Discrete & Computational Geometry最新文献_第3页

A Subdivision Algebra for a Product of Two Simplices via Flow Polytopes 通过流多边形计算两简约积的细分代数

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-06-28 DOI: 10.1007/s00454-024-00671-9

Matias von Bell

引用次数: 0

Transversals to Colorful Intersecting Convex Sets 彩色相交凸集的横截面

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-06-27 DOI: 10.1007/s00454-024-00669-3

Cuauhtemoc Gomez-Navarro, Edgardo Roldán-Pensado

{"title":"Transversals to Colorful Intersecting Convex Sets","authors":"Cuauhtemoc Gomez-Navarro, Edgardo Roldán-Pensado","doi":"10.1007/s00454-024-00669-3","DOIUrl":"https://doi.org/10.1007/s00454-024-00669-3","url":null,"abstract":"Let K be a compact convex set in (mathbb {R}^{2}) and let (mathcal {F}_{1}, mathcal {F}_{2}, mathcal {F}_{3}) be finite families of translates of K such that (A cap B ne emptyset ) for every (A in mathcal {F}_{i}) and (B in mathcal {F}_{j}) with (i ne j). A conjecture by Dol’nikov is that, under these conditions, there is always some (j in { 1,2,3 }) such that (mathcal {F}_{j}) can be pierced by 3 points. In this paper we prove a stronger version of this conjecture when K is a body of constant width or when it is close in Banach-Mazur distance to a disk. We also show that the conjecture is true with 8 piercing points instead of 3. Along the way we prove more general statements both in the plane and in higher dimensions. A related result was given by Martínez-Sandoval, Roldán-Pensado and Rubin. They showed that if (mathcal {F}_{1}, dots , mathcal {F}_{d}) are finite families of convex sets in (mathbb {R}^{d}) such that for every choice of sets (C_{1} in mathcal {F}_{1}, dots , C_{d} in mathcal {F}_{d}) the intersection (bigcap _{i=1}^{d} {C_{i}}) is non-empty, then either there exists (j in { 1,2, dots , n }) such that (mathcal {F}_j) can be pierced by few points or (bigcup _{i=1}^{n} mathcal {F}_{i}) can be crossed by few lines. We give optimal values for the number of piercing points and crossing lines needed when (d=2) and also consider the problem restricted to special families of convex sets.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"9 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141500789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Coarse Embeddability of Wasserstein Space and the Space of Persistence Diagrams 瓦瑟斯坦空间和持久图空间的粗嵌入性

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-06-26 DOI: 10.1007/s00454-024-00674-6

Neil Pritchard, Thomas Weighill

引用次数: 0

Improved Algebraic Degeneracy Testing 改进的代数退化测试

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-06-25 DOI: 10.1007/s00454-024-00673-7

Jean Cardinal, Micha Sharir

{"title":"Improved Algebraic Degeneracy Testing","authors":"Jean Cardinal, Micha Sharir","doi":"10.1007/s00454-024-00673-7","DOIUrl":"https://doi.org/10.1007/s00454-024-00673-7","url":null,"abstract":"In the classical linear degeneracy testing problem, we are given n real numbers and a k-variate linear polynomial F, for some constant k, and have to determine whether there exist k numbers (a_1,ldots ,a_k) from the set such that (F(a_1,ldots ,a_k) = 0). We consider a generalization of this problem in which F is an arbitrary constant-degree polynomial, we are given k sets of n real numbers, and have to determine whether there exists a k-tuple of numbers, one in each set, on which F vanishes. We give the first improvement over the naïve (O^*(n^{k-1})) algorithm for this problem (where the (O^*(cdot )) notation omits subpolynomial factors). We show that the problem can be solved in time (O^*left( n^{k - 2 + frac{4}{k+2}}right) ) for even k and in time (O^*left( n^{k - 2 + frac{4k-8}{k^2-5}}right) ) for odd k in the real RAM model of computation. We also prove that for (k=4), the problem can be solved in time (O^*(n^{2.625})) in the algebraic decision tree model, and for (k=5) it can be solved in time (O^*(n^{3.56})) in the same model, both improving on the above uniform bounds. All our results rely on an algebraic generalization of the standard meet-in-the-middle algorithm for k-SUM, powered by recent algorithmic advances in the polynomial method for semi-algebraic range searching. In fact, our main technical result is much more broadly applicable, as it provides a general tool for detecting incidences and other interactions between points and algebraic surfaces in any dimension. In particular, it yields an efficient algorithm for a general, algebraic version of Hopcroft’s point-line incidence detection problem in any dimension.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"93 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141500790","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Power Mean Inequalities and Sums of Squares 幂均值不等式和平方和

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-06-23 DOI: 10.1007/s00454-024-00652-y

Jose Acevedo, Grigoriy Blekherman

{"title":"Power Mean Inequalities and Sums of Squares","authors":"Jose Acevedo, Grigoriy Blekherman","doi":"10.1007/s00454-024-00652-y","DOIUrl":"https://doi.org/10.1007/s00454-024-00652-y","url":null,"abstract":"We study the limits of the cones of symmetric nonnegative polynomials and symmetric sums of squares, when expressed in power-mean or monomial-mean basis. These limits correspond to forms with stable expression in power-mean polynomials that are globally nonnegative (resp. sums of squares) regardless of the number of variables. We introduce partial symmetry reduction to describe the limit cone of symmetric sums of squares, and reprove a result of Blekherman and Riener (Discrete Comput Geom 65:1–36, 2020) that limits of symmetric nonnegative polynomials and sums of squares agree in degree 4. We use tropicalization of the dual cones, first considered in the context of comparing nonnegative polynomials and sums of squares in Blekherman et al. (Trans Am Math Soc 375(09):6281–6310, 2022), to show differences between cones of symmetric polynomials and sums of squares starting in degree 6, which disproves a conjecture of Blekherman and Riener (Discrete Comput Geom 65:1–36, 2020). For even symmetric nonnegative forms and sums of squares we show that the cones agree up to degree 8, and are different starting with degree 10. We also find, via tropicalization, explicit examples of symmetric forms that are nonnegative but not sums of squares in the limit.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"45 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141527490","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Bounds for the Regularity Radius of Delone Sets 德隆集合正则半径的界限

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-06-22 DOI: 10.1007/s00454-024-00666-6

Nikolay Dolbilin, Alexey Garber, Egon Schulte, Marjorie Senechal

{"title":"Bounds for the Regularity Radius of Delone Sets","authors":"Nikolay Dolbilin, Alexey Garber, Egon Schulte, Marjorie Senechal","doi":"10.1007/s00454-024-00666-6","DOIUrl":"https://doi.org/10.1007/s00454-024-00666-6","url":null,"abstract":"Delone sets are discrete point sets X in ({mathbb {R}}^d) characterized by parameters (r, R), where (usually) 2r is the smallest inter-point distance of X, and R is the radius of a largest “empty ball” that can be inserted into the interstices of X. The regularity radius ({hat{rho }}_d) is defined as the smallest positive number (rho ) such that each Delone set with congruent clusters of radius (rho ) is a regular system, that is, a point orbit under a crystallographic group. We discuss two conjectures on the growth behavior of the regularity radius. Our “Weak Conjecture” states that ({hat{rho }}_{d}={textrm{O}(d^2log _2 d)}R) as (drightarrow infty ), independent of r. This is verified in the paper for two important subfamilies of Delone sets: those with full-dimensional clusters of radius 2r and those with full-dimensional sets of d-reachable points. We also offer support for the plausibility of a “Strong Conjecture”, stating that ({hat{rho }}_{d}={textrm{O}(dlog _2 d)}R) as (drightarrow infty ), independent of r.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"123 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141500791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A Sylvester–Gallai-Type Theorem for Complex-Representable Matroids 复杂可表示矩阵的西尔维斯特-加莱类型定理

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-06-11 DOI: 10.1007/s00454-024-00661-x

Jim Geelen, Matthew E. Kroeker

引用次数: 0

Fragmenting any Parallelepiped into a Signed Tiling 将任意平行六面体分割成有符号平铺法

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-06-11 DOI: 10.1007/s00454-024-00664-8

Joseph Doolittle, Alex McDonough

{"title":"Fragmenting any Parallelepiped into a Signed Tiling","authors":"Joseph Doolittle, Alex McDonough","doi":"10.1007/s00454-024-00664-8","DOIUrl":"https://doi.org/10.1007/s00454-024-00664-8","url":null,"abstract":"It is broadly known that any parallelepiped tiles space by translating copies of itself along its edges. In earlier work relating to higher-dimensional sandpile groups, the second author discovered a novel construction which fragments the parallelepiped into a collection of smaller tiles. These tiles fill space with the same symmetry as the larger parallelepiped. Their volumes are equal to the components of the multi-row Laplace determinant expansion, so this construction only works when all of these signs are non-negative (or non-positive). In this work, we extend the construction to work for all parallelepipeds, without requiring the non-negative condition. This naturally gives tiles with negative volume, which we understand to mean canceling out tiles with positive volume. In fact, with this cancellation, we prove that every point in space is contained in exactly one more tile with positive volume than tile with negative volume. This is a natural definition for a signed tiling. Our main technique is to show that the net number of signed tiles doesn’t change as a point moves through space. This is a relatively indirect proof method, and the underlying structure of these tilings remains mysterious.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141500815","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Random Meander Model for Links 链接的随机蜿蜒模型

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-06-11 DOI: 10.1007/s00454-024-00663-9

Nicholas Owad, Anastasiia Tsvietkova

引用次数: 0

On the Dominant of the Multicut Polytope 论多胞多面体的多面性

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2024-06-04 DOI: 10.1007/s00454-024-00646-w

Markus Chimani, Martina Juhnke, Alexander Nover

{"title":"On the Dominant of the Multicut Polytope","authors":"Markus Chimani, Martina Juhnke, Alexander Nover","doi":"10.1007/s00454-024-00646-w","DOIUrl":"https://doi.org/10.1007/s00454-024-00646-w","url":null,"abstract":"Given a graph (G=(V,E)) and a set (S subseteq left( {begin{array}{c}V 2end{array}}right) ) of terminal pairs, the minimum multicut problem asks for a minimum edge set (delta subseteq E) such that there is no s-t-path in (G -delta ) for any ({s,t}in S). For (|S|=1) this is the well known s-t-cut problem, but in general the minimum multicut problem is NP-complete, even if the input graph is a tree. The multicut polytope (textsc {MultC}^square (G,S)) is the convex hull of all multicuts in G; the multicut dominant is given by (textsc {MultC}(G,S)=textsc {MultC}^square (G,S)+mathbb {R}^E_{{ge 0}}). The latter is the relevant object for the minimization problem. While polyhedra associated to several cut problems have been studied intensively there is only little knowledge for multicut. We investigate properties of the multicut dominant and in particular derive results on liftings of facet-defining inequalities. This yields a classification of all facet-defining path- and edge inequalities. Moreover, we investigate the effect of graph operations such as node splitting, edge subdivisions, and edge contractions on the multicut-dominant and its facet-defining inequalities. In addition, we introduce facet-defining inequalities supported on stars, trees, and cycles and show that the former two can be separated in polynomial time when the input graph is a tree.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"4 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141258295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0