Mathematical Programming最新文献

{"title":"Tight lower bounds for block-structured integer programs.","authors":"Christoph Hunkenschröder, Kim-Manuel Klein, Martin Koutecký, Alexandra Lassota, Asaf Levin","doi":"10.1007/s10107-025-02296-z","DOIUrl":"https://doi.org/10.1007/s10107-025-02296-z","url":null,"abstract":"<p><p>We study fundamental block-structured integer programs called tree-fold and multi-stage IPs. Tree-fold IPs have a constraint matrix with independent blocks linked together by few constraints in a recursive pattern. Transposing this constraint matrix yields the constraint matrix of multi-stage IPs. The state-of-the-art algorithms to solve these IPs have an exponential gap in their running times, making it natural to ask whether this gap is inherent. We answer this question in the affirmative. Assuming the Exponential Time Hypothesis, we prove lower bounds showing that the exponential difference is necessary. This also proves that the known algorithms are essentially optimal. Moreover, we prove unconditional lower bounds on the size of the Graver basis elements, a fundamental building block of all known algorithms to solve these IPs. This shows that none of the current approaches can be improved beyond this bound unconditionally.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"216 1-2","pages":"521-538"},"PeriodicalIF":2.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13124973/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147817113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An FPTAS for Connectivity Interdiction.","authors":"Chien-Chung Huang, Nidia Obscura Acosta, Sorrachai Yingchareonthawornchai","doi":"10.1007/s10107-025-02312-2","DOIUrl":"https://doi.org/10.1007/s10107-025-02312-2","url":null,"abstract":"<p><p>In the connectivity interdiction problem, we are asked to find a global graph cut and remove a subset of edges under a budget constraint, so that the total weight of the remaining edges in this cut is minimized. This problem easily includes the knapsack problem as a special case, hence it is NP-hard. For this problem, Zenklusen [Zenklusen'14] designed a polynomial-time approximation scheme (PTAS) and exact algorithms for the special case of unit edge costs. He posed the question of whether a fully polynomial-time approximation scheme (FPTAS) is possible for the general case. We give an affirmative answer. For the special case of unit edge costs, we also give faster exact and approximation algorithms. Our main technical contribution is to establish a connection with an intermediate graph cut problem, called the <i>normalized</i> min-cut, which, roughly speaking, penalizes the edge weights of the remaining edges more severely, when more edges are taken out for free.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"216 1-2","pages":"605-626"},"PeriodicalIF":2.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13124762/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147817171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A first order method for linear programming parameterized by circuit imbalance.","authors":"Richard Cole, Christoph Hertrich, Yixin Tao, László A Végh","doi":"10.1007/s10107-025-02264-7","DOIUrl":"https://doi.org/10.1007/s10107-025-02264-7","url":null,"abstract":"<p><p>Various first order approaches have been proposed in the literature to solve Linear Programming (LP) problems, recently leading to practically efficient solvers for large-scale LPs. From a theoretical perspective, linear convergence rates have been established for first order LP algorithms, despite the fact that the underlying formulations are not strongly convex. However, the convergence rate typically depends on the Hoffman constant of a large matrix that contains the constraint matrix, as well as the right hand side, cost, and capacity vectors. We introduce a first order approach for LP optimization with a convergence rate depending polynomially on the circuit imbalance measure, which is a geometric parameter of the constraint matrix, and depending logarithmically on the right hand side, capacity, and cost vectors. This provides much stronger convergence guarantees. For example, if the constraint matrix is totally unimodular, we obtain polynomial-time algorithms, whereas the convergence guarantees for approaches based on primal-dual formulations may have arbitrarily slow convergence rates for this class. Our approach is based on a fast gradient method due to Necoara, Nesterov, and Glineur (Math. Prog. 2019); this algorithm is called repeatedly in a framework that gradually fixes variables to the boundary. This technique is based on a new approximate version of Tardos's method, that was used to obtain a strongly polynomial algorithm for combinatorial LPs (Oper. Res. 1986).</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"216 1-2","pages":"339-377"},"PeriodicalIF":2.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13124851/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147817167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Accelerated first-order optimization under nonlinear constraints.","authors":"Michael Muehlebach, Michael I Jordan","doi":"10.1007/s10107-025-02224-1","DOIUrl":"10.1007/s10107-025-02224-1","url":null,"abstract":"<p><p>We exploit analogies between first-order algorithms for constrained optimization and non-smooth dynamical systems to design a new class of accelerated first-order algorithms for constrained optimization. Unlike Frank-Wolfe or projected gradients, these algorithms avoid optimization over the entire feasible set at each iteration. We prove convergence to stationary points even in a nonconvex setting and we derive accelerated rates for the convex setting both in continuous time, as well as in discrete time. An important property of these algorithms is that constraints are expressed in terms of velocities instead of positions, which naturally leads to sparse, local and convex approximations of the feasible set (even if the feasible set is nonconvex). Thus, the complexity tends to grow mildly in the number of decision variables and in the number of constraints, which makes the algorithms suitable for machine learning applications. We apply our algorithms to a compressed sensing and a sparse regression problem, showing that we can treat nonconvex <math><msup><mi>ℓ</mi> <mi>p</mi></msup> </math> constraints ( <math><mrow><mi>p</mi> <mo><</mo> <mn>1</mn></mrow> </math> ) efficiently, while recovering state-of-the-art performance for <math><mrow><mi>p</mi> <mo>=</mo> <mn>1</mn></mrow> </math> .</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"215 1-2","pages":"407-452"},"PeriodicalIF":2.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12790563/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145959590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">A <ns0:math> <ns0:mrow><ns0:mfrac><ns0:mn>4</ns0:mn> <ns0:mn>3</ns0:mn></ns0:mfrac> </ns0:mrow> </ns0:math> -approximation for the maximum leaf spanning arborescence problem in DAGs.","authors":"Meike Neuwohner","doi":"10.1007/s10107-025-02233-0","DOIUrl":"https://doi.org/10.1007/s10107-025-02233-0","url":null,"abstract":"<p><p>The Maximum Leaf Spanning Arborescence problem (MLSA) in directed acyclic graphs (dags) is defined as follows: Given a directed acyclic graph <i>G</i> and a vertex <math><mrow><mi>r</mi> <mo>∈</mo> <mi>V</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo></mrow> </math> from which every other vertex is reachable, find a spanning arborescence rooted at <i>r</i> maximizing the number of leaves (vertices with out-degree zero). The MLSA in dags is known to be APX-hard as reported by Nadine Schwartges, Spoerhase, and Wolff (Approximation and Online Algorithms, Springer, Berlin Heidelberg, 2012) and the best known approximation guarantee of <math><mfrac><mn>7</mn> <mn>5</mn></mfrac> </math> is due to Fernandes and Lintzmayer (J. Comput. Syst. Sci. 135: 158-174,2023): They prove that any <math><mi>α</mi></math> -approximation for the <i>hereditary</i> 3-<i>set packing problem</i>, a special case of weighted 3-set packing, yields a <math><mrow><mo>max</mo> <mo>{</mo> <mfrac><mn>4</mn> <mn>3</mn></mfrac> <mo>,</mo> <mi>α</mi> <mo>}</mo></mrow> </math> -approximation for the MLSA in dags, and provide a <math><mfrac><mn>7</mn> <mn>5</mn></mfrac> </math> -approximation for the hereditary 3-set packing problem. In this paper, we improve upon this result by providing a <math><mfrac><mn>4</mn> <mn>3</mn></mfrac> </math> -approximation for the hereditary 3-set packing problem, and, thus, the MLSA in dags. The algorithm that we study is a simple local search procedure considering swaps of size up to 10 and can be analyzed via a two-stage charging argument. We further provide a clear picture of the general connection between the MLSA in dags and set packing by rephrasing the MLSA in dags as a <i>hereditary set packing problem</i>. With a much simpler proof, we extend the reduction by Fernandes and Lintzmayer and show that an <math><mi>α</mi></math> -approximation for the <i>hereditary</i> <i>k</i>-<i>set packing problem</i> implies a <math><mrow><mo>max</mo> <mo>{</mo> <mfrac><mrow><mi>k</mi> <mo>+</mo> <mn>1</mn></mrow> <mi>k</mi></mfrac> <mo>,</mo> <mi>α</mi> <mo>}</mo></mrow> </math> -approximation for the MLSA dags. On the other hand, we provide lower bound examples proving that our approximation guarantee of <math><mfrac><mn>4</mn> <mn>3</mn></mfrac> </math> is best possible for local search algorithms with constant improvement size.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"216 1-2","pages":"111-133"},"PeriodicalIF":2.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13124901/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147817115","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A better-than-1.6-approximation for prize-collecting TSP.","authors":"Jannis Blauth, Nathan Klein, Martin Nägele","doi":"10.1007/s10107-025-02221-4","DOIUrl":"https://doi.org/10.1007/s10107-025-02221-4","url":null,"abstract":"<p><p>Prize-Collecting TSP is a variant of the traveling salesperson problem where one may drop vertices from the tour at the cost of vertex-dependent penalties. The quality of a solution is then measured by adding the length of the tour and the sum of all penalties of vertices that are not visited. We present a polynomial-time approximation algorithm with an approximation guarantee slightly below 1.6, where the guarantee is with respect to the natural linear programming relaxation of the problem. This improves upon the previous best-known approximation ratio of 1.774. Our approach is based on a known decomposition for solutions of this linear relaxation into rooted trees. Our algorithm takes a tree from this decomposition and then performs a pruning step before doing parity correction on the remainder. Using a simple analysis, we bound the approximation guarantee of the proposed algorithm by <math><mrow><mo>(</mo> <mn>1</mn> <mo>+</mo> <msqrt><mn>5</mn></msqrt> <mo>)</mo> <mrow><mo>/</mo></mrow> <mn>2</mn> <mo>≈</mo> <mn>1.618</mn></mrow> </math> , the golden ratio. With some additional technical care we further improve the approximation guarantee to 1.599. Furthermore, we show that for the path version of Prize-Collecting TSP (known as Prize-Collecting Stroll) our approach yields an approximation guarantee of 1.6662, improving upon the previous best-known guarantee of 1.926.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"216 1-2","pages":"87-109"},"PeriodicalIF":2.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13124855/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147817169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dyadic linear programming and extensions.","authors":"Ahmad Abdi, Gérard Cornuéjols, Bertrand Guenin, Levent Tunçel","doi":"10.1007/s10107-024-02146-4","DOIUrl":"https://doi.org/10.1007/s10107-024-02146-4","url":null,"abstract":"<p><p>A rational number is <i>dyadic</i> if it has a finite binary representation <math><mrow><mi>p</mi> <mo>/</mo> <msup><mn>2</mn> <mi>k</mi></msup> </mrow> </math> , where <i>p</i> is an integer and <i>k</i> is a nonnegative integer. Dyadic rationals are important for numerical computations because they have an exact representation in floating-point arithmetic on a computer. A vector is <i>dyadic</i> if all its entries are dyadic rationals. We study the problem of finding a dyadic optimal solution to a linear program, if one exists. We show how to solve dyadic linear programs in polynomial time. We give bounds on the size of the support of a solution as well as on the size of the denominators. We identify properties that make the solution of dyadic linear programs possible: closure under addition and negation, and density, and we extend the algorithmic framework beyond the dyadic case.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"213 1-2","pages":"473-516"},"PeriodicalIF":2.5,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12402050/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144993066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fast convergence of trust-regions for non-isolated minima via analysis of CG on indefinite matrices.","authors":"Quentin Rebjock, Nicolas Boumal","doi":"10.1007/s10107-024-02140-w","DOIUrl":"https://doi.org/10.1007/s10107-024-02140-w","url":null,"abstract":"<p><p>Trust-region methods (TR) can converge quadratically to minima where the Hessian is positive definite. However, if the minima are not isolated, then the Hessian there cannot be positive definite. The weaker Polyak-Łojasiewicz (PŁ) condition is compatible with non-isolated minima, and it is enough for many algorithms to preserve good local behavior. Yet, TR with an <i>exact</i> subproblem solver lacks even basic features such as a capture theorem under PŁ. In practice, a popular <i>inexact</i> subproblem solver is the truncated conjugate gradient method (tCG). Empirically, TR-tCG exhibits superlinear convergence under PŁ. We confirm this theoretically. The main mathematical obstacle is that, under PŁ, at points arbitrarily close to minima, the Hessian has vanishingly small, possibly negative eigenvalues. Thus, tCG is applied to ill-conditioned, indefinite systems. Yet, the core theory underlying tCG is that of CG, which assumes a positive definite operator. Accordingly, we develop new tools to analyze the dynamics of CG in the presence of small eigenvalues of any sign, for the regime of interest to TR-tCG.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"213 1-2","pages":"343-384"},"PeriodicalIF":2.5,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12401806/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144993056","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Advances on strictly <ns0:math><ns0:mi>Δ</ns0:mi></ns0:math> -modular IPs.","authors":"Martin Nägele, Christian Nöbel, Richard Santiago, Rico Zenklusen","doi":"10.1007/s10107-024-02148-2","DOIUrl":"https://doi.org/10.1007/s10107-024-02148-2","url":null,"abstract":"<p><p>There has been significant work recently on integer programs (IPs) <math><mrow><mo>min</mo> <mo>{</mo> <msup><mi>c</mi> <mi>⊤</mi></msup> <mi>x</mi> <mo>:</mo> <mi>A</mi> <mi>x</mi> <mo>≤</mo> <mi>b</mi> <mo>,</mo> <mspace></mspace> <mi>x</mi> <mo>∈</mo> <msup><mrow><mi>Z</mi></mrow> <mi>n</mi></msup> <mo>}</mo></mrow> </math> with a constraint marix <i>A</i> with bounded subdeterminants. This is motivated by a well-known conjecture claiming that, for any constant <math><mrow><mi>Δ</mi> <mo>∈</mo> <msub><mi>Z</mi> <mrow><mo>></mo> <mn>0</mn></mrow> </msub> </mrow> </math> , <math><mi>Δ</mi></math> -modular IPs are efficiently solvable, which are IPs where the constraint matrix <math><mrow><mi>A</mi> <mo>∈</mo> <msup><mrow><mi>Z</mi></mrow> <mrow><mi>m</mi> <mo>×</mo> <mi>n</mi></mrow> </msup> </mrow> </math> has full column rank and all <math><mrow><mi>n</mi> <mo>×</mo> <mi>n</mi></mrow> </math> minors of <i>A</i> are within <math><mrow><mo>{</mo> <mo>-</mo> <mi>Δ</mi> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>Δ</mi> <mo>}</mo></mrow> </math> . Previous progress on this question, in particular for <math><mrow><mi>Δ</mi> <mo>=</mo> <mn>2</mn></mrow> </math> , relies on algorithms that solve an important special case, namely <i>strictly</i> <math><mi>Δ</mi></math> -<i>modular IPs</i>, which further restrict the <math><mrow><mi>n</mi> <mo>×</mo> <mi>n</mi></mrow> </math> minors of <i>A</i> to be within <math><mrow><mo>{</mo> <mo>-</mo> <mi>Δ</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mi>Δ</mi> <mo>}</mo></mrow> </math> . Even for <math><mrow><mi>Δ</mi> <mo>=</mo> <mn>2</mn></mrow> </math> , such problems include well-known combinatorial optimization problems like the minimum odd/even cut problem. The conjecture remains open even for strictly <math><mi>Δ</mi></math> -modular IPs. Prior advances were restricted to prime <math><mi>Δ</mi></math> , which allows for employing strong number-theoretic results. In this work, we make first progress beyond the prime case by presenting techniques not relying on such strong number-theoretic prime results. In particular, our approach implies that there is a randomized algorithm to check feasibility of strictly <math><mi>Δ</mi></math> -modular IPs in strongly polynomial time if <math><mrow><mi>Δ</mi> <mo>≤</mo> <mn>4</mn></mrow> </math> .</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"210 1-2","pages":"731-760"},"PeriodicalIF":2.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11870991/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143542512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Constant-competitiveness for random assignment Matroid secretary without knowing the Matroid.","authors":"Richard Santiago, Ivan Sergeev, Rico Zenklusen","doi":"10.1007/s10107-024-02177-x","DOIUrl":"https://doi.org/10.1007/s10107-024-02177-x","url":null,"abstract":"<p><p>The Matroid Secretary Conjecture is a notorious open problem in online optimization. It claims the existence of an <i>O</i>(1)-competitive algorithm for the Matroid Secretary Problem (MSP). Here, the elements of a weighted matroid appear one-by-one, revealing their weight at appearance, and the task is to select elements online with the goal to get an independent set of largest possible weight. <i>O</i>(1)-competitive MSP algorithms have so far only been obtained for restricted matroid classes and for MSP variations, including <i>Random-Assignment</i> MSP (RA-MSP), where an adversary fixes a number of weights equal to the ground set size of the matroid, which then get assigned randomly to the elements of the ground set. Unfortunately, these approaches heavily rely on knowing the full matroid upfront. This is an arguably undesirable requirement, and there are good reasons to believe that an approach towards resolving the MSP Conjecture should not rely on it. Thus, both Soto (SIAM Journal on Computing 42(1): 178-211, 2013.) and Oveis Gharan and Vondrák (Algorithmica 67(4): 472-497, 2013.) raised as an open question whether RA-MSP admits an <i>O</i>(1)-competitive algorithm even without knowing the matroid upfront. In this work, we answer this question affirmatively. Our result makes RA-MSP the first well-known MSP variant with an <i>O</i>(1)-competitive algorithm that does not need to know the underlying matroid upfront and without any restriction on the underlying matroid. Our approach is based on first approximately learning the rank-density curve of the matroid, which we then exploit algorithmically.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"210 1-2","pages":"815-846"},"PeriodicalIF":2.2,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11870907/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143542520","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}