Several interior point algorithms have been proposed for solving nonlinear monotone complementarity problems. Some of them have polynomial worst-case complexity but have to connne to short steps, whereas some of the others can take long steps but no polynomial complexity is proven. This paper presents an algorithm which is both long-step and polynomial. In addition, the sequence generated by th...

We first devise moderately exponential exact algorithms for max k-vertex cover, with time-complexity exponential in n but with polynomial space-complexity by developing a branch and reduce method based upon the measure-and-conquer technique. We then prove that, there exists an exact algorithm for max k-vertex cover with complexity bounded above by the maximum among c and γ , for some γ < 2, whe...

We discuss the complexity of computing various graph polynomials of graphs of fixed clique-width. We show that the chromatic polynomial, the matching polynomial and the two-variable interlace polynomial of a graph G of clique-width at most k with n vertices can be computed in time O(n), where f(k) ≤ 3 for the inerlace polynomial, f(k) ≤ 2k + 1 for the matching polynomial and f(k) ≤ 3 · 2 for th...

Despite the impressive progress in the development of algebraic algorithms and computer algebra packages, the inherent computational complexity of even the most basic problems in algebraic geometry is still far from being understood. In [5] a systematic study of the inherent complexity for computing algebraic/topological quantities was launched with the goal of characterizing the complexity of ...

In circuit complexity, the polynomial method is a general approach to proving circuit lower bounds in restricted settings. One shows that functions computed by sufficiently restricted circuits are “correlated” in some way with a low-complexity polynomial, where complexity may be measured by the degree of the polynomial or the number of monomials. Then, results limiting the capabilities of low-c...

We introduce a fast algorithm for determining the linear complexity and the minimal polynomial of a sequence with period p over GF(q) , where p is an odd prime, q is a prime and a primitive root modulo p; and its two generalized algorithms. One is the algorithm for determining the linear complexity and the minimal polynomial of a sequence with period pq over GF(q), the other is the algorithm fo...

The method of partial derivatives is one of the most successful lower bound methods for arithmetic circuits. It uses as a complexity measure the dimension of the span of the partial derivatives of a polynomial. In this paper, we consider this complexity measure as a computational problem: for an input polynomial given as the sum of its nonzero monomials, what is the complexity of computing the ...

We investigate the complexity of 1) computing the characteristic polynomial, the minimal polynomial, and all the invariant factors of an integer matrix, and of 2) verifying them, when the coefficients are given as input. It is known that each coefficient of the characteristic polynomial of a matrix A is computable in GapL, and the constant term, the determinant of A, is complete for GapL. We sh...