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，我们考虑以下一般性问题：检验可行性、计算可行解的数量、寻找最优解以及计算$P \cap Z^n$中最优解的数量，其中$P$是一个多面体，由系统$A x \leq b$或$Ax = b,\, x \geq 0$定义，且矩阵$A$是稀疏的。我们针对这些问题开发了算法，这些算法在元素有界的稀疏实例上优于最先进的整数线性规划（ILP）和计数算法。我们利用已知和新的方法，为图和超图上的边/顶点多重打包/多重覆盖问题设计了新的指数时间算法。该框架包含许多不同的问题，例如稳定多重集、顶点多重覆盖、支配多重集、集合多重覆盖、多重集多重覆盖以及超图多重匹配问题，这些是标准稳定集、顶点覆盖、支配集、集合覆盖和最大匹配问题的自然推广。