Estimating the size of branch-and-bound trees

This paper investigates the problem of estimating the size of branch-and-bound (B&B) trees for solving mixed-integer programs. We first prove that the size of the B&B tree cannot be approximated within a factor of 2 for general binary programs, unless P =NP. Second, we review measures of progress of the B&B search, such as the wellknown gap and the often-overlooked tree weight, and propose a new measure, which we call leaf frequency. We study two simple ways to transform these progress measures into B&B tree-size estimates, either as a direct projection or via double-exponential smoothing, a standard time-series forecasting technique.We then combine different progressmeasures and their trends into nontrivial estimates using machine learning techniques, which yield more precise estimates than any individual measure. The best method that we have identified uses all individual measures as features of a random forest model. In a large computational study, we train and validate all methods on the publicly availableMIPLIB and Coral general purpose benchmark sets. On average, the best method estimates B&B tree sizes within a factor of 3 on the set of unseen test instances, even during the early stage of the search, and improves in accuracy as the search progresses. It also achieves a factor of 2 over the entire search on each of the six additional sets of homogeneous instances that we tested. All techniques are available in version 7 of the branch-and-cut framework SCIP. Summary of Contribution: This manuscript develops a method for online estimation of the size of branch-and-bound trees, thereby combining methods of mixed-integer programming and machine learning. We show that high-quality estimations can be obtained using the presented techniques. The methods are also useful in everyday use of branchand- bound algorithms to obtain approximate search-completion information. The manuscript is accompanied by an extensive online supplement comprising the code used for our simulations and an implementation of all discussed methods in the academic solver SCIP, together with the tools and instructions to train estimators for custominstance sets.