[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82819-en":3,"doc-seo-82819-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82819,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","An Open, Reproducible Branch and Cut for the Capacitated Profitable Tour Problem: A Component Study","An open, reproducible branch-and-cut algorithm is presented for the capacitated profitable tour problem (CPTP) and its open s–t path variant, the capacity-constrained elementary shortest-path problem. The implementation re-creates Jepsen et al.’s formulation and cut families on a fully open mixed-integer programming stack (HiGHS), adding preprocessing, domain propagation, and reduced-cost fixing. The paper provides a rerunnable artifact and a component study benchmarking against a dynamic-programming/labelling reference to identify which cuts drive runtime and solved instances.","arXiv :2607 .04497v1 [math .OC] 5 Jul 2026  \nAN OPEN , REPRODUCIBLE BRANCH-AND-CUT FOR THE CAPACITATED PROFITABLE TOUR PROBLEM: A COMPONENT  \nSTUDY  \nSimon Spoorendonk  \nDenmark  \n[simon@spoorendonk.dk](simon@spoorendonk.dk)  \n[ORCID: 0009-0007-4304-6956](ORCID: 0009-0007-4304-6956)  \nJuly 7, 2026  \nABSTRACT  \nWe 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. [18] on a fully open mixed-integer programming stack (HiGHS [16]), and adds bound-based preprocessing, domain propagation, and reduced-cost variable fixing. We claim no new method; the contribution is twofold.  \nFirst, 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 shortestpath-incompatibility (SPI) cut, a variant of the node-precedence inequalities of García [12], finds no violated inequality on any instance. The solver and all experiments are released as open, reproducible software [27] .  \nKeywords Profitable tour problem · Elementary shortest path · Branch-and-cut · Resource-constrained shortest path · Computational study  \n1 Introduction  \nLet G = (V, E) be an undirected graph with node set V = N ∪ {0}, where 0 is a designated depot and N is a set of customers. Each edge e ∈ E has a cost ce , and each node i ∈ V has a profit pi ≥ 0 and a demand di ≥ 0; the vehicle has capacity Q. The capacitated profitable tour problem (CPTP) asks for a simple cycle through the depot that minimizes total edge cost minus collected node profit, subject to the total demand of the visited nodes not exceeding Q. Because the objective trades travel cost against profit, optimal tours visit a profitable subset of nodes rather than all of them, and the problem is naturally a profitable tour or prize-collecting problem rather than a routing problem with a fixed customer set.  \nCPTP is a special case of the elementary shortest-path problem with resource constraints (ESPPRC): splitting the depot into a source and a sink and pushing node demands and profits onto arcs turns a CPTP instance into an ESPPRC instance, though the reverse reduction does not hold in general [18] . ESPPRC and CPTP arise most prominently as the pricing subproblem in column-generation algorithms for vehicle routing, where negative reduced costs on nodes create  \nexactly the profit-collection incentive modelled here. The problem is NP-hard, and when the underlying cost structure admits negative-cost cycles (the regime that makes elementarity binding), it is the hard core of routing pricing. Two lines of attack dominate the literature. Dynamic-programming / labelling algorithms extend partial paths under dominance and resource bounds; they are the workhorses inside modern column generation and are extremely effective when the resource (here, capacity) keeps the number of non-dominated labels small [7, 11, 21] . Their performance degrades as elementarity, rather than the resource, becomes the binding difficulty. Branch-and-cut (B&C) algorithms instead solve an integer-programming formulation and","cbCairl4iLarlOET","https://ap.wps.com/l/cbCairl4iLarlOET","pdf",520827,1,18,"English","en",105,"# Introduction\n## Problem background and reductions\n## Related work and methodological landscape\n# Abstract\n## Open reproducible artifact and component study\n## Cut impact findings and negative result","[{\"question\":\"What problem does the paper address?\",\"answer\":\"It targets the capacitated profitable tour problem (CPTP) and its open s–t path variant, formulated as the capacity-constrained elementary shortest-path problem.\"},{\"question\":\"What is the main contribution of the work?\",\"answer\":\"It delivers an open, reproducible branch-and-cut implementation on a fully open MIP stack and an experimental component study that measures which cut families and preprocessing steps affect performance.\"},{\"question\":\"Which cuts provide most of the performance benefit?\",\"answer\":\"Capacity-class cuts account for essentially all observed gains: adding them to a connectivity-only baseline raises the number of solved instances substantially and greatly reduces the search tree size, while other components add no measurable 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problem does the paper address?","Question",{"text":75,"@type":76},"It targets the capacitated profitable tour problem (CPTP) and its open s–t path variant, formulated as the capacity-constrained elementary shortest-path problem.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the main contribution of the work?",{"text":80,"@type":76},"It delivers an open, reproducible branch-and-cut implementation on a fully open MIP stack and an experimental component study that measures which cut families and preprocessing steps affect performance.",{"name":82,"@type":73,"acceptedAnswer":83},"Which cuts provide most of the performance benefit?",{"text":84,"@type":76},"Capacity-class cuts account for essentially all observed gains: adding them to a connectivity-only baseline raises the number of solved instances substantially and greatly reduces the search tree size, while other components add no measurable 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