Shortest Paths with Bundles and Non-additive Weights Is Hard

In a standard path auction, all of the edges in a graph are sold as separate entities, each edge having a single cost. We consider a generalisation in which a graph is partitioned and each subset of edges has a unique owner. We show that if the owner is allowed to apply a non-additive pricing structure then the winner determination problem becomes NP-hard (in contrast with the quadratic time algorithm for the standard additive pricing model). We show that this holds even if the owners have subsets of only 2 edges. For subadditive pricing (e.g. volume discounts), there is a trivial approximation ratio of the size of the largest subset. Where the size of the subsets is unbounded then we show that approximation to within a Ω(log n) factor is hard. For the superadditive case we show that approximation with a factor of n for any > 0 is hard even when the subsets are of size at most 2.