Discrete Optimization最新文献

{"title":"Fleet & tail assignment under uncertainty","authors":"Lukas Glomb,&nbsp;Frauke Liers,&nbsp;Florian Rösel","doi":"10.1016/j.disopt.2024.100836","DOIUrl":"https://doi.org/10.1016/j.disopt.2024.100836","url":null,"abstract":"<div><p>Airlines solve many different optimization problems and combine the resulting solutions to ensure smooth, minimum-cost operations. Crucial problems are the Fleet Assignment, which assigns aircraft types to flights of a given schedule, and the Tail Assignment, which determines individual flight sequences to be performed by single aircraft. In order to find a cost-optimal solution, many airlines use mathematical optimization models. For these to be effective, the available data and forecasts must reflect the situation as accurately as possible. However, especially in times of a pandemic, the underlying plan is subject to severe uncertainties: Staff and demand uncertainties can even lead to flight cancellations or result in entire aircraft having to be grounded. Therefore, it is advantageous for airlines to protect their mathematical models against uncertainties in the input parameters. In this work, two computational tractable and cost-efficient robust models and solution approaches are developed: First, we set up a novel mixed integer model for the integrated fleet and tail assignment, which ensures that as few subsequent flights as possible have to be canceled in the event of initial flight cancellations. We then solve this model using a procedure that ensures that the costs of the solution remain considerably low. Our second model is an extended fleet assignment model that allows us to compensate for entire aircraft cancellations in the best possible way, taking into account rescheduling options. We demonstrate the effectiveness of both approaches by conducting an extensive computational study based on real flight schedules of a major German airline. It turns out that both models deliver stable, cost-efficient solutions within less than ten minutes, which significantly reduce follow-up costs in the case uncertainties arise.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S157252862400015X/pdfft?md5=f80dbb301204008a3a9aff4b64f02673&pid=1-s2.0-S157252862400015X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140134200","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Pareto optimal balanced exchanges","authors":"Pavlos Eirinakis ,&nbsp;Ioannis Mourtos ,&nbsp;Michalis Samaris","doi":"10.1016/j.disopt.2024.100835","DOIUrl":"https://doi.org/10.1016/j.disopt.2024.100835","url":null,"abstract":"<div><p>We investigate a market without money in which every agent offers indivisible goods in multiple copies, in exchange for goods of other agents. The exchange must be balanced in the sense that each agent should receive a quantity of good(s) equal to the one she transfers to others. We describe the market in graph-theoretic terms hence we use the notion of circulations to describe a balanced exchange of goods. Each agent has strict preferences over the agents from which she will receive goods and an upper bound on the quantity of each transaction, while a positive integer weight reflects the social importance of each unit exchanged. In this paper, we propose a simple variant of the Top Trading Cycles mechanism that finds a Pareto optimal circulation. We then offer necessary and sufficient conditions for a circulation to be Pareto optimal and, as a consequence, a easy recognition procedure. Last, we show that finding a maximum weight Pareto optimal circulation is NP-hard but becomes polynomial if weights are concordant with preferences.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140041484","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Circuits in extended formulations","authors":"Steffen Borgwardt ,&nbsp;Matthias Brugger","doi":"10.1016/j.disopt.2024.100825","DOIUrl":"https://doi.org/10.1016/j.disopt.2024.100825","url":null,"abstract":"<div><p>Circuits and extended formulations are classical concepts in linear programming theory. The circuits of a polyhedron are the elementary difference vectors between feasible points and include all edge directions. We study the connection between the circuits of a polyhedron <span><math><mi>P</mi></math></span> and those of an extended formulation of <span><math><mi>P</mi></math></span>, i.e., a description of a polyhedron <span><math><mi>Q</mi></math></span> that linearly projects onto <span><math><mi>P</mi></math></span>. It is well known that the edge directions of <span><math><mi>P</mi></math></span> are images of edge directions of <span><math><mi>Q</mi></math></span>. We show that this ‘inheritance’ under taking projections does not extend to the set of circuits, and that this non-inheritance is quite generic behavior. We provide counterexamples with a provably minimal number of facets, vertices, and extreme rays, including relevant polytopes from clustering, and show that the difference in the number of circuits that are inherited and those that are not can be exponentially large in the dimension. We further prove that counterexamples exist for any fixed linear projection map, unless the map is injective. Finally, we characterize those polyhedra <span><math><mi>P</mi></math></span> whose circuits are inherited from all polyhedra <span><math><mi>Q</mi></math></span> that linearly project onto <span><math><mi>P</mi></math></span>. Conversely, we prove that every polyhedron <span><math><mi>Q</mi></math></span> satisfying mild assumptions can be projected in such a way that the image polyhedron <span><math><mi>P</mi></math></span> has a circuit with no preimage among the circuits of <span><math><mi>Q</mi></math></span>. Our proofs build on standard constructions such as homogenization and disjunctive programming.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139714257","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"When greedy gives optimal: A unified approach","authors":"Dmitry Rybin","doi":"10.1016/j.disopt.2024.100824","DOIUrl":"10.1016/j.disopt.2024.100824","url":null,"abstract":"<div><p><span>We present necessary and sufficient conditions when a certain greedy object selection algorithm gives optimal results. Our approach covers known results for the Unbounded Knapsack Problem and Change Making Problem and gives new theoretical results for a variety of </span>packing problems. We also provide connections between packing problems and certain bidirectional capacity installation problems on networks.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139583667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Parametric matroid interdiction","authors":"Nils Hausbrandt ,&nbsp;Oliver Bachtler ,&nbsp;Stefan Ruzika ,&nbsp;Luca E. Schäfer","doi":"10.1016/j.disopt.2024.100823","DOIUrl":"10.1016/j.disopt.2024.100823","url":null,"abstract":"<div><p>We introduce the parametric<span> matroid one-interdiction problem. Given a matroid, each element of its ground set is associated with a weight that depends linearly on a real parameter from a given parameter interval. The goal is to find, for each parameter value, one element that, when being removed, maximizes the weight of a minimum weight basis. The complexity of this problem can be measured by the number of slope changes of the piecewise linear function mapping the parameter to the weight of the optimal solution of the parametric matroid one-interdiction problem. We provide two polynomial upper bounds as well as a lower bound on the number of these slope changes. Using these, we develop algorithms that require a polynomial number of independence tests and analyse their running time in the special case of graphical matroids.</span></p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139551674","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On polytopes with linear rank with respect to generalizations of the split closure","authors":"Sanjeeb Dash ,&nbsp;Yatharth Dubey","doi":"10.1016/j.disopt.2023.100821","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100821","url":null,"abstract":"<div><p><span>In this paper we study the rank of polytopes contained in the 0-1 cube with respect to </span><span><math><mi>t</mi></math></span>-branch split cuts and <span><math><mi>t</mi></math></span>-dimensional lattice cuts for a fixed positive integer <span><math><mi>t</mi></math></span>. These inequalities are the same as split cuts when <span><math><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow></math></span> and generalize split cuts when <span><math><mrow><mi>t</mi><mo>&gt;</mo><mn>1</mn></mrow></math></span>. For polytopes contained in the <span><math><mi>n</mi></math></span>-dimensional 0-1 cube, the work of Balas implies that the split rank can be at most <span><math><mi>n</mi></math></span>, and this bound is tight as Cornuéjols and Li gave an example with split rank <span><math><mi>n</mi></math></span>. All known examples with high split rank – i.e., at least <span><math><mrow><mi>c</mi><mi>n</mi></mrow></math></span> for some positive constant <span><math><mrow><mi>c</mi><mo>&lt;</mo><mn>1</mn></mrow></math></span> – are defined by exponentially many (as a function of <span><math><mi>n</mi></math></span><span>) linear inequalities. For any fixed integer </span><span><math><mrow><mi>t</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>, we give a family of polytopes contained in <span><math><msup><mrow><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow><mrow><mi>n</mi></mrow></msup></math></span> for sufficiently large <span><math><mi>n</mi></math></span> such that each polytope has empty integer hull, is defined by <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> inequalities, and has rank <span><math><mrow><mi>Ω</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> with respect to <span><math><mi>t</mi></math></span>-dimensional lattice cuts. Therefore the split rank of these polytopes is <span><math><mrow><mi>Ω</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span><span><span>. It was shown earlier that there exist generalized branch-and-bound proofs, with logarithmic depth, of the nonexistence of </span>integer points in these polytopes. Therefore, our lower bound results on split rank show an exponential separation between the depth of branch-and-bound proofs and split rank.</span></p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139487029","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Tighter bounds on the minimum broadcast time","authors":"Dag Haugland","doi":"10.1016/j.disopt.2024.100822","DOIUrl":"https://doi.org/10.1016/j.disopt.2024.100822","url":null,"abstract":"<div><p>Given a connected graph and a subset of its vertices referred to as the sources, the minimum broadcast time problem asks for the shortest time necessary for communicating a message from the sources to all other vertices in the graph. Information exchange is possible only between neighbors, and each vertex can transmit the message to at most one neighbor at a time. Since early works on complexity theory, the problem has been known to be NP-hard. Contributions from the current text to the understanding of the minimum broadcast time problem are threefold. Through considerations of the shortest distances between sources and other vertices, a new lower bound on the broadcast time is derived. Analytical expressions of this bound are given in the single source instances of several graph classes. Fast procedures for computing upper bounds are studied next, including both the construction of feasible solutions, and the improvement of existing ones. Finally, with a focus on a new stable-set interpretation of the problem, integer programming formulations are studied, and for their theoretical interest, associated facet-defining valid inequalities are given. The computational performance of the novel methodology is evaluated in numerical experiments applied to standard benchmark instances and to instances larger than those studied in preceding recent works.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139434347","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Characterizing linearizable QAPs by the level-1 reformulation-linearization technique","authors":"Lucas Waddell ,&nbsp;Warren Adams","doi":"10.1016/j.disopt.2023.100812","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100812","url":null,"abstract":"<div><p>The quadratic assignment problem (QAP) is an extremely challenging NP-hard combinatorial optimization program. Due to its difficulty, a research emphasis has been to identify special cases that are polynomially solvable. Included within this emphasis are instances which are <em>linearizable</em><span><span>; that is, which can be rewritten as a linear assignment problem having the property that the objective function value is preserved at all feasible solutions. Various known sufficient conditions for identifying linearizable instances have been explained in terms of the continuous relaxation of a weakened version of the level-1 reformulation-linearization-technique (RLT) form that does not enforce nonnegativity<span> on a subset of the variables. Also, conditions that are both necessary and sufficient have been given in terms of decompositions of the objective coefficients. The main contribution of this paper is the identification of a relationship between polyhedral theory and linearizability that promotes a novel, yet strikingly simple, necessary and sufficient condition for identifying linearizable instances; specifically, an instance of the QAP is linearizable if and only if the continuous relaxation of the same weakened RLT form is bounded. In addition to providing a novel perspective on the QAP being linearizable, a consequence of this study is that every linearizable instance has an optimal solution to the (polynomially-sized) continuous relaxation of the level-1 RLT form that is binary. The converse, however, is not true so that the continuous relaxation can yield binary optimal solutions to instances of the QAP that are not linearizable. Another consequence follows from our defining a maximal </span></span>linearly independent set of equations in the lifted RLT variable space; we answer a recent open question that the theoretically best possible linearization-based bound cannot improve upon the level-1 RLT form.</span></p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138327976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Packing mixed hyperarborescences","authors":"Zoltán Szigeti","doi":"10.1016/j.disopt.2023.100811","DOIUrl":"10.1016/j.disopt.2023.100811","url":null,"abstract":"<div><p><span>The aim of this paper is twofold. We first provide a new orientation theorem which gives a natural and simple proof of a result of Gao and Yang (2021) on matroid-reachability-based packing of mixed arborescences in mixed graphs by reducing it to the corresponding theorem of Király (2016) on directed graphs. Moreover, we extend another result of Gao and Yang (2021) by providing a new theorem on mixed hypergraphs having a packing of mixed hyperarborescences such that their number is at least </span><span><math><mi>ℓ</mi></math></span> and at most <span><math><msup><mrow><mi>ℓ</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>, each vertex belongs to exactly <span><math><mi>k</mi></math></span> of them, and each vertex <span><math><mi>v</mi></math></span> is the root of least <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span> and at most <span><math><mrow><mi>g</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span> of them.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136102757","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The maximum number of short paths in a Halin graph","authors":"Shunhai He ,&nbsp;Huiqing Liu","doi":"10.1016/j.disopt.2023.100809","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100809","url":null,"abstract":"<div><p>A Halin graph <span><math><mi>G</mi></math></span> is a plane graph consisting of a plane embedding of a tree <span><math><mi>T</mi></math></span> of order at least 4 containing no vertex of degree 2, and of a cycle <span><math><mi>C</mi></math></span> connecting all leaves of <span><math><mi>T</mi></math></span>. Let <span><math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>h</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the maximum number of copies of <span><math><mi>G</mi></math></span> in a Halin graph on <span><math><mi>n</mi></math></span> vertices. In this paper, we give exact values of <span><math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>h</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> when <span><math><mi>G</mi></math></span> is a path on <span><math><mi>k</mi></math></span> vertices for <span><math><mrow><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mn>5</mn></mrow></math></span>. Moreover, we develop a new graph transformation preserving the number of vertices, so that the resulting graph has a monotone behavior with respect to the number of short paths.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91729833","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}