In our paper, we consider the following general problems: check feasibility, count the number of feasible solutions, find an optimal solution, and count the number of optimal solutions in $P \cap Z^n$, assuming that $P$ is a polyhedron, defined by systems $A x \leq b$ or $Ax = b,\, x \geq 0$ with a sparse matrix $A$. We develop algorithms for these problems that outperform state of the art ILP and counting algorithms on sparse instances with bounded elements. We use known and new methods to develop new exponential algorithms for Edge/Vertex Multi-Packing/Multi-Cover Problems on graphs and hypergraphs. This framework consists of many different problems, such as the Stable Multi-set, Vertex Multi-cover, Dominating Multi-set, Set Multi-cover, Multi-set Multi-cover, and Hypergraph Multi-matching problems, which are natural generalizations of the standard Stable Set, Vertex Cover, Dominating Set, Set Cover, and Maximal Matching problems.
翻译:在我们的论文中,我们考虑了以下一般性问题:检查可行性、计算可行解决方案的数量、找到最佳解决方案,并以$P =cap =n$计算最佳解决方案的数量,假设美元是多元面值,由系统 $A x\leq b$或$Ax = b,\, x\geq 0$ 定义为稀薄的矩阵 $A 。我们为这些问题开发了优于艺术 ILP 状态的算法,并在有约束元素的稀有实例上计算算法。我们使用已知的和新的方法来为 Edge/ Vertex 多包/多包/多包问题开发新的指数算法。这个框架由许多不同的问题组成,如Stagable 多重设置、 Vertex 多重覆盖、 支配性多重设置、 多重覆盖、 超版多重匹配问题, 这些问题是标准的Staget、 Vertex Cover、 Doming Set、 Set Cover 和 Maximal Matching 等的自然概括问题。