Note on Structural Properties and Sizes of Eigenspaces of Min-max Functions

In this paper, we study relationships among structural properties of min-max functions and complexity on deciding the sizes of the eigenspaces of minmax functions. We show that strong connectivity implies inseparability; Olsder’s separated systems are separable. A relaxed condition under which inseparability remains equivalent to the balance condition is also presented. The decision problem of whether the size of the eigenspace of a min-max function is greater than one is then report to be NP-hard based on the connection between the separability of a min-max function and the size of the eigenspace of its skeleton.