ﻻ يوجد ملخص باللغة العربية
Consider a graph with a rotation system, namely, for every vertex, a circular ordering of the incident edges. Given such a graph, an angle cover maps every vertex to a pair of consecutive edges in the ordering -- an angle -- such that each edge participates in at least one such pair. We show that any graph of maximum degree 4 admits an angle cover, give a poly-time algorithm for deciding if a graph with no degree-3 vertices has an angle-cover, and prove that, given a graph of maximum degree 5, it is NP-hard to decide whether it admits an angle cover. We also consider extensions of the angle cover problem where every vertex selects a fixed number $a>1$ of angles or where an angle consists of more than two consecutive edges. We show an application of angle covers to the problem of deciding if the 2-blowup of a planar graph has isomorphic thickness 2.
The Turaev-Viro invariants are a powerful family of topological invariants for distinguishing between different 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them require exponential time. The invaria
Ailon et al. [SICOMP11] proposed self-improving algorithms for sorting and Delaunay triangulation (DT) when the input instances $x_1,cdots,x_n$ follow some unknown emph{product distribution}. That is, $x_i$ comes from a fixed unknown distribution $ma
The classic Ham-Sandwich theorem states that for any $d$ measurable sets in $mathbb{R}^d$, there is a hyperplane that bisects them simultaneously. An extension by Barany, Hubard, and Jeronimo [DCG 2008] states that if the sets are convex and emph{wel
We initiate a study of algorithms with a focus on the computational complexity of individual elements, and introduce the fragile complexity of comparison-based algorithms as the maximal number of comparisons any individual element takes part in. We g
$ ewcommand{eps}{varepsilon} $In learning theory, the VC dimension of a concept class $C$ is the most common way to measure its richness. In the PAC model $$ ThetaBig(frac{d}{eps} + frac{log(1/delta)}{eps}Big) $$ examples are necessary and sufficien