A compact mixed integer linear formulation for safe set problems

Abstract

The Safe Set Problem is a type of partitioning problem with particular constraints on the weight of adjacent partition components. Several theoretical results exist in the literature along with a mixed integer linear formulation. We propose a new, more compact, Mixed Integer Linear Program for the (Weighted) Safe Set Problem and Connected Safe Set Problem with a polynomial number of variables and constraints, based on a flow from an additional fictitious node. The model is enriched by symmetry-breaking considerations and a variable reduction subproblem. We test the model on a benchmark of small instances which underline the difficulty of solving the Safe Set Problem through mathematical programming approaches. We also compare it with the only existing formulation in the literature and find that our approach is usually more efficient on a set of benchmark instances.

Appendices

Appendix A: Results over small world networks

In this appendix we present detailed results over small networks generated through the mechanism described in [22]. These networks typically have a small diameter and represent social networks fairly well. They are known to be notoriously difficult to solve for some partitioning problems, such as [4, 23]. We generated instances with a number of nodes in the set $n\in \{20,25,30,35,40\}$ as in Sect. 4. The parameters needed to generate small world graphs are the average number of neighbours k and a rewiring probability of edges which we choose as $p=0.05$, similar to [19]. We vary parameter $k\in \{4,6,8,10\}$ to obtain instances with different characteristics and generate five instances for each value of n and k. The different average coefficients over those sets of five instances are displayed in Table 5 in a manner similar to Table 1, except for the additional column $\bar{\delta }_e$ which represents the average density of the instances. We can clearly see that the clustering coefficient is usually much larger for small world graphs while the ratio of the dispersion (standard deviation) in the nodes degree over the average degree (last column) is much smaller, denoting the fact that two different nodes tend to have similar degree in a small world graph.

Table 5 Average characteristics for the sets of five instances, for the small world graphs

Table 6 Average results for the unweighted SSP over sets of five instances of small world graphs for parameter $k=4$ and $k=10$ and number of nodes n

As for the results in Sect. 4, we first display the effect of the MILP improvements for small world graphs in Table 6, for unweighted instances with parameter $k=4$ and $k=10$. The results are similar to those of Sect. 4, in the sense that instances which are less dense tend to profit from the MILP improvements while denser instances are not solved faster on average, while there even are instance sizes where the improvements slightly worsen the performance of the model. This underlines the dependency of the performance of said improvements on the graph topology.

We display in Tables 7 and 8 the results of model $\mathcal {M}_{SS}$ with symmetry breaking constraints (40) and the variable reduction preprocessing on small world graphs, similar to Sect. 4. The SSP is still very hard to solve for such instances, even for very small graphs. As for random graphs, we can see that instances with a larger k parameter (and hence with a larger edge density) are usually easier to solve. Contrary to Sect. 4, however, no instance with 40 nodes is ever solved to optimality since the density for such instances do not even reach density 0.3 or 0.4, even with $k=10$.

Table 7 Average results of model $\mathcal {M}_{SS}$ with symmetry breaking constraints (40) and the variable reduction preprocessing for the SSP, over sets of five instances for small world graphs, for each average number of neighbours k and graph size n

Table 8 Results of model $\mathcal {M}_{SS}$ with symmetry breaking constraints (40) and the variable reduction preprocessing for the Connected SSP on small world graphs, both unweighted and weighted

Appendix B Comparison with the model of [16]

We compare in the following results of our model $\mathcal {M}_{SS}$ with those of the MILP approach in [16] on a set of benchmark instances they introduced. As discussed in Sect. 1, the algorithm advocated in [16] is a branch-and-cut algorithm based on a model with an exponential number of constraints, implemented as lazy constraints. The results of both models are reported in Table 9 for unweighted instances of SSP and in Table 10 for weighted instances. The instances range from 10 to 30 nodes with densities 0.3, 0.5 and 0.7. We report the upper bound (UB) and running time of both models. For the model of [16], when the running time is 7200 s, the algorithm reached its time limit and the instance is not solved at optimality, however the LB was not reported.

Table 9 UB and running time for the benchmark instances of [16] for unweighted instances with densities 0.3, 0.5 and 0.7, for model $\mathcal {M}_{SS}$ with improvements and the branch-and-cut algorithm of [16]

Table 10 UB and running time for the benchmark instances of [16] for weighted instances with densities 0.3, 0.5 and 0.7, for model $\mathcal {M}_{SS}$ with improvements and the branch-and-cut algorithm of [16]

The results for the model of [16] are those reported in the tables of [16]. Even though those results were obtained on a different computer and using a somewhat different framework (e.g., we make use of the heuristics of CPLEX, while [16] does not), we believe the important quantitative differences in the running times of both models is enough to assert the relative computational efficiency of our approach. Indeed, in most instances with 20 nodes or more, model $\mathcal {M}_{SS}$ closes the instances at least ten times faster. On the smallest instances, and contrary to the numerical analysis in Sect. 4 and Appendix A, $\mathcal {M}_{SS}$ does not always solve denser instances faster for a given graph size. The same pattern, however, can be seen again on the largest unweighted instances. The results confirm that model $\mathcal {M}_{SS}$ (with MILP improvements) is able to solve instances of up to 30 nodes without problem.