The cardinality and precedence constrained maximum value sub-hypergraph problem and its applications Original Research Article

Given a hypergraph with values on the hyperedges, the problem of finding a subhypergraph of maximum value subject to cardinality and precedence constraints on the nodes has applications ranging from partitioning fields in a database, to loading tools on a machine in a flexible manufacturing environment, to investment selection. For any instance of such a problem, consider the corresponding undirected graph G′ = (V, E′) with E′ containing all pairs of nodes that either have a direct precedence relationship or are common to at least one hyperedge. Let n = ¦V¦, and let s be the number of connected components of G′. We present an exact, O(n2 log n) dynamic programming based algorithm for the case where G′ is a forest (¦E′¦ = n − s). By extending the result, we derive an exact, polynomial-time algorithm for cases where ¦E′¦ ⩽ n − s + α log n, for any constant α. These algorithms significantly improved the complexity and enlarged the application scope of the best existing algorithms. Finally, we show that the case where ¦E′¦ = O(n − s + nε) is NP-hard, even when G′ is connected and bipartite and the hyperedges all have unit value.