Pascale Bendotti , Luca Brunod Indrigo , Philippe Chrétienne , Bruno Escoffier
{"title":"Anchor-robust project scheduling with non-availability periods","authors":"Pascale Bendotti , Luca Brunod Indrigo , Philippe Chrétienne , Bruno Escoffier","doi":"10.1016/j.disopt.2024.100864","DOIUrl":"10.1016/j.disopt.2024.100864","url":null,"abstract":"<div><div>In large-scale scheduling applications, it is often decisive to find reliable schedules prior to the execution of the project. Most of the time however, data is affected by various sources of uncertainty. Robust optimization is used to overcome this imperfect knowledge. Anchor robustness, as introduced in the literature for processing time uncertainty, makes it possible to guarantee job starting times for a subset of jobs. In this paper, anchor robustness is extended to the case where uncertain non-availability periods must be taken into account. Three problems are considered in the case of budgeted uncertainty: checking that a given subset of jobs is anchored in a given schedule, finding a schedule of minimal makespan in which a given subset of jobs is anchored and finding an anchored subset of maximum weight in a given schedule. Polynomial time algorithms are proposed for the first two problems while an inapproximability result is given for the third one.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"54 ","pages":"Article 100864"},"PeriodicalIF":0.9,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552680","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":"Circuit and Graver walks and linear and integer programming","authors":"Shmuel Onn","doi":"10.1016/j.disopt.2024.100862","DOIUrl":"10.1016/j.disopt.2024.100862","url":null,"abstract":"<div><div>We show that a circuit walk from a given feasible point of a given linear program to an optimal point can be computed in polynomial time using only linear algebra operations and the solution of the single given linear program. We also show that a Graver walk from a given feasible point of a given integer program to an optimal point is polynomial time computable using an integer programming oracle, but without such an oracle, it is hard to compute such a walk even if an optimal solution to the given program is given as well. Combining our oracle algorithm with recent results on sparse integer programming, we also show that Graver walks from any point are polynomial time computable over matrices of bounded tree-depth and subdeterminants.</div></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"54 ","pages":"Article 100862"},"PeriodicalIF":0.9,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425831","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}
Ismail Naderi, Mohsen Rezapour, Mohammad R. Salavatipour
{"title":"Approximation schemes for Min-Sum k-Clustering","authors":"Ismail Naderi, Mohsen Rezapour, Mohammad R. Salavatipour","doi":"10.1016/j.disopt.2024.100860","DOIUrl":"10.1016/j.disopt.2024.100860","url":null,"abstract":"<div><p>We consider the Min-Sum <span><math><mi>k</mi></math></span>-Clustering (<span><math><mi>k</mi></math></span>-MSC) problem. Given a set of points in a metric which is represented by an edge-weighted graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> and a parameter <span><math><mi>k</mi></math></span>, the goal is to partition the points <span><math><mi>V</mi></math></span> into <span><math><mi>k</mi></math></span> clusters such that the sum of distances between all pairs of the points within the same cluster is minimized.</p><p>The <span><math><mi>k</mi></math></span>-MSC problem is known to be APX-hard on general metrics. The best known approximation algorithms for the problem obtained by Behsaz et al. (2019) achieve an approximation ratio of <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mo>log</mo><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow><mo>)</mo></mrow></mrow></math></span> in polynomial time for general metrics and an approximation ratio <span><math><mrow><mn>2</mn><mo>+</mo><mi>ϵ</mi></mrow></math></span> in quasi-polynomial time for metrics with bounded doubling dimension. No approximation schemes for <span><math><mi>k</mi></math></span>-MSC (when <span><math><mi>k</mi></math></span> is part of the input) is known for any non-trivial metrics prior to our work. In fact, most of the previous works rely on the simple fact that there is a 2-approximate reduction from <span><math><mi>k</mi></math></span>-MSC to the balanced <span><math><mi>k</mi></math></span>-median problem and design approximation algorithms for the latter to obtain an approximation for <span><math><mi>k</mi></math></span>-MSC.</p><p>In this paper, we obtain the first Quasi-Polynomial Time Approximation Schemes (QPTAS) for the problem on metrics induced by graphs of bounded treewidth, graphs of bounded highway dimension, graphs of bounded doubling dimensions (including fixed dimensional Euclidean metrics), and planar and minor-free graphs. We bypass the barrier of 2 for <span><math><mi>k</mi></math></span>-MSC by introducing a new clustering problem, which we call min-hub clustering, which is a generalization of balanced <span><math><mi>k</mi></math></span>-median and is a trade off between center-based clustering problems (such as balanced <span><math><mi>k</mi></math></span>-median) and pair-wise clustering (such as Min-Sum <span><math><mi>k</mi></math></span>-clustering). We then show how one can find approximation schemes for Min-hub clustering on certain classes of metrics.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"54 ","pages":"Article 100860"},"PeriodicalIF":0.9,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1572528624000392/pdfft?md5=0298fb9c3c75e407870e412a1aae1a26&pid=1-s2.0-S1572528624000392-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142242680","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}
Michael A. Henning , Johannes Pardey , Dieter Rautenbach , Florian Werner
{"title":"Mostar index and bounded maximum degree","authors":"Michael A. Henning , Johannes Pardey , Dieter Rautenbach , Florian Werner","doi":"10.1016/j.disopt.2024.100861","DOIUrl":"10.1016/j.disopt.2024.100861","url":null,"abstract":"<div><p>Došlić et al. defined the Mostar index of a graph <span><math><mi>G</mi></math></span> as <span><math><mrow><mi>M</mi><mi>o</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></munder><mrow><mo>|</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow><mo>−</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span>, where, for an edge <span><math><mrow><mi>u</mi><mi>v</mi></mrow></math></span> of <span><math><mi>G</mi></math></span>, the term <span><math><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span> denotes the number of vertices of <span><math><mi>G</mi></math></span> that have a smaller distance in <span><math><mi>G</mi></math></span> to <span><math><mi>u</mi></math></span> than to <span><math><mi>v</mi></math></span>. For a graph <span><math><mi>G</mi></math></span> of order <span><math><mi>n</mi></math></span> and maximum degree at most <span><math><mi>Δ</mi></math></span>, we show <span><math><mrow><mi>M</mi><mi>o</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>Δ</mi></mrow></msub><mi>n</mi><mo>log</mo><mrow><mo>(</mo><mo>log</mo><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math></span> where <span><math><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>Δ</mi></mrow></msub><mo>></mo><mn>0</mn></mrow></math></span> only depends on <span><math><mi>Δ</mi></math></span> and the <span><math><mrow><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> term only depends on <span><math><mi>n</mi></math></span>. Furthermore, for integers <span><math><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and <span><math><mi>Δ</mi></math></span> at least 3, we show the existence of a <span><math><mi>Δ</mi></math></span>-regular graph of order <span><math><mi>n</mi></math></span> at least <span><math><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> with <span><math><mrow><mi>M</mi><mi>o</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mfrac><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><msubsup><mrow><mi>c</mi></mrow><mrow><mi>Δ</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mi>n</mi><mo>log</mo><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span> where <span><mat","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"54 ","pages":"Article 100861"},"PeriodicalIF":0.9,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1572528624000409/pdfft?md5=e34071dea61722ee4baab21c7039f3bf&pid=1-s2.0-S1572528624000409-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142228672","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":"Easy and hard separation of sparse and dense odd-set constraints in matching","authors":"Brady Hunsaker, Craig Tovey","doi":"10.1016/j.disopt.2024.100849","DOIUrl":"10.1016/j.disopt.2024.100849","url":null,"abstract":"<div><p>We investigate polytopes intermediate between the fractional matching and the perfect matching polytopes, by imposing a strict subset of the odd-set (blossom) constraints. For sparse constraints, we give a polynomial time separation algorithm if only constraints on all odd sets bounded by a given size (e.g. <span><math><mrow><mo>≤</mo><mn>9</mn><mo>+</mo><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow><mo>/</mo><mn>6</mn></mrow></math></span>) are present. Our algorithm also solves the more general problem of finding a T-cut subject to upper bounds on the cardinality of its defining node set and on its cost. In contrast, regarding dense constraints, we prove that for every <span><math><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span>, it is NP-complete to separate over the class of constraints on odd sets of size <span><math><mrow><mn>2</mn><mrow><mo>⌊</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>α</mi><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow><mo>)</mo></mrow><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow><mo>−</mo><mn>1</mn></mrow></math></span> or <span><math><mrow><mo>≥</mo><mi>α</mi><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow></mrow></math></span>.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"54 ","pages":"Article 100849"},"PeriodicalIF":0.9,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142228671","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":"Two-set inequalities for the binary knapsack polyhedra","authors":"Todd Easton , Jennifer Tryon , Fabio Vitor","doi":"10.1016/j.disopt.2024.100859","DOIUrl":"10.1016/j.disopt.2024.100859","url":null,"abstract":"<div><p>This paper presents two-set inequalities, a class of valid inequalities for knapsack and multiple knapsack problems. Two-set inequalities are generated from two arbitrary sets of variables from a knapsack constraint. This class of cutting planes is not a traditional type of lifting since a valid inequality over a restricted space is not required to start. Furthermore, they cannot be derived using any existing lifting technique. The paper presents a quadratic algorithm to efficiently generate many two-set inequalities. Conditions for facet-defining two-set inequalities are also derived. Computational experiments tested these inequalities as pre-processing cuts versus CPLEX, a high-performance mathematical programming solver, at default settings. Overall, two-set inequalities reduced the time to solve some benchmark multiple knapsack instances to up to 80%. Computational results also showed the potential of this new class of cutting planes to solve computationally challenging binary integer programs.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"54 ","pages":"Article 100859"},"PeriodicalIF":0.9,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142172518","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":"Revisiting some classical linearizations of the quadratic binary optimization problem and linkages with constraint aggregations","authors":"Abraham P. Punnen, Navpreet Kaur Dhanda","doi":"10.1016/j.disopt.2024.100858","DOIUrl":"10.1016/j.disopt.2024.100858","url":null,"abstract":"<div><p>In this paper, we examine explicit linearization models associated with the quadratic unconstrained binary optimization problem (QUBO). After reviewing some well-known basic linearization techniques, we present a new explicit linearization technique (PK) for QUBO with interesting properties. In particular, PK yields the same LP relaxation value as that of the standard linearization of QUBO while employing less number of constraints. We then show that using weighted aggregations of selected constraints of the basic linearization models, several new and effective linearization models of QUBO can be developed. Although aggregation of constraints has been studied in the past for solving systems of Diophantine equations in non-negative variables, none of them resulted in practical models, particularly due to the large size of the associated multipliers. For our aggregation based models, the multipliers can be of any positive real number. Moreover, we show that by choosing the multipliers appropriately, the LP relaxations of the resulting linearizations have optimal objective function values identical to that of the corresponding non-aggregated models. Theoretical and experimental comparisons of the new and existing explicit linearization models are also provided. Although our discussions were focused primarily on QUBO, the models and results we obtained extend naturally to the quadratic binary optimization problem with linear constraints.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"54 ","pages":"Article 100858"},"PeriodicalIF":0.9,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1572528624000379/pdfft?md5=88f5ed7ffa74e8ecf9bb69fd52845013&pid=1-s2.0-S1572528624000379-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142012067","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}
Victoria Kaial, Hervé Kerivin , Annegret K. Wagler
{"title":"On non-superperfection of edge intersection graphs of paths","authors":"Victoria Kaial, Hervé Kerivin , Annegret K. Wagler","doi":"10.1016/j.disopt.2024.100857","DOIUrl":"10.1016/j.disopt.2024.100857","url":null,"abstract":"<div><p>The routing and spectrum assignment problem in modern flexgrid elastic optical networks asks for assigning to given demands a route in an optical network and a channel within an optical frequency spectrum so that the channels of two demands are disjoint whenever their routes share a link in the optical network. This problem can be modeled in two phases: firstly, a selection of paths in the network and, secondly, an interval coloring problem in the edge intersection graph of these paths. The interval chromatic number equals the smallest size of a spectrum such that a proper interval coloring is possible, the weighted clique number is a natural lower bound. Graphs where both parameters coincide for all possible non-negative integral weights are called superperfect. Therefore, the occurrence of non-superperfect edge intersection graphs of routing paths can provoke the need of larger spectral resources. In this work, we examine the question which minimal non-superperfect graphs can occur in the edge intersection graphs of routing paths in different underlying networks: when the network is a path, a tree, a cycle, or a sparse planar graph with small maximum degree. We show that for any possible network (even if it is restricted to a path) the resulting edge intersection graphs are not necessarily superperfect. We close with a discussion of possible consequences and of some lines of future research.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"54 ","pages":"Article 100857"},"PeriodicalIF":0.9,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141979335","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 packing number of cubic graphs","authors":"Wayne Goddard , Michael A. Henning","doi":"10.1016/j.disopt.2024.100850","DOIUrl":"10.1016/j.disopt.2024.100850","url":null,"abstract":"<div><p>A packing in a graph is a set of vertices that are mutually distance at least 3 apart. By using optimization and linear programming to help analyze the greedy algorithm, we improve on a result of Favaron and show that every connected cubic graph of order <span><math><mi>n</mi></math></span> has a packing of size at least <span><math><mrow><mfrac><mrow><mn>17</mn></mrow><mrow><mn>132</mn></mrow></mfrac><mi>n</mi><mo>−</mo><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"53 ","pages":"Article 100850"},"PeriodicalIF":0.9,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938261","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}
Jannis Blauth , Stephan Held , Dirk Müller , Niklas Schlomberg , Vera Traub , Thorben Tröbst , Jens Vygen
{"title":"Vehicle routing with time-dependent travel times: Theory, practice, and benchmarks","authors":"Jannis Blauth , Stephan Held , Dirk Müller , Niklas Schlomberg , Vera Traub , Thorben Tröbst , Jens Vygen","doi":"10.1016/j.disopt.2024.100848","DOIUrl":"10.1016/j.disopt.2024.100848","url":null,"abstract":"<div><p>We develop theoretical foundations and practical algorithms for vehicle routing with time-dependent travel times. We also provide new benchmark instances and experimental results.</p><p>First, we study basic operations on piecewise linear arrival time functions. In particular, we devise a faster algorithm to compute the pointwise minimum of a set of piecewise linear functions and a monotonicity-preserving variant of the Imai–Iri algorithm to approximate an arrival time function with fewer breakpoints.</p><p>Next, we show how to evaluate insertion and deletion operations in tours efficiently and update the underlying data structure faster than previously known when a tour changes. Evaluating a tour also requires a scheduling step which is non-trivial in the presence of time windows and time-dependent travel times. We show how to perform this in linear time.</p><p>Based on these results, we develop a local search heuristic to solve real-world vehicle routing problems with various constraints efficiently and report experimental results on classical benchmarks. Since most of these do not have time-dependent travel times, we generate and publish new benchmark instances that are based on real-world data. This data also demonstrates the importance of considering time-dependent travel times in instances with tight time windows.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"53 ","pages":"Article 100848"},"PeriodicalIF":0.9,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1572528624000276/pdfft?md5=22a97d8c380cf8927372907e85523ccf&pid=1-s2.0-S1572528624000276-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938257","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}
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