The subset sum problem is one of the simplest and most fundamental NP-hard problems in combinatorial optimization. We consider two extensions of this problem: The subset sum problem with digraph constraint (SSG) and subset sum problem with weak digraph constraint (SSGW). In both problems there is given a digraph with sizes assigned to the vertices. Within SSG we want to find a subset of vertices whose total size does not exceed a given capacity and which contains a vertex if at least one of its predecessors is part of the solution. Within SSGW we want to find a subset of vertices whose total size does not exceed a given capacity and which contains a vertex if all its predecessors are part of the solution. SSG and SSGW have been introduced recently by Gourvès et al. who studied their complexity for directed acyclic graphs and oriented trees. We show that both problems are NP-hard even on oriented co-graphs and minimal series-parallel digraphs. Further, we provide pseudo-polynomial solutions for SSG and SSGW with digraph constraints given by directed co-graphs and series-parallel digraphs.

子集和问题是组合优化中最简单，最基本的NP难题之一。我们考虑此问题的两个扩展：带图约束的子集和问题（SSG）和带弱图约束的子集和问题（SSGW）。在这两个问题中，都给出了具有指定给顶点大小的有向图。在SSG中，我们希望找到一个顶点的子集，其总大小不超过给定的容量，并且如果至少有其前任之一是该解决方案的一部分，则它包含一个顶点。在SSGW内，我们希望找到一个顶点的子集，其总大小不超过给定的容量，并且如果其所有前代都是解决方案的一部分，则包含一个顶点。Gourvès等人最近引入了SSG和SSGW。他们研究了有向无环图和有向树的复杂性。我们表明，即使在定向co-graph和最小级数平行-digraph上，这两个问题都是NP难题。此外，我们为SSG和SSGW提供伪多项式解，并且有向图约束由有向协同图和串并行图赋予。