We study the following decision problem: given an exponential equation $a_1g_1^{x_1}a_2g_2^{x_2}\dots a_ng_n^{x_n}=1$ over a recursively presented group G, decide if it has a solution with all $x_i$ in $\mathbb {Z}$. We construct a finitely presented group G where this problem is decidable for equations with one variable and is undecidable for equations with two variables. We also study functions estimating possible solutions of such an equation through the lengths of its coefficients with respect to a given generating set of G. Another result concerns Turing degrees of some natural fragments of the above problem.

The Bilevel Knapsack Problem (BKP) is a hierarchical optimization problem in which thefeasible set is determined by the set of optimal solutions of parametric Knapsack Problem.In this paper, we propose two stages exact method for solving the BKP. In the first stage,a dynamic programming algorithm is used to compute the set of reactions of the follower.The second stage consists in solving an integer program reformulation of BKP. We show thatthe integer program reformulation is equivalent to the BKP. Numerical results show theefficiency of our method compared with those obtained by the algorithm of Moore andBard