Subset sum problems with digraph constraints

Abstract

We introduce and study four optimization problems that generalize the well-known subset sum problem. Given a node-weighted digraph, select a subset of vertices whose total weight does not exceed a given budget. Some additional constraints need to be satisfied. The (weak resp.) digraph constraint imposes that if (all incoming nodes of resp.) a node x belongs to the solution, then the latter comprises all its outgoing nodes (node x itself resp.). The maximality constraint ensures that a solution cannot be extended without violating the budget or the (weak) digraph constraint. We study the complexity of these problems and we present some approximation results according to the type of digraph given in input, e.g. directed acyclic graphs and oriented trees.

Approximation Schemes for Subset Sum Ratio Problems

“…In [8] there were introduced digraph constraints for the subset sum problem which can easily be modeled via our framework. Generally we are able demand as constraints of S 1 and S 2 to be a specific property considering the vertices of the graph, for example we may demand that the solution consists of independent sets or dominant sets etc.…”