In this paper, we describe an algorithm for solving systems of linear Diophantine equations based on a generalization of an algorithm for solving one equation due to Fortenbacher [3]. It can solve a system as a whole, or be used incrementally when the system is a sequential accumulation of several subsystems. The proof of termination of the algorithm is difficult, whereas the proofs of completeness and correctness are straightforward generalizations of Fortenbacher 's proof. 1 Introduction Linear Diophantine equations occur frequently in mathematics and computer science, namely in the decision procedure of the accessibility property for Petri-Nets, in associative-commutative unification [1], in constrained logic programming, in the vectorization of FORTRAN programs, etc. By definition, a Diophantine equation has the form Q = 0, where Q is a 1 polynomial with integer coefficients. The equation is homogeneous if Q has no constant and linear if all monomial have the form kx i where k ...

The problem of quickly determining whether a given large integer is prime or composite has been of interest for centuries, if not longer. The past 30 years has seen a great deal of progress, leading up to the recent deterministic, polynomial-time algorithm of Agrawal, Kayal, and Saxena [2]. This new “AKS test ” for the primality of n involves verifying the

We show that generating all negative cycles of a weighted graph is a hard enumeration problem, in both the directed and undirected cases. More precisely, given a family of negative (directed) cycles, it is an NP-complete problem to decide whether this family can be extended or there are no other negative (directed) cycles in the graph, implying that (directed) negative cycles cannot be generated in polynomial output time, unless P = NP. As a corollary, we solve in the negative two well-known generating problems from linear programming: (i) Given an infeasible system of linear inequalities, generating all minimal infeasible subsystems is hard. Yet, for generating maximal feasible subsystems the complexity remains open. (ii) Given a feasible system of linear inequalities, generating all vertices of the corresponding polyhedron is hard. Yet, in the case of bounded polyhedra the complexity remains

We present some new Farkas-type results for inequality systems involving a finite as well as an infinite number of convex constraints. For this, we use two kinds of conjugate dual problems, namely an extended Fenchel-type dual problem and the recently introduced Fenchel-Lagrange dual problem. For the latter, which is a ”combination” of the classical Fenchel and Lagrange duals, the strong duality is established.

ly, a constraint is a relation of the form a 1 v 1 + · · · +a n v n k, where v 1 , . . . , v n are variables, a 1 , . . . , a n # Q , k # Z, and is either #, #, or =. Constraints are represented in several different ways, according to what variables and coefficients are allowed: 1. When constraints appear in commands, the variables are arbitrary (within the confines of the language), and the coefficients are in Z. 2. These constraints are internally represented (using the "constraint" class) in the same way, except that Q -coefficients are allowed. This is because some internal operations may result in constraints whose coefficients are not integers. (This can happen when the "incorporate" command is used.) 3. For split linear programming calculations, constraints are first converted to constraints involving v variables and having coefficients in Q (class qvconstraint). 4. Clearing denominators yields constraints having v variables and coefficients in Z (class vconstra...

In this paper, we discuss the decision situation where the (vague) preferences are represented by fuzzy relations. There are two ways to interpret this situation: (i) an individual decision maker chooses perfect rankings from a set and uses that combination of rankings as her/his preference ranking. (ii) or a group of decision makers are reporting perfect rankings but their group decision is a combination of individual rankings. Rationality of decisions is measured by the degree they match (explain) observed choice behaviour. We work with this well-established notion of rationality and extend it to situations where the observed choice behaviour is partial and incomplete. We derive a condition which, if satisfied, rationalizes the observed choice behaviour when the underlying preferences are vague. 1 Introduction We consider the decision making situation in which a decision maker has to choose an alternative based on her/his preferences. In general, the preferences are represented as a...

In this paper we investigate the class NP " co-NP (or the class of problems permitting a good characterisation) from the point of view of morphisms of oriented matroids. We prove several morphism-duality theorems for oriented matroids. These generalize LP-duality (in form of Farkas' Lemma) and Minty's Painting Lemma. Moreover, we characterize all morphism duality theorems, thus proving the essential unicity of Farkas' Lemma. This research helped to isolate perhaps the most natural definition of strong maps for oriented matroids.

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In this paper we give an elementary introduction into the theory of variational inequalities and their application to general equilibrium analysis. Furthermore we incorporate some results on sensitivity analysis of variational inequalities on polyhedral sets.

The first purpose of this paper is to tell the history of John von Neumann’s development of the minimax theorem for two-person zero-sum games from his first proof of the theorem in 1928 until 1944 when he gave a completely different proof in the first coherent book on game theory. I will argue that von Neumann’s conception of this theorem