∀ v i ∈ V k − 1 X d =0 x k − 1 − d i x d j = 0 , ∀ ( v i , v j ) ∈ E Pro of Idea. If the graph G is k -colorable, then there exists a p rop er k - coloring of graph G . Denote these set of k colors b y k th ro ots of unity . Consider a p oin t p ∈ k n suc h that i th co-ordinate of p (d enoted b y p ( i ) ) is the same as the color assigned to the v ertex x i . The equations corresp onding to eac h v ertex (of the form x k i − 1 = 0) are satisfied at p oint p . The equations corresp ondin g to the edges can b e rewritten as x k i − x k j x i − x j = 0. S ince x k i = x k j = 1 and x i 6 = x j , ev en the edge equation is satisfied at p . Assume th at the system of equ ations h av e a solution p . It can b e seen that p cannot hav e more than k distinct co-ordinates. W e color the vertice s of th e graph G as follo ws: color the verte x v i with the v alue p ( i ) . It can b e sho wn that if the system is satisfied then in the ed ge equations, x i and x j need to take d ifferen t v alues. In other words, if ( v i , v j ) is an edge then p ( i ) AN ALGEBRAIC CHARA CTERIZA TION OF RAINBOW CONNECTIVITY 3 and p ( j ) are d ifferen t. Hence, the ve rtex coloring of G is a pr op er coloring. A stable set (ind ep endent set) in a graph is a sub set of v ertices such that no tw o v ertices in the su bset are adjacen t. The stable set problem is de- fined as the problem of fi nding th e maximum stable set in the graph. The cardinalit y of the largest s table set in the graph is termed as the indep en- dence num b er of G . The enco ding of the decision ve rsion of th e stable set problem is presented. The decision v ersion of the stable set pr oblem deals with determining whether a graph G has a stable set of size at least k . The follo win g result is due to Lo v´ asz [9]. Lemma 2.2. A gr aph G = ( V , E ) has an indep endent set of size ≥ k if and only if the fol lowing zer o-dimensional system of e quations has a solution x 2 i − x i = 0 , ∀ i ∈ V x i x j = 0 , ∀{ i, j } ∈ E n X i =1 x i − k = 0 . The numb er of solutions e quals th e numb er of distinct indep endent sets of size k . The p r o of of th e ab o ve result can b e found in [10]. 2.2. NulLA algorithm. De Lo era et al. [6] p rop osed the Nullstellensatz Linear Algebra Algorithm (NulLA) whic h is an approac h to ascertain whether the system has a s olution or not. Their m etho d relies on the one of the most imp ortant theorems in algebraic geometry , namely the Hilb ert Nullstellen- satz. The Hilb ert Nullstellensatz theorem states that the v ariet y of an ideal is empty o ve r an algebraica lly closed field iff the elemen t 1 b elongs to the ideal. More formally , Theorem 2.3. L e t a b e a pr op er ide al of k [ x 1 , . . . , x n ] . If k is algebr aic al ly close d, then ther e exists ( a 1 , . . . , a n ) ∈ k n such that f ( a 1 , . . . , a n ) = 0 for al l f ∈ a . Th us, to determine w hether a system of equations f 1 = 0 , . . . , f s = 0 h as a solution or not is the same as determining whether there exists p olynomials h i where i ∈ { 1 , . . . , s } such that P s i =1 h i f i = 1. A result b y Koll´ ar [7] sho ws that the degree of the co efficien t p olynomials h i can b e b oun d ed ab o v e by deg { 3 , d } n where n is the n umber of indeterminates. Hence, eac h h i can b e expressed as a sum of monomials of degree at most deg { 3 , d } n , w ith un- kno wn co efficien ts. By expanding the summation P s i =1 h i f i , a system of linear equations is obtained with the un k n o wn co efficien ts b eing the v ari- ables. Solving this sy s tem of linear equations will yield us the p olynomials h i suc h that P s i =1 h i f i = 1. Th e equation P s i =1 h i f i = 1 is kno wn as Null- stellensatz certificate and is said to b e of degree d if max 1 ≤ i ≤ s { deg( h i ) } = d . There ha ve b een efforts to determine the b ound s on th e degree of th e Null- stellensatz certificate whic h in turn has an impact on the running time of NulLA algorithm. The descrip tion of the NulLA algorithm can b e found in [10]. The r unning time of the algorithm dep ends on the d egree b ounds on 4 P . ANANTH AND A. DUK KIP A TI the p olynomials in the Nullstellensatz certificate. It was sho wn in [4 ] that if f 1 = 0 , . . . , f s = 0 is an infeasible system of equations then there exists p olynomials h 1 , . . . , h s suc h that P s i =1 h i f i = 1 and d eg( h i ) ≤ n ( d − 1) where d = max { deg ( f i ) } . Thus with th is b ound , the running time of th e ab o v e algorithm in the w orst case is exp onent ial in n ( d − 1). Eve n though this is still far b eing practical, for some sp ecial cases of p olynomial systems this appr oac h seems to b e promising. More sp ecifically this p ro v ed to b e b eneficial for the system of p olynomial equ ations arising from com binatorial optimization problems [10]. Also using NulLA, p olynomial-time pro cedures w ere designed to solv e the com binatorial p r oblems for some sp ecial class of graphs [8]. 2.3. Rain b o w connectivit y . Consider an edge colored graph G . A rain- b o w path is a path consisting of distinctly colored edges. T he graph G is said to b e rain b ow connected if b et w een ev ery t wo ve rtices there exists a rain b ow p ath. Th e least n umb er of colors required to edge color the graph G su c h that G is rain b ow connected is called the rain b o w conn ection num b er of the graph, denoted b y r c ( G ). The problem of determining r c ( G ) for a graph G is termed as the r ain b o w connectivit y p roblem. The corresp onding decision ve rsion, termed as the k -rain b ow connectivit y problem is defined as f ollo ws: Giv en a graph G , decide w hether rc ( G ) ≤ k . The k -rain b ow connectivit y problem is NP-complete ev en for the case k = 2. 3. Rainbow connectivity as an ideal members hip p roblem Com binatorial optimization problems like vertex coloring [2, 5] were f or- m ulated as a memb ership pr ob lem in p olynomial ideals. The ge neral ap- proac h is to asso ciate a p olynomial to eac h graph an d then consider an ideal which con tains all and only those graph p olynomials that ha ve some prop erty (for example, c hromatic num b er of the corresp onding graph is less than or equal to k ). T o test wh ether th e graph has a requir ed pr op ert y , we just need to c hec k whether the corresp onding graph p olynomial b elongs to the id eal. In this section, we d escrib e a p r o cedure of solving the k -rain b ow connectivit y problem by form u lating it as an id eal mem b er s hip problem. By this, w e mean that a solution to the ideal memb er s hip problem yields a solution to the k -rain b ow conn ectivit y problem. W e restrict our atten tion to the case w hen k = 2. In ord er to form ulate the 2-rainbow conn ectivit y p roblem as a m emb ership problem, we fir st consider an ideal I m, 3 ⊂ Q [ x 1 , . . . , x m ]. Then the problem of deciding whether th e give n graph G can b e rain b ow connected with 2 colors or not is reduced to the problem of d eciding whether a p olynomial f G b elongs to the ideal I m, 3 or not. The ideal I m, 3 is defined as the ideal v anishing on V m, 3 , where V m, 3 is d efined as the set of all p oin ts wh ic h hav e at most 2 distinct co ordinates. More formally , V m, 3 ∈ Q m is the un ion of S ( m, 2) (S tirling n umb er of the second kind) linear subspaces of dimen sion 2. The follo wing theorem was p ro v ed by De Lo era [5]: AN ALGEBRAIC CHARA CTERIZA TION OF RAINBOW CONNECTIVITY 5 Theorem 3.1. The set of p olynomials G m, 3 = { Q 1
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