Minimum Cell Connection in Line Segment Arrangements

We study the complexity of the following cell connection problems in segment arrangements. Given a set of straight-line segments in the plane and two points a and b in different cells of the induced arrangement: [(i)] compute the minimum number of segments one needs to remove so that there is a path connecting a to b that does not intersect any of the remaining segments; [(ii)] compute the minimum number of segments one needs to remove so that the arrangement induced by the remaining segments has a single cell. We show that problems (i) and (ii) are NP-hard and discuss some special, tractable cases. Most notably, we provide a near-linear-time algorithm for a variant of problem (i) where the path connecting a to b must stay inside a given polygon P with a constant number of holes, the segments are contained in P, and the endpoints of the segments are on the boundary of P. The approach for this latter result uses homotopy of paths to group the segments into clusters with the property that either all segments in a cluster or none participate in an optimal solution

The complexity of separating points in the plane

We study the following separation problem: given n connected curves and two points s and t in the plane, compute the minimum number of curves one needs to retain so that any path connecting s to t intersects some of the retained curves. We give the first polynomial (O(n3)) time algorithm for the problem, assuming that the curves have reasonable computational properties. The algorithm is based on considering the intersection graph of the curves, defining an appropriate family of closed walks in the intersection graph that satisfies the 3-path-condition, and arguing that a shortest cycle in the family gives an optimal solution. The 3-path-condition has been used mainly in topological graph theory, and thus its use here makes the connection to topology clear. We also show that the generalized version, where several input points are to be separated, is NP-hard for natural families of curves, like segments in two directions or unit circles

The generalized Kochen-Specker theorem

A proof of the generalized Kochen-Specker theorem in two dimensions due to Cabello and Nakamura is extended to all higher dimensions. A set of 18 states in four dimensions is used to give closely related proofs of the generalized Kochen-Specker, Kochen-Specker and Bell theorems that shed some light on the relationship between these three theorems.Comment: 5 pages, 1 Table. A new third paragraph and an additional reference have been adde

Implications of quantum automata for contextuality

We construct zero-error quantum finite automata (QFAs) for promise problems which cannot be solved by bounded-error probabilistic finite automata (PFAs). Here is a summary of our results: - There is a promise problem solvable by an exact two-way QFA in exponential expected time, but not by any bounded-error sublogarithmic space probabilistic Turing machine (PTM). - There is a promise problem solvable by an exact two-way QFA in quadratic expected time, but not by any bounded-error $o(\log \log n)$-space PTMs in polynomial expected time. The same problem can be solvable by a one-way Las Vegas (or exact two-way) QFA with quantum head in linear (expected) time. - There is a promise problem solvable by a Las Vegas realtime QFA, but not by any bounded-error realtime PFA. The same problem can be solvable by an exact two-way QFA in linear expected time but not by any exact two-way PFA. - There is a family of promise problems such that each promise problem can be solvable by a two-state exact realtime QFAs, but, there is no such bound on the number of states of realtime bounded-error PFAs solving the members this family. Our results imply that there exist zero-error quantum computational devices with a \emph{single qubit} of memory that cannot be simulated by any finite memory classical computational model. This provides a computational perspective on results regarding ontological theories of quantum mechanics \cite{Hardy04}, \cite{Montina08}. As a consequence we find that classical automata based simulation models \cite{Kleinmann11}, \cite{Blasiak13} are not sufficiently powerful to simulate quantum contextuality. We conclude by highlighting the interplay between results from automata models and their application to developing a general framework for quantum contextuality.Comment: 22 page

Non-covalent interactions at electrochemical interfaces : one model fits all?

Acknowledgements Funding from the DGI (Spanish Ministry of Education and Science) through Project CTQ2009-07017 is gratefully acknowledged. E.P.M.L. wishes to thank the Universidad Nacional de Co´rdoba, Argentina, for a grant within the ‘‘Programa de Movilidad Internacional de Profesores Cuarto Centenario’’.Peer reviewedPublisher PD

Experimental Bell inequality violation without the postselection loophole

We report on an experimental violation of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality using energy-time entangled photons. The experiment is not free of the locality and detection loopholes, but is the first violation of the Bell-CHSH inequality using energy-time entangled photons which is free of the postselection loophole described by Aerts et al. [Phys. Rev. Lett. 83, 2872 (1999)].Comment: 4 pages, 3 figures, v2 minor correction

Upward Point-Set Embeddability

We study the problem of Upward Point-Set Embeddability, that is the problem of deciding whether a given upward planar digraph $D$ has an upward planar embedding into a point set $S$. We show that any switch tree admits an upward planar straight-line embedding into any convex point set. For the class of $k$-switch trees, that is a generalization of switch trees (according to this definition a switch tree is a $1$-switch tree), we show that not every $k$-switch tree admits an upward planar straight-line embedding into any convex point set, for any $k \geq 2$. Finally we show that the problem of Upward Point-Set Embeddability is NP-complete

A Universal Point Set for 2-Outerplanar Graphs

A point set $S \subseteq \mathbb{R}^2$ is universal for a class $\cal G$ if every graph of ${\cal G}$ has a planar straight-line embedding on $S$. It is well-known that the integer grid is a quadratic-size universal point set for planar graphs, while the existence of a sub-quadratic universal point set for them is one of the most fascinating open problems in Graph Drawing. Motivated by the fact that outerplanarity is a key property for the existence of small universal point sets, we study 2-outerplanar graphs and provide for them a universal point set of size $O(n \log n)$.Comment: 23 pages, 11 figures, conference version at GD 201

Parity proofs of the Kochen-Specker theorem based on the 24 rays of Peres

A diagrammatic representation is given of the 24 rays of Peres that makes it easy to pick out all the 512 parity proofs of the Kochen-Specker theorem contained in them. The origin of this representation in the four-dimensional geometry of the rays is pointed out.Comment: 14 pages, 6 figures and 3 tables. Three references have been added. Minor typos have been correcte

Solving the liar detection problem using the four-qubit singlet state

A method for solving the Byzantine agreement problem [M. Fitzi, N. Gisin, and U. Maurer, Phys. Rev. Lett. 87, 217901 (2001)] and the liar detection problem [A. Cabello, Phys. Rev. Lett. 89, 100402 (2002)] is introduced. The main advantages of this protocol are that it is simpler and is based on a four-qubit singlet state already prepared in the laboratory.Comment: REVTeX4, 4 page