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Incorporating bounds from decision diagrams into integer programming

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Abstract

Decision diagrams have been successfully used to help solve several classes of discrete optimization problems. We explore an approach to incorporate them into integer programming solvers, motivated by the wide adoption of integer programming technology in practice. The main challenge is to map generic integer programming models to a recursive structure that is suitable for decision diagram compilation. We propose a framework that opportunistically constructs decision diagrams for suitable substructures, if present. In particular, we explore the use of a prevalent substructure in integer programming solvers known as the conflict graph, which we show to be amenable to decision diagrams. We use Lagrangian relaxation and constraint propagation to consider constraints that are not represented directly by the substructure. We use the decision diagrams to generate dual and primal bounds to improve the pruning process of the branch-and-bound tree of the solver. Computational results on the independent set problem with side constraints indicate that our approach can provide substantial speedups when conflict graphs are present.

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Notes

  1. As a technicality, this requires that Theorem 2 holds when there are arcs between the two nodes of a same variable. Despite assuming a standard implication graph, the proof from Aspvall et al. [6] is also valid with same-variable arcs.

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Authors and Affiliations

  1. Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA

    Christian Tjandraatmadja & Willem-Jan van Hoeve

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  1. Christian Tjandraatmadja
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  2. Willem-Jan van Hoeve
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Correspondence to Christian Tjandraatmadja.

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Appendices

Additional computational observations

In this section, we report three additional experimental observations on the runs performed in Sect. 7:

  1. 1.

    Percentage of time spent generating bounds and compiling decision diagrams (the former is the latter plus any Lagrangian relaxation or primal bound computation);

  2. 2.

    Number of improving feasible solutions found by the primal heuristic.

  3. 3.

    Number of times that the bound generator was called (i.e. it met the criterion on subproblem size), that a bound resulted in an improvement over the LP bound at a node, and that the improvement resulted in the node being pruned by bound;

Tables 4, 5, 6, 7, 8 and 9 in this section report the values described above. All averages of the values above are done with arithmetic mean across the instances and random seeds examined in Sect. 7. Note that the percentage of time spent on each portion of the method are calculated via ratios of times summed up across all seeds and instances (or equivalently, via ratios of arithmetic means).

Table 4 Percentage of time used for bound generation and number of times an improving primal feasible solution was found for independent set instances of size 150 from Sect. 7.1
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Table 5 Number of times an improving bound was found or resulted in pruning for independent set instances of size 150 from Sect. 7.1
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Table 6 Percentage of time used for bound generation and number of times an improving primal feasible solution was found for independent set instances of size 300 from Sect. 7.1
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Table 7 Number of times an improving bound was found or resulted in pruning for independent set instances of size 300 from Sect. 7.1
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Table 8 Percentage of time used for bound generation and number of times an improving primal feasible solution was found for independent set + knapsack instances from Sect. 7.2
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Table 9 Number of times an improving bound was found or resulted in pruning for independent set + knapsack instances from Sect. 7.2
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Experiments on MIPLIB 2017

We performed the following simple experiment on MIPLIB 2017 instances [32]. We took the 109 instances with the tags ‘binary’ and ‘benchmark_suitable’ from MIPLIB 2017. For these instances, we ran SCIP with bounds from relaxed decision diagrams, using a width of 100 and generating them at subproblems where number of variables is at most 1000. We chose a threshold that is not problem-dependent to avoid building large decision diagrams in large problems.

We ran a first pass with the intention to filter out instances where our approach is very unlikely to help. For this pass, we used a time limit of 30 min and one random seed. Out of the 109 instances, there were only 38 of them where the bound generator was called, or in other words, the solver did not observe a subproblem of size at most 1000 within the time limit of 30 min. Out of these 38 instances, in only 9 instances the bound generator found a bound better than the LP bound at least 20 times. These were the following 9 instances: bnatt400, eil33-2, eilC76-2, mine-166-5, mine-90-10, neos18, ponderthis0517-inf, reblock115, reblock166.

In the second pass, we reran the same experiment for these 9 instances, but with five random seeds and a higher time limit of 3 hours. We analyze the 4 instances of this subset that did not hit the time limit for the runs with bounds from decision diagrams. We report the solving times and number of nodes in Table 10, and number of bound improvements (over LP) and pruning in Table 11.

In all cases, the overhead of generating bounds increases the solving time, sometimes substantially. However, for the instances mine-166-5 and mine-90-10, we do obtain relevant reductions in the number of nodes by 25.97% and 23.77% respectively. Moreover, we observe that in all four instances, a significant portion of bounds generated improve over the corresponding LP bounds.

Table 10 Solving time in seconds and number of nodes for a subset of MIPLIB 2017 instances
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Table 11 Number of times an improving bound was found or resulted in pruning for a subset of MIPLIB 2017 instances
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Tjandraatmadja, C., van Hoeve, WJ. Incorporating bounds from decision diagrams into integer programming. Math. Prog. Comp. 13, 225–256 (2021). https://doi.org/10.1007/s12532-020-00191-6

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