Related papers: Link Crossing Number is NP-hard
In 2013, Adams introduced the $n$-crossing number of a knot $K$, denoted by $c_n(K)$. Inequalities between the $2$-, $3$-, $4$-, and $5$-crossing numbers have been previously established. We prove $c_9(K)\leq c_3(K)-2$ for all knots $K$…
The paper provides bounds for the ropelength of a link in terms of the crossing numbers of its split components. As in earlier papers, the bounds grow with the square of the crossing number; however, the constant involved is a substantial…
An $n$-crossing projection of a link $L$ is a projection of $L$ onto a plane such that $n$ points on $L$ are superimposed on top of each other at every crossing. We prove that for all $k \in \mathbb{N}$ and all links $L$, the inequality…
We give a new, elementary proof of what we believe is the simplest known example of a ``natural'' problem in computational 3-dimensional topology that is $\mathsf{NP}$-hard -- namely, the \emph{Trivial Sublink Problem}: given a diagram $L$…
We prove some necessary conditions for a link to be either concordant to a quasi-positive link, quasi-positive, positive, or the closure of a positive braid. The main applications of our results are a characterisation of positive links with…
Complex networks are at the core of an intense research activity. However, in most cases, intricate and costly measurement procedures are needed to explore their structure. In some cases, these measurements rely on link queries: given two…
The effective graph resistance, also known as the Kirchhoff index, is metric that is used to quantify the robustness of a network. We show that the optimisation problem of minimizing the effective graph resistance of a graph by adding a…
A quantum channel is said to be a mixed-unitary channel if it can be expressed as a convex combination of unitary channels. We prove that, given the Choi representation of a quantum channel, it is NP-hard with respect to polynomial-time…
We study the problem of connecting the parts of a multipartite graph using a minimum number of edges under a matching constraint. We introduce interconnection trees, defined as matchings whose projections onto the quotient graph form a…
Link prediction is one of the fundamental problems in network analysis. In many applications, notably in genetics, a partially observed network may not contain any negative examples of absent edges, which creates a difficulty for many…
Recent cognitive experiments have shown that the negative impact of an edge crossing on the human understanding of a graph drawing, tends to be eliminated in the case where the crossing angles are greater than 70 degrees. This motivated the…
Ropelength and embedding thickness are related measures of geometric complexity of classical knots and links in Euclidean space. In their recent work, Freedman and Krushkal posed a question regarding lower bounds for embedding thickness of…
IV-matching is a generalization of perfect bipartite matching. The complexity of finding IV-matching in a graph was posted as an open problem at the ICALP 2014 conference. In this note, we resolve the question and prove that, contrary to…
This technical note extends recent results on the computational complexity of globally minimizing the error of piecewise-affine models to the related problem of minimizing the error of switching linear regression models. In particular, we…
Recently a large number of graph separator problems have been proven to be \textsc{NP-Hard}. Amazingly we have found that $\alpha$-Subgraph-Balanced-Vertex-Separator, an important variant, has been overlooked. In this work ``Yet Another…
A problem that arises in drawings of transportation networks is to minimize the number of crossings between different transportation lines. While this can be done efficiently under specific constraints, not all solutions are visually…
The crossing number of a graph $G$ is the least number of crossings over all possible drawings of $G$. We present a structural characterization of graphs with crossing number one.
How hard is it to invert NP-problems? We show that all superlinearly certified inverses of NP problems are coNP-hard. To do so, we develop a novel proof technique that builds diagonalizations against certificates directly into a circuit.
Link prediction is a paradigmatic problem in network science, which aims at estimating the existence likelihoods of nonobserved links, based on known topology. After a brief introduction of the standard problem and metrics of link…
We explore Cluster Editing and its generalization Correlation Clustering with a new operation called permissive vertex splitting which addresses finding overlapping clusters in the face of uncertain information. We determine that both…