Precedence-Constrained arborescences

Compact Models for the Precedence-Constrained Minimum-Cost Arborescence Problem

“…Experiments are based on the benchmark instances of TSPLIB [13], SOPLIB [14] and COMPILERS [15] originally proposed for the Sequential Ordering Problem (SOP) [16]. The benchmark instances are the same instances previously adopted in [10,11].…”

“…A violated constraint (10) or (17), can be detected by computing a minimum-cut in the graph where the weight of an arc is equal to the value of its corresponding x i j variable. On the other hand, a violated constraint (11) or (18), can be detected by finding a violated s − t path using a DFS algorithm. For more details on how violated constraints are detected and dynamically added to the model, please see [10].…”

“…Constraints (10) are the connectivity constraints that enforce every set of vertices S ⊆ V \{r} must be reachable from the root r. The size of the set of constraints ( 10) is exponential with a size of O(2 |A| ). Constraints (11) propagate the value of u st i down to u st j if x i j = 1. Constraints ( 12) and ( 13) set the values of u st s and u st t to 0 and 1 respectively, for all (s,t) ∈ R. Finally, constraints ( 14) and ( 15) define the domains of the variables.…”

“…A model for the PCMCA has been proposed in [10] based on a well-know formulation for the MCA, and enforces precedence relationships by propagating a value on every path in the arborescence. An alternative model has been discussed in [11]. Figure 1 presents an example that shows the difference between the classic MCA and the PCMCA.…”

Compact Models to Solve the Precedence-Constrained Minimum-Cost Arborescence Problem with Waiting Times

“…Several models for the problem, all based on Mixed Integer Linear Programming (MILP), were proposed. We refer the interested reader to Chou et al [38].…”