Max Edge Coloring of Trees

Max Edge Color ing of T rees Giorgio Lucarelli ∗ Ioannis Milis ∗ V angelis Th. Pasc hos † June 29, 2018 Abstract W e study the w eigh ted generalization of the edge coloring p roblem where th e weig ht of eac h color class (matc hing) equals to the w eigh t of its heaviest e dge and the goal is to minimize the sum of the colors’ weigh ts. W e present a 3 / 2-appro ximation algorithm for trees. 1 In tro duction In the standard e dge c oloring problem we ask for a partition S = { M 1 , M 2 , . . . , M s } of the edg e set of G into color cla sses (matchings) suc h that s is minimized. In this pa rer we study the following generalization of the standard edge co loring pro ble m which arises in the domain of optica l communication systems (see for example [4 ]): A positive integer weigh t is asso ciated with each edge of G and we no w ask for a partition S = { M 1 , M 2 , . . . , M s } o f the edg e s of G into color cla sses, each one o f weight w i = ma x { w ( e ) | e ∈ M i } , such that the sum of the co lo rs’ weigh ts W = P s i =1 w i is minim ized. The analogo us g eneralizatio n for the standard vertex color ing pr oblem, where weigh ts are as- so ciated to the vertices of a gra ph and the weight o f ea ch co lor class (indepe nden t set) equals to the weight of its heaviest vertex, has been also addressed in the liter ature and it is known as Ma x (V ertex) Coloring ( MVC ) pr oblem [8, 7]. Resp ectively to this we refer to our problem as Max Edge Coloring ( MEC ) problem. It is k nown tha t the MEC problem is str ongly NP-ha r d and 7 / 6 inapproximable even for cubic planar bipartite g raphs with edg e weights w ( e ) ∈ { 1 , 2 , 3 } [1]. On the other hand, the MEC problem is kno wn to b e polynomia l fo r a few sp ecial cases including bipartite graphs with edge weigh ts w ( e ) ∈ { 1 , 2 } [2], c hains [3] (in fact, this algo rithm can b e a lso applied for graphs of ∆ = 2 ), stars of c hains and bounded deg ree trees [5]. Concerning the approximability of the MEC problem, a natural gre e dy 2-appr oximation al- gorithm for ge ne r al graphs has b een prop osed b y K e sselman and Kog an [4]. The ratio of this algorithm has b een slightly improved to 2 − 1 ∆ and 2 − 2 ∆+2 in [6]. E sp ecially for bipartite graphs of maximum degre e ∆ = 3 a n alg orithm that attains the 7 / 6 inapproximability b ound has been presented in [1]. F or bipartite gr aphs hav e be e n also pr esented algorithms improving the b est known 2 − 2 ∆+2 approximation ratio for ge neral gra phs. In fac t, algor ithms pr esented in [3] and [5] achieve b etter ratios for bipa rtite gr aphs of ∆ ≤ 7, and ∆ ≤ 1 2, resp ectively . Moreover, tw o algorithms of approximation ratio s 2 − 2 ∆+1 and 2∆ 3 ∆ 3 +∆ 2 +∆ − 1 hav e b een presented in [6]. It is interesting that no algor ithm of appr oximation r a tio 2 − δ , for a ny small cons tant δ > 0, is known for the ME C pro blem on bipartite graphs or e ven o n tre e s. Recall that the ME C problem on bipartite gra phs is 7 / 6 inappr oximable and notice that neither the complexity of the MEC pr oblem on trees is known. On the other hand, for the MVC problem this gap is close d fo r bipartite g r aphs and it is very narrow for trees. In fa ct, an alg orithm whic h matches the inappr oximabilit y b ound of 8/7 is known for bipartite graphs [2, 1, 7] while a PT AS is known for trees [7]. Howev er , the complexity of the MVC pr oblem on trees remains also unknown. In this note we decr ease this gap for the MEC problem on tr e es by presenting a 3 / 2-appr oximation algor ithm. ∗ Departmen t of Informatics, Athe ns Universit y of Economics and Business, Greece. { gluc,mil is } @aueb.g r . † LAMSADE, Universit ´ e P aris-Dauphine, F rance. paschos@lamsade. dauphine.fr . 1 2 Definitions a nd Preliminaries W e co nsider the MEC problem o n a weigh ted tree T = ( V , E ). By d ( v ) we deno te the degree o f vertex v ∈ V and by ∆ the ma x im um degre e o f T . By S ∗ = { M ∗ 1 , M ∗ 2 , . . . , M ∗ s ∗ } we denote an optimal solution to the MEC problem of weight OP T = w ∗ 1 + w ∗ 2 + . . . + w ∗ s ∗ . F or each vertex u ∈ V , we denote by E u : e u 1 , e u 2 , . . . , e u d ( u ) an order ing of its adjacent edg es in non increasing weigh ts, i.e. w ( e u 1 ) ≥ w ( e u 2 ) ≥ . . . ≥ w ( e u d ( u ) ). F urthermo re, w e define y i , 1 ≤ i ≤ ∆ , to b e the weigh t of the heaviest edge b etw een those ranked i in each o rdering E u , u ∈ V , i.e. y i = max u ∈ V { w ( e u i ) } . It is clear that y 1 ≥ y 2 ≥ . . . ≥ y ∆ . Prop ositio n 1 F or al l 1 ≤ i ≤ ∆ , it holds that w ∗ i ≥ y i . Pro of: Let e = ( u, v ) b e the heaviest edge with rank equal to i i.e., y i = w ( e ). F o r at lea st one of the e ndp oints of e , as sume w.l.o.g. for u , it holds tha t e is ranked i in E u , tha t is y i = w ( e u i ). Therefore, there exist i edges adjacent to vertex v of weigh t at lea st y i . These i edges b elong in i different matchings in an o ptima l solution, since they sha re vertex u as a common endp oint. Th us, the i -th ma tch ing in an optimal solution is of w eight at least y i . In [4], K esselman and Kog an present the mo st interesting and gener al result we hav e for the MEC problem. This is a gr eedy 2 -approximation alg o rithm fo r genera l graphs , to which we r efer as Algorithm KK. A slightly better analysis of this algo r ithm pres ent ed in [6] leads to the following approximation ra tio which also matches exa ctly the ratio of the tightn ess counterexample given in [4]. Lemma 1 [6] Algorithm KK achieves a tight appr oximation r atio of 2 − w ∗ 1 OP T < 2 − 1 ∆ . 3 Appro ximation algorithm In this section, we firs t pres ent a (1 + w ∗ 1 − w ∗ ∆ OP T )-approximation alg orithm for the MEC problem on trees. Our a lgorithm ro ots the tree in an arbitrary vertex r a nd constructs a so lution a s following: F or e a ch vertex v ∈ V , co nsider the edges to the children of v in non increasing o rder and insert them in to the first matching they fit. Algorithm 1 1. Root the tree on an arbi trary v ertex r ; 2. For each vertex v in p re-ord er d o 3. Sort the childr en e dges of v i n non-incr easing or der, i.e. w ( e v 1 ) ≥ w ( e v 2 ) ≥ . . . ≥ w ( e v d ( v ) ) ; 4. Using this order, insert each edge into the first matchin g that fits; Prop ositio n 2 Algorithm 1 c onstru cts a solution of exactly ∆ matchings. F or the weight, w i of the i -th, 2 ≤ i ≤ ∆ , matching it holds that w i ≤ y i − 1 . Pro of: F or the first part o f the lemma co ns ider first the ro ot vertex o f the tree. It ha s at most ∆ adjac e nt edges which the algorithm ins erts into at most ∆ different matchings. Consider, next, any other vertex v and let e b e the edge b etw een v and its parent. This edge e has b een alre ady inserted by the algo rithm into a match ing, say M k . The r e st, but e , adjacen t to vertex v edg es are at most ∆ − 1 which the alg orithm inserts in to a t most ∆ − 1 matchings different than M k . Therefore, the algor ithm will use exactly ∆ matchings M 1 , M 2 , . . . , M ∆ . W e sha ll pr ov e the seco nd par t o f the lemma by induction on the vertices in the order they ar e pro cessed by the algor ithm. 2 F or the ro ot r , the a lg orithm so r ts all adjacent edges to r a nd inserts e r 1 int o matching M 1 , e r 2 int o matching M 2 , and so on. Thus, after the first iteration it holds that w i = w ( e r i ) ≤ y i ≤ y i − 1 , 2 ≤ i ≤ ∆. Assume that the s tatement of the lemma holds b efore the iter ation pro cessing the vertex v ∈ V , that is w i ≤ y i − 1 , 2 ≤ i ≤ ∆. Consider, now, the iteration in which the a lgorithm pro ce s ses the vertex v . Let e b e the edg e betw een v and its parent, j be the rank of the edge e in E v and M k be the matching wher e the algorithm has alrea dy inser ted edge e . Let us also denote by w ′ i the weight of the matching M i , 2 ≤ i ≤ ∆, after pro ces sing the vertex v . W e distinguish b etw een thr ee cases , and for each one we prov e that w ′ i ≤ y i − 1 , 2 ≤ i ≤ ∆. (i) If k = j , then after this iteration ea ch edge e v i belo ngs to matching M i , 1 ≤ i ≤ d ( v ). By the inductive h yp othesis it follows that w ′ i = max { w i , w ( e v i ) } , where w i ≤ y i − 1 . Since w ( e v i ) ≤ y i , it holds tha t w ′ i ≤ y i − 1 , 2 ≤ i ≤ ∆. (ii) If k > j , then after this iter ation: for 1 ≤ i ≤ j − 1 and k + 1 ≤ i ≤ d ( v ) each edge e v i belo ngs to matching M i ; fo r j + 1 ≤ i ≤ k each edge e v i belo ngs to matching M i − 1 . F or the for mer case we conclude as in Case (i). F or the later case b y the inductiv e hyp o thesis it follows that w ′ i = max { w i , w ( e v i +1 ) } where w i ≤ y i − 1 . Since w ( e v i +1 ) ≤ y i +1 ≤ y i − 1 , it holds that w ′ i ≤ y i − 1 . (iii) If k < j , then after this iter ation: for 1 ≤ i ≤ k − 1 a nd j + 1 ≤ i ≤ d ( v ) each edge e v i belo ngs to matching M i ; fo r k ≤ i ≤ j − 1 ea ch edge e v i belo ngs to matching M i +1 . F or the for mer case we conclude as in Ca se (i). F or the later c a se by the inductive hypo thesis it follows that w ′ i = max { w i , w ( e v i − 1 ) } where w i ≤ y i − 1 . Since w ( e v i − 1 ) ≤ y i − 1 , it holds tha t w ′ i ≤ y i − 1 . Lemma 2 Al gorithm 1 achieves an appr oximation r atio e qu al to 1 + w ∗ 1 − w ∗ ∆ OP T for the MEC pr oblem on t r e es. This is an asymptotic al ly tight 2 appr oximation r atio. Pro of: F o r the w eight of the fir st ma tching o btained by Algorithm 1 it ho lds that w 1 ≤ y 1 = w ∗ 1 , since b oth y 1 and w ∗ 1 are equal to the weight of heaviest edge of the tree. By P rop osition 2 it holds that w i ≤ y i − 1 , 2 ≤ i ≤ ∆ and by P rop osition 1 it holds that y i ≤ w ∗ i , 1 ≤ i ≤ ∆. Therefore, the weight of the solution o btained by Alg orithm 1 is W = ∆ X i =1 w i ≤ y 1 + ∆ X i =2 y i − 1 = y 1 + ∆ − 1 X i =1 y i ≤ w ∗ 1 + ∆ − 1 X i =1 w ∗ i ≤ w ∗ 1 + O P T − w ∗ ∆ , that is W OP T ≤ 1 + w ∗ 1 − w ∗ ∆ OP T < 2 . The counterexample in Figur e 1(a) shows that this is a n asymptotically tight 2 approximation ratio. The weight of an o ptimal solution to this instance is C + 2 ǫ (Figure 1(b)) and the weight of the s olution obtained by Algorithm 1 is 2 C + ǫ (Figure 1(b)). Thus, the approximation ra tio for this instance beco mes 2 C + ǫ C +2 ǫ . ǫ ǫ ǫ ǫ ǫ ǫ C C ǫ ( a ) C C M ∗ 1 ǫ ǫ ǫ M ∗ 2 ǫ ǫ M ∗ 3 ǫ ( b ) M 1 ǫ C ǫ M 2 ǫ ǫ M 3 ( c ) ǫ C ǫ ǫ ǫ Figure 1: (a) A instance of the ME C pr o blem where C >> ǫ . (b) An optimal solution. (c) The solution obtained by Algorithm 1. 3 Next, we com bine Algorithm KK [4] with Algorithm 1 i.e., we r un bo th algorithms and w e select the b est of the tw o solutions found. F or this new a lgorithm the next theo rem ho lds. Theorem 1 Ther e is a t ight 3 2 -appr oximation algorithm for the MEC pr oblems on tr e es. Pro of: Let W b e weight o f the b est of the tw o s olutions found by Algorithm KK a nd Algorithm 1 . By Lemma 1 it ho lds that W OP T ≤ 2 − w ∗ 1 OP T and by L e mma 2 that W OP T ≤ 1 + w ∗ 1 − w ∗ ∆ OP T . As the first b o und is incre asing and the se c ond one is decreasing with r esp ect to O P T , it follows that the ratio W OP T is maximized when 2 − w ∗ 1 OP T = 1 + w ∗ 1 − w ∗ ∆ OP T , that is O P T = 2 · w ∗ 1 − w ∗ ∆ . Therefor e, W OP T ≤ 2 − w ∗ 1 OP T = 2 − w ∗ 1 2 · w ∗ 1 − w ∗ ∆ ≤ 2 − w ∗ 1 2 · w ∗ 1 = 3 2 . F or the tightness of this ratio co ns ider the counterexample shown in Figure 2(a). The weigh t o f an optimal solution to this instance is 2 C + 2 ǫ (Fig ure 2(b)), the weight of the solutio n cr eated by Algorithm 1 is 3 C (Figure 2 (c)) and the weigh t of the solution cr eated by Algorithm KK (Figure 2(d)) is 3 C − ǫ . Our a lgorithm selects the s olution obta ined by Algorithm KK of weight 3 C − ǫ and the approximation ratio for this instance bec o mes 3 C − ǫ 2 C + 2 ǫ . C − ǫ C − ǫ ǫ ǫ ǫ ǫ C C ǫ ( a ) C C C − ǫ M ∗ 1 C C − ǫ C M ∗ 2 ǫ ǫ M ∗ 3 ǫ M ∗ 4 ( b ) M 1 ǫ C C − ǫ M 2 ǫ ǫ M 3 ( c ) C − ǫ C ǫ C C ǫ ǫ C C ǫ C C M 1 C − ǫ C M 2 ǫ M 3 ( d ) ǫ ǫ C ǫ ǫ C − ǫ Figure 2: (a) A instance of the ME C pr o blem where C >> ǫ . (b) An optimal solution. (c) The solution obtained by Algorithm 1. (d) The so lutio n obta ined by Algor ithm KK . References [1] D. de W erra, M. Demange, B. Es coffier, J. Mo nnot, and V. Th. Paschos. W eighted coloring on planar, bipartite and split gr aphs: Complexity and improv ed a pproximation. In 15th Intern a- tional Symp osium on Algori thms and Computation (ISAAC’04 ) , volume 3341 of LNCS , pa g es 896–9 07. Springer , 2004 . [2] M. Demange, D. de W err a, J. Monno t, and V. Th. Pasc hos. W eig hted no de colo ring: When stable sets are ex p ensive. In 28th Workshop on Gr aph-The or etic Conc epts in Computer Scienc e (WG’02) , volume 2573 of LNCS , pages 114– 1 25. Springer, 2002 . 4 [3] B. Esco ffier, J. Monnot, and V. 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