Subset Sum Problems With Digraph Constraints

TL;DR

This paper explores generalized subset sum problems with digraph constraints, analyzing their complexity and approximation solutions across different graph types such as DAGs and trees.

Contribution

It introduces new subset sum variants with digraph constraints, studies their computational complexity, and provides approximation algorithms for specific graph classes.

Findings

Complexity results vary with graph type.

Approximation schemes like PTAS are developed for certain cases.

New problem formulations extend classical subset sum 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. Key words. Subset Sum, Maximal problems, digraph constraints, complexity, directed acyclic graphs, oriented trees, PTAS.