K. ?. , L. ?. , and H. }. Now, Observe that there exist O(n2 k ) values of the form P, order to compute value P [v, K ? ], we have to check at most O(nk) other values P [u, K ?? ]. Hence the time complexity is O(n 2 k2 k )

{. H. {1-f, F. }. , and H. }. , Consider two different perfect families of hash functions: a family H from M to , k H }, as we have previously introduced in this section, and a family F from the set V to By the property of the family of perfect hash functions, we know that there is a function f ? F such that the vertices of G that belong to a solution of size k are associated to distinct labels of , k F }. Similarly, we know that there is a function h ? H such that the occurrences of colors of M that belong to an optimal solution, are associated to different labels of Observe that each family of perfect hash functions consists of O(log n) 2 O(k) functions. Hence, we can combine all the possible pairs (f, h) of functions, with f ? F and h ? H, in O(log 2 n) 4 O(k) time. Recall that, for each color c i ? M, S H (c i ) denotes the set of labels associated to occurrences of color c i by function h, and that, given that v is colored c i , S H (v) = S(c i ). Now, for each v ? V and for each subset, define M L (v) as the family of all sets of labels H ? ? {1 H , . . . , k H } such that there exists an occurrence V ? , with v ? V ? , where the set of labels in {1 F , . . . , k F } that f assigns to V ? is exactly L and such that C(S H , V ? , H ? ) is feasible. Now, we present a method called the Batch procedure for computing M L (v), similar to that introduced in [8, 12]. Assume that we have computed the family of sets M L ? (v), with L ? ? L \ f (v), we apply the following procedure. Batch Procedure(L, v)

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