Lower Bounds on the Size of General Branch-and-Bound Trees

A Theoretical and Computational Analysis of Full Strong-Branching

Full strong-branching is a well-known variable selection rule that is known experimentally to produce significantly smaller branch-and-bound trees in comparison to all other known variable selection rules. In this paper, we attempt an analysis of the performance of the strong-branching rule both from a theoretical and a computational perspective. On the positive side for strong-branching we identify vertex cover as a class of instances where this rule provably works well. In particular, for vertex cover we present an upper bound on the size of the branch-and-bound tree using strong-branching a…

Lower bound on size of branch-and-bound trees for solving lot-sizing problem

We show that there exists a family of instances of the lot-sizing problem, such that any branch-and-bound tree that solves them requires an exponential number of nodes, even in the case when the branchings are performed on general split disjunctions.

Learning to branch with Tree MDPs

State-of-the-art Mixed Integer Linear Program (MILP) solvers combine systematic tree search with a plethora of hard-coded heuristics, such as the branching rule. The idea of learning branching rules from data has received increasing attention recently, and promising results have been obtained by learning fast approximations of the strong branching expert. In this work, we instead propose to learn branching rules from scratch via Reinforcement Learning (RL). We revisit the work of Etheve et al. (2020) and propose tree Markov Decision Processes, or tree MDPs, a generalization of temporal MDPs th…

压缩分支定界树

Compressing Branch-and-Bound Trees

本文提出分支定界树压缩问题 TCP，分析其复杂性与局限，并用精确及启发式算法评估常用分支策略所生成树的可压缩性。

Shaping and Trimming Branch-and-bound Trees

We present a new branch-and-bound type search method for mixed integer linear optimization problems based on the concept of offshoots (introduced in this paper). While similar to a classic branch-and-bound method, it allows for changing the order of the variables in a dive (shaping) and removing unnecessary branching variables from a dive (trimming). The regular branch-and-bound algorithm can be seen as a special case of our new method. We also discuss extensions to our new method such as choosing to branch from the top or the bottom of an offshoot. We present several numerical experiments to …

Matrices of optimal tree-depth and a row-invariant parameterized algorithm for integer programming

A long line of research on fixed parameter tractability of integer programming culminated with showing that integer programs with n variables and a constraint matrix with dual tree-depth d and largest entry D are solvable in time g(d,D)poly(n) for some function g. However, the dual tree-depth of a constraint matrix is not preserved by row operations, i.e., a given integer program can be equivalent to another with a smaller dual tree-depth, and thus does not reflect its geometric structure. We prove that the minimum dual tree-depth of a row-equivalent matrix is equal to the branch-depth of the …