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DOI: 10.4230/LIPIcs.SEA.2025.24

A New Relaxation for Tree-Based Problems and Minimum Power-Cost Spanning Trees

Authors: Luzie Marianczuk, Ernst Althaus, Stefan Irnich, and Marc E. Pfetsch

Published in: LIPIcs, Volume 338, 23rd International Symposium on Experimental Algorithms (SEA 2025)

Abstract

In this paper, we investigate lower bounds of tree-based optimization problems in order to obtain effective exact algorithms, such as branch-and-bound algorithms. Our new approach inherits the development of dynamic programming algorithms for constrained shortest path problems, as they occur as subproblems in Lagrangian relaxation algorithms and column generation-based algorithms for variants of the Vehicle Routing Problem. In the q-route relaxation, paths must satisfy a capacity constraint while the elementarity constraint is relaxed, that is, paths may contain cycles. An analogue of q-routes for tree optimization problems are q-arbs, a structure that relaxes elementarity for arborescences. We introduce a generalized formulation of q-arbs for a broad class of tree-based problems, and apply the neighborhood restrictions of so-called ng-routes to them to obtain tighter bounds. We apply the new dynamic programming approach to the Minimum Power-Cost Spanning Tree Problem and show empirically that the resulting bounds are often better than traditional LP-based lower bounds of (mixed) integer programming models.

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Luzie Marianczuk, Ernst Althaus, Stefan Irnich, and Marc E. Pfetsch. A New Relaxation for Tree-Based Problems and Minimum Power-Cost Spanning Trees. In 23rd International Symposium on Experimental Algorithms (SEA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 338, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{marianczuk_et_al:LIPIcs.SEA.2025.24,
  author =	{Marianczuk, Luzie and Althaus, Ernst and Irnich, Stefan and Pfetsch, Marc E.},
  title =	{{A New Relaxation for Tree-Based Problems and Minimum Power-Cost Spanning Trees}},
  booktitle =	{23rd International Symposium on Experimental Algorithms (SEA 2025)},
  pages =	{24:1--24:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-375-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{338},
  editor =	{Mutzel, Petra and Prezza, Nicola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2025.24},
  URN =		{urn:nbn:de:0030-drops-232620},
  doi =		{10.4230/LIPIcs.SEA.2025.24},
  annote =	{Keywords: lower bounds, symmetric connectivity, power range assignment, dynamic programming, optimal substructure}
}

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DOI: 10.4230/DagSemProc.09261.18

New Models and Methods for Arc Routing

Authors: Stefan Irnich

Published in: Dagstuhl Seminar Proceedings, Volume 9261, Models and Algorithms for Optimization in Logistics (2009)

Abstract

The talk presents two non-standard extensions for single-vehicle arc-routing problems a.k.a. postman problems: First, street segments that require a service on both sides of the street can be covered either by two separate services or by a single zigzag service. Instead of deciding the type of service beforehand, we propose to take into account the zigzagging option when designing a tour. We present MIP models for the extension of Undirected Chinese and Rural Postman Problem (UCPP, URPP). We show that these models can be solved reasonable well using a cutting-plane or branch-and-cut algorithm. Second, capacitated postman problems occur as subproblems in column-generation and Lagrangian-relaxation approaches of the capacitated arc-routing problem. In order to model these and similar subproblems or submodels, we present the Profitable Capacitated Rural Postman Problem (PCRPP): In the PCRPP, edges that are serviced give a profit, but deadheading through edges generates costs. Both service and deadheading consume time. The task is to find a tour that maximizes the difference of profits and costs, while the overall duration of the tour must not exceed a given bound. The solution approach for this problem is again based on branch-and-cut.

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Stefan Irnich. New Models and Methods for Arc Routing. In Models and Algorithms for Optimization in Logistics. Dagstuhl Seminar Proceedings, Volume 9261, pp. 1-5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{irnich:DagSemProc.09261.18,
  author =	{Irnich, Stefan},
  title =	{{New Models and Methods for Arc Routing}},
  booktitle =	{Models and Algorithms for Optimization in Logistics},
  pages =	{1--5},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2009},
  volume =	{9261},
  editor =	{Cynthia Barnhart and Uwe Clausen and Ulrich Lauther and Rolf H. M\"{o}hring},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.09261.18},
  URN =		{urn:nbn:de:0030-drops-21635},
  doi =		{10.4230/DagSemProc.09261.18},
  annote =	{Keywords: Postman problems, branch-and-cut}
}

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Documents authored by Irnich, Stefan

A New Relaxation for Tree-Based Problems and Minimum Power-Cost Spanning Trees

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New Models and Methods for Arc Routing

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