This paper is concerned with factor left prime factorization problems for multivariate polynomial matrices without full row rank. We propose a necessary and sufficient condition the existence of factorizations class matrices, then design an algorithm to compute all if they exist. implement on computer algebra system Maple, two examples are given illustrate effectiveness algorithm. The results p...

We present a first theoretical analysis of the power of polynomial-time preprocessing for important combinatorial problems from various areas in AI. We consider problems from Constraint Satisfaction, Global Constraints, Satisfiability, Nonmonotonic and Bayesian Reasoning. We show that, subject to a complexity theoretic assumption, none of the considered problems can be reduced by polynomial-tim...

Abstract We show that adjustable robust linear programs with affinely box data uncertainties under separable polynomial decision rules admit exact sums of squares (SOS) reformulations. These problems share the same optimal values and a one-to-one correspondence between solutions. A sum representation non-negativity non-convex over plays key role in reformulation. This reformulation allows us to...

The class UP of ‘ultimate polynomial time’ problems over C is introduced; it contains the class P of polynomial time problems over C. The τ -Conjecture for polynomials implies that UP does not contain the class of non-deterministic polynomial time problems definable without constants over C. This latest statement implies that P 6 = NP over C. A notion of ‘ultimate complexity’ of a problem is su...

In this paper, we study the relationship between the approximability and the parameterized complexity of NP optimization problems. We introduce a notion of polynomial fixed-parameter tractability and prove that, under a very general constraint, an NP optimization problem has a fully polynomial time approximation scheme if and only if the problem is polynomial fixed-parameter tractable. By enfor...

In this paper, we develop necessary conditions for global optimality that apply to non-linear programming problems with polynomial constraints which cover a broad range of optimization problems that arise in applications of continuous as well as discrete optimization. In particular, we show that our optimality conditions readily apply to problems where the objective function is the difference o...