Complexity of majority monopoly and signed domination problems

被引:5
作者
Mishra, Sounaka [1 ]
机构
[1] Indian Inst Technol Madras, Dept Math, Chennai 600036, Tamil Nadu, India
关键词
Approximation algorithms; Dominating set; Signed domination;
D O I
10.1016/j.jda.2011.12.019
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We consider approximability of two natural variants of classical dominating set problem, namely minimum majority monopoly and minimum signed domination. In the minimum majority monopoly problem, the objective is to find a smallest size subset X subset of V in a given graph G = (V, E) such that |N[v] boolean AND X| >= 1/2 |N[v]|, for at least half of the vertices in V. On the other hand, given a graph G = (V, E), in the signed domination problem one needs to find a function f :V -> {- 1, 1} such that f (N[v]) >= 1, for all v. V, and the cost f (V) = Sigma(v is an element of V) f (v) is minimized. We show that minimum majority monopoly and minimum signed domination cannot be approximated within a factor of (1/2 - is an element of) lnn and (1/3 - is an element of) lnn, respectively, for any is an element of > 0, unless NP subset of Dtime (n(O(loglogn))). We also prove that, if Delta is the maximum degree of a vertex in the graph, then both problems cannot be approximated within a factor of ln Delta - D ln ln Delta, for some constant D, unless P = NP. On the positive side, we give ln(Delta + 1)- factor approximation algorithm for minimum majority monopoly problem for general graphs. We show that minimum majority monopoly problem is APX-complete for graphs with degree at most 3 and at least 2 and minimum signed domination problem is APX-complete, for 3-regular graphs. For 3-regular graphs, these two problems are approximable within a factor of 4/3 (asymptotically) and 1.6, respectively. (C) 2011 Elsevier B.V. All rights reserved.
引用
收藏
页码:49 / 60
页数:12
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