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In this paper we study the 0-1 inverse maximum stable set problem, denoted by IS"{"0","1"}. Given a graph and a fixed stable set, it is to delete the minimum number of vertices to make this stable set maximum in the new graph. We also consider IS"{"0","1"} against a specific algorithm such as Greedy and 2opt, aiming to delete the minimum number of vertices so that the algorithm selects the given stable set in the new graph; we denote them by IS"{"0","1"}","g"r"e"e"d"y and IS"{"0","1"}","2"o"p"t, respectively. Firstly, we show that they are NP-hard, even if the fixed stable set contains only one vertex. Secondly, we achieve an approximation ratio of 2-@Q(1log@D) for IS"{"0","1"}","2"o"p"t. Thirdly, we study the tractability of IS"{"0","1"} for some classes of perfect graphs such as comparability, co-comparability and chordal graphs. Finally, we compare the hardness of IS"{"0","1"} and IS"{"0","1"}","2"o"p"t for some other classes of graphs.