Solving the maximum edge-weight clique problem in sparse graphs with compact formulations

Abstract

This paper studies the behavior of compact formulations for solving the maximum edge-weight clique (MEWC) problem in sparse graphs. The MEWC problem has long been discussed in the literature, but mostly addressing complete graphs, with or without a cardinality constraint on the clique. Yet, several real-world applications are defined on sparse graphs, where the missing edges are due to some threshold process or because they are not even supposed to be in the graph, at all. Such situations often arise in cell’s metabolic networks, where the amount of metabolites shared among reactions is an important issue to understand the cell’s prevalent elements. We propose new node-discretized formulations for the problem, which are more compact than other models known from the literature. Computational experiments on benchmark and real-world instances are conducted for discussing and comparing the models. These tests indicate that the node-discretized formulations are more efficient for solving large size sparse graphs. Additionally, we also address a new variant of the MEWC problem where the objective to be maximized includes the neighboring edges of the clique.

Appendix

Table 2 summarizes information on the characteristics of the instances used in Sect. 5. It indicates the number of nodes, number of edges and the density of each graph. It also includes heuristic solution values for the MEWC problem (denoted by $W_{e}(G))$, reported in Pullan (2008). The solutions for the RTN and SC instances were taken from the branch-and-bound executions. The $W_{e}(G)$ values are lower bounds for the MEWC optimums.

Table 2 Instances characteristics and lower bounds for the optimums, reported in Pullan (2008)

Results for setting parameter $q_{max}$, defining an upper bound for $\omega (G)$, are shown in Table 3. As mentioned before, these upper bounds were calculated using the sequential elimination algorithm described in Gendron et al. (2008) with the DSATUR greedy procedure (proposed by Brélaz (1979)). These bounds were used in the tests conducted in Sect. 5. The clique number ($\omega (G))$ of the graph is also shown. CPU times are reported in seconds.

Table 3 Upper bounds for the clique number $\omega (G)$, generated by the sequential elimination algorithm with the DSATUR procedure, described in Gendron et al. (2008)

Table 4 gives the linear programming relaxation percent gaps. It reports the results taken over the DIMACS, RTN and SC instances, using the F1, F2, F5 and F6 based models described in Sects. 2 and 3, and discussed in Sect. 5. Lowest gaps are given in bold.

Table 4 LP relaxation percent gaps using the models under discussion

Table 5 indicates the CPU times (in seconds) for running the linear programming relaxation tests reported in Table 4. Lowest times are given in bold.

Table 5 CPU times (in seconds) associated to the LP relaxation tests reported in Table 4

Table 6 reports the branch-and-bound execution times (in seconds) for the models under discussion. Column 2, with heading “Opt.”, represents the optimum solution value returned by branch and bound. The tests for the RTN instances with more than 10,000 nodes were omitted. Smallest CPU time values are given in bold.

Table 6 Execution times (in seconds) of the branch and bound using the models under discussion

Table 7 provides the same information as in Table 6 but considers “Strong branching” variable selection within branch and bound, and the automatic generation of global cuts provided by CPLEX for strengthening at the root node of the branch-and-bound tree. The table only focuses on the RTN and NIP classes, to which improvements were observed.

Table 7 Branch-and-bound execution times (in seconds) using branch on variable with maximum infeasibility strategy and the automatic generation of global cuts, for the RTN and NIP classes

Table 8 presents MEWC problem optimum solutions attained for the RTN class instances.

Table 8 MEWC problem optimum solutions for the RTN class instances

Table 9 presents optimum solutions for the MEWNC problem for the RTN class instances.

Table 9 MEWNC problem optimum solutions for the RTN class instances