Béla Vizvári

An application of gomory cuts in number theory

AbstractFrobenius has stated the following problem. Suppose thata1, a2, ⋯, an are given positive integers and g.c.d. (a1, ⋯ , an) = 1. The problem is to determine the greatest positive integerg so that the equation

On Pleasant Knapsack Problems

\sum\limits_{i = 1}^n {a_i x_i = g}

New upper bounds on the probability of events based on graph structures

has no nonnegative integer solution. Showing the interrelation of the original problem and discrete optimization we give lower bounds for this number using Gomory cuts which are tools for solving discrete programming problems.In the first section an important theorem is cited after some remarks. In Section 2 we state a parametric knapsack problem. The Frobenius problem is equivalent with finding the value of the parameter where the optimal objective function value is maximal. The basis of this reformulation is the above mentioned theorem. Gomorys cutting plane method is applied for the knapsack problem in Section 3. Only one cut is generated and we make one dual simplex step after cutting the linear programming optimum of the knapsack problem. Applying this result we gain lower bounds for the Frobenius problem in Section 4. In the last section we show that the bounds are sharp in the sense that there are examples with arbitrary many coefficients where the lower bounds and the exact solution of the Frobenius problem coincide.

Application of decision support methods in preparation for a Hungarian bio-fuel programme

Leta1,a2, ...,an be relative prime positive integers. The Frobenius problem is to determine the greatest integer not belonging to the set {Σj=1najxj :x∈Z+n}. The Frobenius problem belongs to the combinatorial number theory, which is very rich in methods. In this paper the Frobenius problem is handled by integer programming which is a new tool in this field. Some new upper bounds and exact solutions of subproblems are provided. A lot of earlier results obtained with very different methods can be discussed in a unified way.

ON THE ESTIMATION OF THE SUPPLY FUNCTION OF THE HUNGARIAN PORK MARKET

In this paper the following knapsack problems will be considered

VERSENYKÉPES ÉLELMISZERGAZDASÁG – ÉLHETŐ VIDÉK (Négy tézis egy lehetséges fejlesztési politika körvonalainak meghatározásához)

\max (\min )\{ \sum\nolimits_{j = 1}^n {cjx} j:\sum\nolimits_{j = 1}^n {ajxj} = b,x \in z_ + ^n\}