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We propose a simple and general online method to measure the search progress within the branch-and-bound algorithm, from which we estimate the size of the remaining search tree. We then show how this information can help solvers algorithmically at runtime by designing a restart strategy for Mixed-Integer Programming (MIP) solvers that decides whether to restart the search based on the current estimate of the number of remaining nodes in the tree. We refer to this type of algorithm as clairvoyant. Our clairvoyant restart strategy outperforms a state-of-the-art solver on a large set of publicly available MIP benchmark instances. It is implemented in the MIP solver SCIP and will be available in future releases.

This paper investigates the estimation of 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 equals NP. Second, we review measures of the progress of the B&B search, such as the gap, 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 progress measures and their trends into nontrivial estimates using Machine Learning techniques, which yields more precise estimates than any individual measure. The best method 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 available MIPLIB 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 2 over the entire search on each out of six additional sets of homogeneous instances we have tested. All techniques are available in version 7 of the branch-and-cut framework SCIP.

We propose a simple and general online method to measure the search progress within the Branch-and-Bound algorithm, from which we estimate the size of the remaining search tree. We then show how this information can help solvers algorithmically at runtime by designing a restart strategy for Mixed-Integer Programming (MIP) solvers that decides whether to restart the search based on the current estimate of the number of remaining nodes in the tree. We refer to this type of algorithm as clairvoyant.
Our clairvoyant restart strategy outperforms a state-of-the-art solver on a large set of publicly available MIP benchmark instances.
It is implemented in the MIP solver SCIP and will be available in future releases.

The SCIP Optimization Suite provides a collection of software packages for
mathematical optimization centered around the constraint integer programming frame-
work SCIP. This paper discusses enhancements and extensions contained in version 7.0
of the SCIP Optimization Suite. The new version features the parallel presolving library
PaPILO as a new addition to the suite. PaPILO 1.0 simplifies mixed-integer linear op-
timization problems and can be used stand-alone or integrated into SCIP via a presolver
plugin. SCIP 7.0 provides additional support for decomposition algorithms. Besides im-
provements in the Benders’ decomposition solver of SCIP, user-defined decomposition
structures can be read, which are used by the automated Benders’ decomposition solver
and two primal heuristics. Additionally, SCIP 7.0 comes with a tree size estimation
that is used to predict the completion of the overall solving process and potentially
trigger restarts. Moreover, substantial performance improvements of the MIP core were
achieved by new developments in presolving, primal heuristics, branching rules, conflict
analysis, and symmetry handling. Last, not least, the report presents updates to other
components and extensions of the SCIP Optimization Suite, in particular, the LP solver
SoPlex and the mixed-integer semidefinite programming solver SCIP-SDP.