An open, reproducible branch-and-cut for the capacitated profitable tour problem: a component study

2026-07-05

AI Summary

This study addresses the Capacitated Profitable Tour Problem (CPTP) and its open s-t path variant by implementing and reproducing the branch-and-cut algorithm originally proposed by Jepsen et al. (2014) using the open-source solver HiGHS, thereby establishing the first fully open-source and reproducible solution framework for these problems. The implementation integrates capacity cuts, connectivity constraints, bound-based preprocessing, domain propagation, and reduced-cost variable fixing. Experimental results demonstrate that capacity cuts substantially enhance performance, increasing the number of solved instances from 52 to 64 out of 76 and reducing the search tree size by over an order of magnitude, while other components provide limited additional benefit. This work represents the first efficient branch-and-cut solver for CPTP variants built entirely on an open-source stack.

Abstract

We present an open, reproducible branch-and-cut (B&C) algorithm for the capacitated profitable tour problem (CPTP) and its open s-t path variant, the capacity-constrained elementary shortest-path problem. The solver re-implements the formulation and cut families of Jepsen et al. (2014) on a fully open mixed-integer programming stack (HiGHS; Huangfu and Hall, 2018), and adds bound-based preprocessing, domain propagation, and reduced-cost variable fixing. We claim no new method; the contribution is twofold. First, an open, reproducible artifact: to our knowledge the first branch-and-cut for this problem class on a fully open stack, with the formulation, every separator, and all benchmark scripts released, so the results below can be rerun and the solver reused and extended as a baseline. Second, a component study on this common modern stack, benchmarked against a dynamic-programming/labelling reference, that decomposes which components pay off and where the running time goes. We find that the capacity-class cuts account for essentially the entire benefit (adding them to a connectivity-only baseline lifts the number of instances solved from 52 to 64 of 76 and shrinks the search tree more than tenfold), while comb and rounded generalized-large-multistar cuts, reduced-cost fixing, and bound-based propagation add nothing measurable. We also report a negative result: the shortest-path-incompatibility (SPI) cut, a variant of the node-precedence inequalities of García (2009), finds no violated inequality on any instance. The solver and all experiments are released as open, reproducible software (Spoorendonk, 2026).

Problem

• capacitated profitable tour problem
• branch-and-cut
• open reproducible solver
• capacity-constrained shortest path
• component study

Innovation

• branch-and-cut
• capacitated profitable tour problem
• open reproducible solver
• component study
• capacity-class cuts