Checking copositivity of a matrix is a co-NP-complete problem. This paper studies copositive matrices with certain spectral properties. It shows that an indefinite matrix with exactly one positive eigenvalue is copositive if and only if the matrix is nonnegative. Moreover, it shows that finding out if a matrix with exactly one negative eigenvalue is strictly copositive or not can be formulated ...

In contrast to the extremal variants of coNP-complete problems, which are frequently DP-complete, many extremal variants of NP-complete problems are in P. We investigate the extremal variants of two NP-complete problems, the extremal colorability problem with restricted degree and the extremal unfrozen non-implicant problem, and show that both of them are DP-complete. As far as we know, no extr...

Recently, some significant work has been done to analyze the complexity class of classic electronic games. In these works, the rules of the game are generalized and then the generalized games are classified by formulating them as a decision problem and reducing them to a problem known to be NP-complete [1, 2]. We use a similar approach to prove the complexity class of a particular class of mode...

Monotone 3-Sat-4 is a variant of the satisfiability problem for boolean formulae in conjunctive normal form. In this variant, each clause contains exactly three literals—either all or none of them are positive, i. e., no clause contains both a positive and a negative literal—and every variable appears at most four times in the formula. Moreover, every clause consists of three distinct literals....

Let M be a real r × c matrix and let k be a positive integer. In the column subset selection problem (CSSP), we need to minimize the quantity M − SA, where A can be an arbitrary k × c matrix, and S runs over all r × k submatrices of M. This problem and its applications in numerical linear algebra are being discussed for several decades, but its algorithmic complexity remained an open issue. We ...

An irreversible k-threshold process is a process on a graph where vertices change color from white to black if they have at least k black neighbors. An irreversible k-conversion set is a subset S of vertices of a graph G such that the irreversible k-threshold process changes all vertices of G to black if S is the initial set of black vertices. We show that deciding the existence of an irreversi...

A fragment of linear time temporal logic (LTL) is presented. It is proved that the satisfiability problem for this fragment is NPcomplete. The fragment is larger than previously known NP-complete fragments. It is obtained by prohibiting the use of until operator and requiring to use only next operators indexed by a letter.

combinatorial optimization is an active field of research in Neural Networks. Since the first attempts to solve the travelling salesman problem with Hopfield nets several progresses have been made. I will present some Neural Network approximate solutions for NP-complete problems that have a sound mathematical foundation and that, beside their theoretical interest, are also numerically encouragi...

We study the computational complexity of quantum discord (a measure of quantum correlation beyond entanglement), and prove that computing quantum discord is NP-complete. Therefore, quantum discord is computationally intractable: the running time of any algorithm for computing quantum discord is believed to grow exponentially with the dimension of the Hilbert space so that computing quantum disc...