A fundamental challenge in the development of defence capability is to decide on a collection of projects that represent the best value within a given budget constraint. A complicating factor in doing this is taking account of the inter-relationships between projects when assessing value. We investigate two such models. The first assigns value based on subsets of projects that come together to provide effects, the second assigns value through the intermediary of scenarios. In terms of recognized combinatorial optimization problems the first is a form of the Set Union Knapsack problem while the second appears to be a new problem we call the Budget Scenario problem. We analyse the known results about the algorithmic complexity of these problems by showing their relationships with existing problems and known approximation and inapproximability results. We also provide new approximation results for both problems. The main results of this paper are summarised as follows. When prospective projects mature into operational capabilities they typically come together in subsets to provide joint effects. Thus capabilities that work together and depend upon eachother provide extra value than that provided by each capability in isolation. It seems natural then to balance the cost of a collection of projects against the value provided by the subsets in the collection. This problem is a form of the Set Union Knapsack Problem which generalises a number of well known combinatorial problems in two broad classes. The Knapsack, Subset-Sum and Partition problems have a polynomial time approximation scheme PTAS (see Garey and Johnson (1979) and Vazirani (2003)). That is for any given epsilon > 0 there is an algorithms that approximates Knapsack to within 1 - epsilon of the optimal with a run time that bounded by a polynomial of the input size. Further it has a fully polynomial time approximation scheme FPTAS with an approximation algorithm with run time bounded by a polynomial of both the input size and 1/epsilon. On the other hand Quadratic Knapsack, Weighted Clique, maximum Edge Weighted Clique, and Densest k-Subgraph does not admit a PTAS assuming that random 3-SAT formulas are hard to refute Feige (2002), or if NP does not have randomized algorithms that run in sub-exponential time Knot (2004). On the positive side there is an approximation algorithms for DkS that is within the ratio O (n(1/4 + epsilon)) of the optimal and runs in time n O-(1/epsilon) Bhaskara et al. (2010). We show that the The Set Union Knapsack Problem with subsets of size at most m has an approximation algorithm with run time of at most O (n(m +) (2)/epsilon) with an approximation ratio of at least m!(1 - epsilon)/n(m - 1) In the Budget Scenario problem a list of initiatives is provided each with an anticipated cost. Each initiative is scored against a number of scenarios with a value indicating how useful the initiative is against that scenario. For a collection of initiatives the total value is calculated by summing the best value obtained by any initiative for that scenario in the collection. This initiative can be thought of as the best tool in the toolbox (collection) for the particular job (scenario), while the total value reflects the ability of the toolbox (initiative collection) to address any single job (scenario). The Budget Scenario problem is shown to generalise the established problems of Budgeted Maximum Coverage, Maximum Coverage, Weighted Set Cover and Set Cover. In Khuller, Moss and Naor (1999) an approximation algorithm for Budgeted Maximum Coverage is presented and shown to provide an approximation factor of (1 - 1/e) of the optimal solution. On the other hand all of the problems in Figure 2 are NP-hard. Also Khuller, Moss and Naor (1999) and Feige (1996) show that if Maximum Coverage, respectively Set Cover is approximable within a factor of (1 - epsilon) log n for any epsilon > 0 then N P subset of D T I M E (n(loglogn)). It follows that the same must be true of Budget Scenario, Budgeted Maximum Coverage, and Weighted Set Cover. We prove that for any given epsilon > 0 the Budget Scenario Problem G (X; Y) has a polynomial approximation algorithm that achieves a factor of (1 - 1/e - epsilon) of the optimal solution.
