Overview of approach
Our basic LP/DEE approach is to: (1) formulate an instance of motif finding as a graph problem (2) apply the DEE techniques described above in the order of increasing complexity so as to prune the graph (3) use mathematical programming to find a solution to the smaller graph problem and (4) evaluate statistical significance.
While applying DEE, if the size of the graph becomes small enough (set at 800 vertices for the described experiments), we submit the appropriate LP to the CPLEX LP solver and, if necessary, to the ILP solver. To reduce the graph to that necessary small size, we apply the DEE variants, running each one of them until either the specified graph size has been reached, or to convergence so that no further pruning is possible. In particular, we first attempt to prune the graph using basic clique-bounds DEE, then we consider tighter bound computations, and lastly we employ graph decomposition in conjunction with the DEE methods.
In rare cases the optimality-preserving DEE procedures are unable to prune the graph, and we perform what we call speculative pruning using higher C* values, which do not necessarily correspond to known cliques in graph G. Three outcomes of such pruning are possible: (i) The graph is eliminated completely. This guarantees that a clique of value C* does not exist in G. (ii) The pruning proves once again insufficient to reduce the graph. (iii) The pruning procedure converges to a small graph. We search the space of possible C* values until we find one that produces outcome (iii). To identify such a value we first translate the possible clique scores into their corresponding e-values, and then perform binary search on the e-value exponent. This method converges quickly, typically locating an appropriate C* in fewer than 10 iterations. If the optimal solution for the final reduced graph is better than the C* used in pruning, then it is also optimal for the original graph. Otherwise, the e-value corresponding to C* provides us with a lower bound on the significance of the actual optimal solution.