In many branch-and-price algorithms, the column generation subproblem consists of computing feasible constrained paths. In the capacitated arc-routing problem (CARP), elementarity constraints concerning the edges to be serviced and additional constraints resulting from the branch-and-bound process together impose two types of loop-elimination constraints. To fulfill the former constraints, it is common practice to rely on a relaxation where loops are allowed. In a k-loop elimination approach all loops of length k and smaller are forbidden. Following Bode and Irnich (2012) for solving the CARP, branching on followers and non-followers is the only known approach to guarantee integer solutions within branch-and-price. However, it comes at the cost of additional task-2-loop elimination constraints. In this paper, we show that a combined ( k , 2 ) -loop elimination in the shortest-path subproblem can be accomplished in a computationally efficient way. Overall, the improved branch-and-price often allows the computation of tighter lower bounds and integer optimal solutions for several instances from standard benchmark sets. 
The shortest-path problem with resource constraints with ( k , 2 ) -loop elimination and its application to the capacitated arc-routing problem