具有特殊有向图约束的子集和问题的解
Solutions for Subset Sum Problems with Special Digraph Constraints
摘要：
子集和问题是组合优化中最简单、最基本的NP-难问题之一。我们考虑了这个问题的两个扩展：有向图约束的子集和问题（SSG）和弱有向图约束的子集和问题（SSGW）。在这两个问题中，都给出了一个有向图，其顶点具有指定的大小。在SSG中，我们希望找到一个顶点子集，其总大小不超过给定的容量，并且如果至少有一个其前任是解决方案的一部分，则包含一个顶点。在SSGW中，我们希望找到一个顶点子集，其总大小不超过给定的容量，并且如果其所有前任都是解决方案的一部分，则包含一个顶点。SSG和SSGW是最近由Gourves等人提出的，他们研究了有向无环图和有向树的复杂性。我们证明了这两个问题是NP-困难的，即使在有向余图和最小串并行有向图。此外，我们提供了伪多项式的解决方案，SSG和SSGW的有向图的限制，给出了有向图和串并行有向图。
英文摘要：
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 Gourves 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.
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