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, and finds that the capacity-class cuts account for essentially the entire benefit.
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 Garc\'ia (2009), finds no violated inequality on any instance. The solver and all experiments are released as open, reproducible software (Spoorendonk, 2026).
The Job Sequencing and Tool Switching Problem (SSP) is a well-known combinatorial optimization problem arising in the context of flexible manufacturing. Since the seminal work of Tang and Denardo (1988), the SSP has received significant attention in the literature, leading to the development of numerous exact and heuristic approaches. Despite these efforts, several benchmark instances proposed decades ago and containing only 20 jobs have remained unsolved to proven optimality. In this work, we propose an exact algorithm for the SSP, namely the Combinatorial Branch-and-Bound (C-B\&B) algorithm, which combines two distinct branch-and-bound algorithms, each introducing novel features compared with the existing literature. The former relies on a new branching scheme designed to reduce the size of the implicit enumeration tree, together with a collection of new bounding functions. The latter builds on the branching scheme introduced by Laporte et al. (2004) and strengthens it with a new bounding function and two dominance rules. Within C-B\&B, these exact algorithms are complemented by a preprocessing phase that incorporates a new branch-and-bound-based heuristic capable of rapidly generating a high-quality initial incumbent solution. Extensive computational experiments show that C-B\&B represents a strong breakthrough over previously published approaches, proving optimality for more instances with significantly less computational effort and closing several benchmark instances that have remained open for decades.
Alberto Locatelli, Jean-François Côté, Leandro C. Coelho· 0 citations
The length-constrained cycle partition problem (LCCP) is a graph optimization problem in which a set of nodes must be partitioned into a minimum number of cycles. Every node is associated with a critical time and the length of every cycle must not exceed the critical time of any node in the cycle. We formulate LCCP as a set partitioning model and solve it using an exact branch-price-and-cut approach. Our dynamic programming-based pricing algorithm to generate improving cycles exploits the particular structure of the pricing problem for efficient bidirectional search and symmetry breaking. Computational results show that the LP relaxation of the set partitioning model produces very strong dual bounds and our branch-price-and-cut method improves significantly over the state of the art. It is able to solve previously solved instances in a fraction of the time and closes 14 previously unsolved instances with numerically safe bounds, one of which has 76 nodes, a notable improvement over the previous limit of 52 nodes.
Mohammed Ghannam, Ambros M. Gleixner, Gioni Mexi et al.· arXiv.org· 0 citations
We present an open-source software package that implements a provably convergent Benders-type decomposition algorithm for multistage stochastic integer programs. In addition to standard cut families, such as Benders, strengthened Benders, and Lagrangian cuts, the algorithm incorporates rectified linear unit (ReLU) cuts, which provide convergence guarantees for general mixed-integer state variables. However, the dual problems used to generate these cuts often admit multiple optimal solutions. Although each solution yields a valid cut that separates the incumbent, the resulting cuts can differ in how well they approximate the subproblem cost. To strengthen these cuts, our package implements and evaluates two cut-selection strategies based on normalization and regularization of the dual problem. We also incorporate an alternating-cut criterion that uses cheaper Benders cuts when they are effective and invokes more expensive tight cuts only when necessary. Computational experiments on four classes of multistage stochastic integer programs benchmark these methods and provide insights on how problem structure affects their practical performance.
Mixed-integer programming (MIP) is a cornerstone in applied optimization, both in industry and academia. Recently, there has been increased attention to finding strong primal solutions quickly. This is reflected, for example, in the development of the NVIDIA cuOpt solver and, most recently, in the new MIPFEAS benchmark, which has a tight time limit of 600 seconds and evaluates solvers based on how quickly they find high-quality primal solutions. This article introduces a MIP portfolio parallelization scheme, focusing on efficiently exchanging information between its workers. We present two implementations of this scheme: one built directly into the open-source MIP solver SCIP, and an external one, which we call ReXi. ReXi is currently the fastest non-commercial solver in the MIPFEAS benchmark, followed by the SCIP-integrated implementation. Moreover, we present new versions of both implementations that considerably outperform their predecessors on the MIPFEAS benchmark.
The Stop Number Minimization Problem (SNMP), inspired by an autonomous vehicle service from France, arises when a homogeneous fleet of autonomous vehicles transports cargo and personnel across a circuit of stations. The objective is to satisfy all client requests while minimizing the total number of pickup/dropoff stops made by the vehicles, without violating their maximum capacity. We modeled the SNMP as a Pickup and Delivery Problem on an expanded graph with an ordered structure and solved it via a branch-and-price algorithm. The pricing problem was solved using a labeling algorithm with two distinct dominance rules. Additionally, three types of path elimination were introduced in the pricing algorithm. When branching, two strategies were implemented: SOS and Ryan-Foster. We tested our approach in a setting where each vehicle may perform up to 4 laps on the circuit. Computational experiments indicate that this approach solves instances of equivalent size to those tested in previous works in less time. Furthermore, our approach solved instances with up to 80 requests in the single-lap setting, contrasting previous works that solved instances with up to 50 requests.
Victor Nascimento, Luidi Simonetti· Journal of combinatorial opt...· 0 citations
A generic solver for computing integral flows in decision hypergraphs, subject to upper bound constraints on some hyperarcs, is proposed, which outperforms the best algorithm known thus far and is the first to close the optimality gap for all instances of several well-known benchmarks.
Arthur Léonard, F. Clautiaux· INFORMS journal on computing· 0 citations
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