分支定界法求解Polytime中的随机二元IP问题
Branch-and-Bound Solves Random Binary IPs in Polytime
摘要：
分支定界是所有最先进的混合整数线性规划（MILP）求解器的主力。分支定界的这些实现通常使用变量分支，也就是说，通过将某个变量在一个节点中固定为整数值$v$并在另一个节点中固定为$v + 1$来获得子节点。尽管现代MILP求解器能够有效地解决非常大规模的实例，但相对较少注意理解为什么底层的分支定界算法表现如此出色。在本文中，我们的目标是从理论上分析的标准变量分支的分支定界算法的性能。为了避免指数最坏情况下的下界，我们遵循考虑随机实例的常见思想。更确切地说，我们考虑随机整数规划的系数矩阵和目标函数的条目是随机抽样的。我们的主要结果是，具有良好的概率分支定界与变量分支探索只有多项式数量的节点来解决这些情况下，为固定数量的约束。据我们所知，这是第一个已知的标准版本的分支定界这样的结果。我们相信，这个结果提供了一个令人信服的指示，为什么分支和绑定与变量分支在实践中工作得如此之好。
英文摘要：
Branch-and-bound is the workhorse of all state-of-the-art mixed integer linear programming (MILP) solvers. These implementations of branch-and-bound typically use variable branching, that is, the child nodes are obtained by fixing some variable to an integer value $v$ in one node and to $v + 1$ in the other node. Even though modern MILP solvers are able to solve very large-scale instances efficiently, relatively little attention has been given to understanding why the underlying branch-and-bound algorithm performs so well. In this paper our goal is to theoretically analyze the performance of the standard variable branching based branch-and-bound algorithm. In order to avoid the exponential worst-case lower bounds, we follow the common idea of considering random instances. More precisely, we consider random integer programs where the entries of the coefficient matrix and the objective function are randomly sampled. Our main result is that with good probability branch-and-bound with variable branching explores only a polynomial number of nodes to solve these instances, for a fixed number of constraints. To the best of our knowledge this is the first known such result for a standard version of branch-and-bound. We believe that this result provides a compelling indication of why branch-and-bound with variable branching works so well in practice.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
