An abstract model for branch and cut

Abstract

Branch and cut is the dominant paradigm for solving a wide range of mathematical programming problems—linear or nonlinear—combining efficient search (via branch and bound) and relaxation-tightening procedures (via cutting planes, or cuts). While there is a wealth of computational experience behind existing cutting strategies, there is simultaneously a relative lack of theoretical explanations for these choices, and for the tradeoffs involved therein. Recent papers have explored abstract models for branching and for comparing cuts with branch and bound. However, to model practice, it is crucial to understand the impact of jointly considering branching and cutting decisions. In this paper, we provide a framework for analyzing how cuts affect the size of branch-and-cut trees, as well as their impact on solution time. Our abstract model captures some of the key characteristics of real-world phenomena in branch-and-cut experiments, regarding whether to generate cuts only at the root or throughout the tree, how many rounds of cuts to add before starting to branch, and why cuts seem to exhibit nonmonotonic effects on the solution process.

Appendices

Computational results with selected MIPLIB instances

Figure 6 shows the linear relaxation bound, predicted bound (using the harmonically-worsening cuts model of Sect. 5.2), linear relaxation resolve time, and cumulative number of Gomory cuts added after up to 100 rounds of cuts have been applied to ten additional instances, using the same computational setup described in Sect. 5. The same general trends are observed as in the two plots in Fig. 2. For several instances, such as air05, binkar10, and swath3, the predicted bound—which is calculated based only on the improvement from the first round of cuts—is quite close to the actual bound changes after tens of rounds. The prediction tends to be inaccurate (a large overestimate) as more significant tailing in bound improvement occurs, but occasionally underestimates the bound improvement, such as for eil33-2.

Fig. 6

Applying rounds of cuts on assorted MIPLIB 2017 instances (after preprocessing) typically yields diminishing bound improvement. Several instances show a linear tendency in LP resolve time. The overlayed bar plot for each instance shows the cumulative number of cuts added after each round. The predicted bound using the improvement from the first round follows a logarithmic function that is similar to the actual bound evolution until cut strength exhibits more pronounced tailing off

Experiments with optimal proportion of cut rounds in SVBHC

Theorem 15 proves that, in the SVBHC model of Sect. 5, the number of cuts prescribed by Algorithm 1 is approximately optimal in the sense that the resulting tree is at most a multiplicative factor larger than the optimal tree size. Theorem 21 shows that using this approximately-optimal number of cuts proves a constant proportion of the overall bound, in the limit when the target bound goes to infinity. However, since the multiplicative factor in Theorem 15 may be quite large, it is not clear if the same type of limit exists for minimal-size trees. In Fig. 7, we address this question computationally, showing that the proportion of bound proved by cut nodes tends to the same limit in a minimal tree for four artificial instances of the SVBHC model. Experiments with more instances have shown the same behavior and therefore are omitted.

Fig. 7

Fraction of bound proved by cut nodes in an (exactly-)optimal SVBHC tree, exhibiting convergence to the bound from Theorem 21 provided by the approximately-optimal number of cut nodes prescribed by Algorithm 1