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Graph of number of issues dominating set

Us ∈ c, has a x'Sfu. Problem 2 for the ordinary integer c, find the smallest in x in a subset of x makes Us ∈ s-u satisfying: For any element P ∈ u, x, at least in a C +1 S contains e. Question 3 in the X,[link widoczny dla zalogowanych], find the smallest subset of X makes the USI ~ X'S-U and satisfies: for any element X in S,[link widoczny dla zalogowanych], in X, there s an element for the s,[link widoczny dla zalogowanych],, so SnSJ ≠. Approximation algorithm for the convenience of the back of the approximate analysis, we give the (u, X) multi-definition set covering optimization problem: find the smallest in the X,[link widoczny dla zalogowanych], a subset of X makes a 【UsS. Alternatively, you can see the problem 2 is the multiple expansion set covering optimization problem that is multi-set multi-cover optimization problem. If constructed for each point u set S = {\Set the optimization problem can be transformed into questions 1 and 3 of the optimization problem is easy to see questions 1 and 2, these two problems are equivalent. irani for Question 2 is a multiple set of multiple coverage optimization in [3 ] gives the approximation ratio for this problem is 1 + lna where (a = maxl <i <ISI) approximation algorithm. Therefore, a collection of multiple coverage based on multi-c strongly dominating set approximation algorithm optimization for approximate degree 1 + lna. As for question 3, we give a similar problem with the set covering greedy algorithm, the algorithm is described as follows: GREEDY (U.X) 1if (for any element x in s, both in x there is a not for S elements, we have to Si ¨ Sj ≠) 2thengoto3elsereturnfalse3Zq, 4X +-5whileZ ≠ 6 in X can be selected one of the largest element 『snzI s7Z + Z-S ・】 86 ・ l5endwhile16return. GREEDY algorithm is easy to see that a number of Type time to complete. Now to analyze the degree of approximation algorithms. that the optimal solution of Problem 3 is OPT, GREEDY algorithm to the solution by the end of the size of the t. lie in an instance (u, x) on the multi-set cover the optimal solution is the OP, GREEDY algorithm in step 9 X generated by the end of the size of the t. 'According to finally get GREEDY algorithm is that the solution x has two parts; first, 5 to 9 in the algorithm Step-generated; followed by 15 in the algorithm generated in Step 11. According to Section l1 ~ 15-step algorithm description we can see the number of elements in the second part of generated no more than the first part (as in the first part of the generated each element of the algorithm 11 and 12, according to the description of steps up to a new element will be generated), so easy that ? ≤ 2t (1) under similar [4] in the set cover approximation algorithm analysis, ? ≤ (Ina +1) OP (2) itself is also set covering the OPT, it is OP7 \1) OlPT, so GREEDY algorithm approximation ratio for the 2 (1na +1). Conclusion In this paper, the two graph dominating set problem has applications in the deformation of the background research. on how to get a better approximation ratio and lower bounds require further research.
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