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Breaking the Quadratic Barrier for Matroid Intersection [article]

Joakim Blikstad, Jan van den Brand, Sagnik Mukhopadhyay, Danupon Nanongkai
2021 arXiv   pre-print
The matroid intersection problem is a fundamental problem that has been extensively studied for half a century.  ...  The only progress towards breaking the quadratic barrier requires either approximation algorithms or a more powerful rank oracle query [CLS+ FOCS 2019].  ...  algorithms to break the quadratic barrier.  ... 
arXiv:2102.05548v2 fatcat:rez2h7vtbngrjjrm2bd3sa4rfu

Faster Matroid Intersection [article]

Deeparnab Chakrabarty, Yin Tat Lee, Aaron Sidford, Sahil Singla, and Sam Chiu-wai Wong
2019 arXiv   pre-print
In this paper we consider the classic matroid intersection problem: given two matroids _1=(V,_1) and _2=(V,_2) defined over a common ground set V, compute a set S∈_1∩_2 of largest possible cardinality,  ...  The rank oracle is of particular interest as the matroid intersection problem with this oracle is a special case of the submodular function minimization problem with an evaluation oracle.  ...  Acknowledgment We are thankful to Kent Quanrud for pointing us to Karger's paper [23] . We are also thankful to Troy Lee for pointing us to [32] , and for his comments on a previous version.  ... 
arXiv:1911.10765v1 fatcat:seloq5nt4jgdxibgeyuwo6y44y

The Central Curve in Linear Programming

Jesús A. De Loera, Bernd Sturmfels, Cynthia Vinzant
2012 Foundations of Computational Mathematics  
These invariants, along with the degree of the Gauss image of the curve, are expressed in terms of the matroid of the input matrix.  ...  The central curve of a linear program is an algebraic curve specified by linear and quadratic constraints arising from complementary slackness.  ...  Todd for useful suggestions, clarifications, and references. Bernd Sturmfels and Cynthia Vinzant acknowledge support by the U.S. National Science Foundation (DMS-0757207 and DMS-0968882).  ... 
doi:10.1007/s10208-012-9127-7 fatcat:sxxuv325bfeyvaorjelucc4yhq

Pivot rules for linear programming: A survey on recent theoretical developments

Tamás Terlaky, Shuzhong Zhang
1993 Annals of Operations Research  
The last class has the common feature that the rules all have close connections to certain interior point methods. Finally, we mention some open problems for future research.  ...  In this paper we discuss three classes of recently developed pivot rules for linear programming.  ...  Acknowledgements The authors are grateful to the anonymous referees and Hans Frenk for the constructive comments on the first version of the paper.  ... 
doi:10.1007/bf02096264 fatcat:foj3h7ql6bavddvblb5pg4u4fu

Rigidity through a Projective Lens

Anthony Nixon, Bernd Schulze, Walter Whiteley
2021 Applied Sciences  
to the spherical metric, slide joints and pure conditions for singular configurations.  ...  Part III widens the lens to include the closely related cofactor matroids arising from multivariate splines, which also exhibit the projective invariance.  ...  Acknowledgments: We would like to thank the referees for their careful reading and numerous helpful suggestions which improved the presentation of the paper.  ... 
doi:10.3390/app112411946 fatcat:25pxvf5mezavdi7fb46gramgby

Online User-AP Association with Predictive Scheduling in Wireless Caching Networks [article]

Xi Huang and Shuang Zhao and Xin Gao and Ziyu Shao and Hua Qian and Yang Yang
2020 arXiv   pre-print
For wireless caching networks, the scheme design for content delivery is non-trivial in the face of the following tradeoff.  ...  By exploiting its unique structure, we transform the problem into a series of modular maximization sub-problems with matroid constraints.  ...  Moreover, with prediction, PUARA can further break the [O(1/V ), O(V )] performance barrier with a notable reduction in the average queue backlog size of the system.  ... 
arXiv:2008.00161v1 fatcat:gulnb4vs2zd5riskap3nk5ij2m

14th International Symposium on Mathematical Programming

1990 Mathematical programming  
It is shown that for the success of the variant dom must ful ll a regularity property and that the choice of the normal vectors must meet some demands.Both requirements are ful lled if dom is polyhedral  ...  well known serious steps und null steps of bundle methods a third type of steps is used to generate the model of dom .  ...  The matroid parity problem, which contains the matching and the matroid intersection problem, is intractable in general, but solvable if the matroid in question is linearly represented.  ... 
doi:10.1007/bf01580875 fatcat:3jtclwmntzgjxkqs5uecombdaa

Full Spark Frames [article]

Boris Alexeev, Jameson Cahill, Dustin G. Mixon
2012 arXiv   pre-print
In applications like sparse signal processing, data transmission with robustness to erasures, and reconstruction without phase, there is a pressing need for deterministic constructions of frames with the  ...  following property: every size-M subcollection of the M-dimensional frame elements is a spanning set.  ...  For the moment, we assume that F succeeds in representing the matroid.  ... 
arXiv:1110.3548v2 fatcat:wbom7hkrabd5xj5xrdcnirnwta

A Faster Cutting Plane Method and its Implications for Combinatorial and Convex Optimization [article]

Yin Tat Lee, Aaron Sidford, Sam Chiu-wai Wong
2015 arXiv   pre-print
Matroid Intersection: Our runtimes are O(nrT_rank n (nM) +n^3^O(1)(nM)) and O(n^2 (nM) T_ind+n^3 ^O(1) (nM)), achieving the first quadratic bound on the query complexity for the independence and rank oracles  ...  In the unweighted case, this is the first improvement since 1986 for independence oracle.  ...  Acknowledgments We thank Matt Weinberg for insightful comments about submodular minimization and minimizing the intersection of convex sets that were deeply influential to our work.  ... 
arXiv:1508.04874v2 fatcat:74i57p5lardbnmstsdml66ovta

Sensor Planning for Large Numbers of Robots [article]

Micah Corah
2021 arXiv   pre-print
Please refer to the thesis for the full abstract.* After a disaster, locating and extracting victims quickly is critical because mortality rises rapidly after the first two days.  ...  Specifically, consider the problem of how to select motions for a team of robots to maximize a notion of sensing quality (the sensing objective) over the near future, say by maximizing the amount of unknown  ...  [194] demonstrated that this approach also applies to problems with more complex constraints such as intersections of matroids.  ... 
arXiv:2102.04054v1 fatcat:zgg75lo6wfdwvekn5twqpmi7oe

Cardinality Minimization, Constraints, and Regularization: A Survey [article]

Andreas M. Tillmann, Daniel Bienstock, Andrea Lodi, Alexandra Schwartz
2021 arXiv   pre-print
often regarded as impractical due to commonly unsatisfactory behavior of black-box solvers applied to generic problem formulations, can in fact produce provably high-quality or even optimal solutions for  ...  We survey optimization problems that involve the cardinality of variable vectors in constraints or the objective function.  ...  The girth of a matroid M * equals the cogirth of the associated dual matroid M * .  ... 
arXiv:2106.09606v1 fatcat:jmsvxuqycvfz3gha4jv4f5seze

Minimum color‐degree perfect b ‐matchings

Mariia Anapolska, Christina Büsing, Martin Comis, Tabea Krabs
2020 Networks  
The last speaker of the session will chair the session, with two exceptions for PhD-only sessions: Combinatorial Optimization (Mon 18, Room PP, 16-17) chaired by R.  ...  Schedule The seminar rooms are the Paul Painlevé (PP), the Robert Faure (Z) and the Jean-Baptiste Say (Y) amphitheatres, located in Access 1, lower ground floor.  ...  We then show a kernelization algorithm for γ-matching parameterized by the size of the solution. Our process produces quadratic kernels.  ... 
doi:10.1002/net.21974 fatcat:4qgtutmxrrcy7anvxks4fivcii

A Glimpse at Paul G. Spirakis [chapter]

Ioannis Chatzigiannakis, Dimitris Fotakis, Spyros Kontogiannis, Othon Michail, Sotiris Nikoletseas, Grammati Pantziou, Christos Zaroliagis
2015 Lecture Notes in Computer Science  
We are indebted to Eleni Spiraki (Paul's sister) for her great help in the preparation of this article.  ...  The main idea was based on the polynomial-time tractability of convex quadratic programs, along with a parameterization of the quadratic-program formulation of Mangasarian and Stone for Nash equilibria  ...  In [73] , he suggested the employment of probabilistic choice for interprocess communication (and symmetry breaking).  ... 
doi:10.1007/978-3-319-24024-4_1 fatcat:vmn7swilvjdcdjs6lir335pch4

A survey of the algorithmic aspects of modular decomposition

Michel Habib, Christophe Paul
2010 Computer Science Review  
Since the first polynomial time algorithm in the early 1970's, the algorithmic of the modular decomposition has known an important development.  ...  The notion of a module naturally appears in the proofs of many graph theoretical theorems.  ...  and matroids, among others.  ... 
doi:10.1016/j.cosrev.2010.01.001 fatcat:5ylx2ohitbarpkkr3jatb3hqwu

A survey on algorithmic aspects of modular decomposition [article]

Michel Habib, Christophe Paul
2009 arXiv   pre-print
Since the first polynomial time algorithm in the early 70's, the algorithmic of the modular decomposition has known an important development.  ...  The modular decomposition is a technique that applies but is not restricted to graphs. The notion of module naturally appears in the proofs of many graph theoretical theorems.  ...  and matroids among others.  ... 
arXiv:0912.1457v2 fatcat:oqxqjjnoejgqrlr72plmzeisga
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