Flexible Object Hierarchies in Polymake

Gawrilow, Ewgenij; Joswig, Michael

doi:10.1007/11832225_20

Cited by 14 publications

(18 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this research, we have made extensive use of both of the following software packages: LattE [1], and polymake [7,8].…”

Section: Ehrhart Quasi-polynomials and Period Collapsementioning

confidence: 99%

On the period collapse of a family of Ehrhart quasi-polynomials

Fernandes,

de Pina,

Alfonsín

et al. 2021

Preprint

View full text Add to dashboard Cite

A graph whose nodes have degree 1 or 3 is called a {1, 3}-graph. Liu and Osserman associated a polytope to each {1, 3}-graph and studied the Ehrhart quasipolynomials of these polytopes. They showed that the vertices of these polytopes have coordinates in the set {0, 1 4 , 1 2 , 1}, which implies that the period of their Ehrhart quasi-polynomials is either 1, 2, or 4. We show that the period of the Ehrhart quasipolynomial of these polytopes is at most 2 if the graph is a tree or a cubic graph, and it is equal to 4 otherwise.In the process of proving this theorem, several interesting combinatorial and geometric properties of these polytopes were uncovered, arising from the structure of their associated graphs. The tools developed here may find other applications in the study of Ehrhart quasi-polynomials and enumeration problems for other polytopes that arise from graphs. Additionally, we have identified some interesting connections with triangulations of 3-manifolds.

show abstract

“…In this research, we have made extensive use of both of the following software packages: LattE [1], and polymake [7,8].…”

Section: Ehrhart Quasi-polynomials and Period Collapsementioning

confidence: 99%

On the period collapse of a family of Ehrhart quasi-polynomials

Fernandes,

de Pina,

Alfonsín

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…In this way it is avoided to repeat costly computations. More details on polymake's big object concept are found in [20].…”

Section: Tropical Hypersurfaces and Dual Subdivisions Letmentioning

confidence: 99%

Tropical Computations in polymake

Hampe

Joswig

2017

Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory

Self Cite

View full text Add to dashboard Cite

show abstract

“…We used both the multiparametric linear programming approach (Theorem ) and the boundary triangulation method (Theorem ) to obtain optimal routing controllers . In particular, we used Multi‐Parametric Toolbox (MPT, ) for Matlab to solve (21)–(23) and polymake to compute boundary triangulations. The objective function for (21)–(23) was set for minimizing the total cost of the routing.…”

Section: Numerical Evaluationsmentioning

confidence: 99%

Rate-adaptive multipath routing: Distributed, centralized, and hybrid architectures

Németh¹,

Rétvári

2015

NETWORKS

View full text Add to dashboard Cite

With the increasing volume and volatility of Internet traffic, the need for adaptive routing algorithms has become compelling lately. An adaptive routing algorithm controls the rate at which traffic is placed on forwarding paths in concert with the actual user demands, making it possible to avoid congestion even when no information on expected traffic is available. In this article, we present a new model for rate‐adaptive multipath routing, which allows one to analyze distributed, centralized, and hybrid routing architectures within a single framework, and to develop quantitative as well as qualitative arguments regarding their optimality, stability, and realizability. By a novel generalization of oblivious routing, we present a centralized algorithm with provable optimality, and we arrive at the conclusion that congestion can be completely eliminated even if routing decisions are completely precomputed. We find, although, that the complexity of the centralized scheme can become exponential. Therefore, we develop a hybrid distributed‐centralized algorithm that combines the simplicity of distributed algorithms with the efficiency of centralized ones, and we provide numerical studies demonstrating that the hybrid scheme performs well in a broad selection of realistic scenarios. © 2015 Wiley Periodicals, Inc.NETWORKS, Vol. 66(2), 118–135 2015

show abstract

Flexible Object Hierarchies in Polymake

Cited by 14 publications

References 1 publication

On the period collapse of a family of Ehrhart quasi-polynomials

On the period collapse of a family of Ehrhart quasi-polynomials

Tropical Computations in polymake

Rate-adaptive multipath routing: Distributed, centralized, and hybrid architectures

Contact Info

Product

Resources

About