Computing Partitions with Applications to Capital Budgeting Problems

Abstract

We consider the following capital budgeting problem. A firm is given a set of investment opportunities $X=\{x_1,\ldots ,x_n\}$ and a number m of portfolios. Every investment $x_i$, $1\le i\le n$, has a return of $r_i$ and a price of $p_{i}$. Further for every portfolio j there is capacity $c_j$. The task is to choose m disjoint portfolios $X'_1,\ldots , X'_m$ from X such that for every $1\le j\le m$ the prices in $X'_j$ do not exceed the capacity $c_j$ and the total return of this selection is maximized. From a computational point of view this problem is intractable, even for $m=1$ [8]. Since the problem is defined on inputs of various informations, in this paper we consider the fixed-parameter tractability for several parameterized versions of the problem. For a lot of small parameter values we obtain efficient solutions for the partitioning capital budgeting problem. We also consider the connection to pseudo-polynomial algorithms.