Ramsey's theorem and self-complementary graphs

AbstractIt is proved that, given any positive integer k, there exists a self-complementary graph with more than 4·214k vertices which contains no complete subgraph with k+1 vertices. An application of this result to coding theory is mentioned

Reconstruction of Causal Networks by Set Covering

We present a method for the reconstruction of networks, based on the order of nodes visited by a stochastic branching process. Our algorithm reconstructs a network of minimal size that ensures consistency with the data. Crucially, we show that global consistency with the data can be achieved through purely local considerations, inferring the neighbourhood of each node in turn. The optimisation problem solved for each individual node can be reduced to a Set Covering Problem, which is known to be NP-hard but can be approximated well in practice. We then extend our approach to account for noisy data, based on the Minimum Description Length principle. We demonstrate our algorithms on synthetic data, generated by an SIR-like epidemiological model.Comment: Under consideration for the ECML PKDD 2010 conferenc

Generating Bounds for the Ground State Energy of the Infinite Quantum Lens Potential

Moment based methods have produced efficient multiscale quantization algorithms for solving singular perturbation/strong coupling problems. One of these, the Eigenvalue Moment Method (EMM), developed by Handy et al (Phys. Rev. Lett.{\bf 55}, 931 (1985); ibid, {\bf 60}, 253 (1988b)), generates converging lower and upper bounds to a specific discrete state energy, once the signature property of the associated wavefunction is known. This method is particularly effective for multidimensional, bosonic ground state problems, since the corresponding wavefunction must be of uniform signature, and can be taken to be positive. Despite this, the vast majority of problems studied have been on unbounded domains. The important problem of an electron in an infinite quantum lens potential defines a challenging extension of EMM to systems defined on a compact domain. We investigate this here, and introduce novel modifications to the conventional EMM formalism that facilitate its adaptability to the required boundary conditions.Comment: Submitted to J. Phys.

Designing cost-sharing methods for Bayesian games

We study the design of cost-sharing protocols for two fundamental resource allocation problems, the Set Cover and the Steiner Tree Problem, under environments of incomplete information (Bayesian model). Our objective is to design protocols where the worst-case Bayesian Nash equilibria, have low cost, i.e. the Bayesian Price of Anarchy (PoA) is minimized. Although budget balance is a very natural requirement, it puts considerable restrictions on the design space, resulting in high PoA. We propose an alternative, relaxed requirement called budget balance in the equilibrium (BBiE).We show an interesting connection between algorithms for Oblivious Stochastic optimization problems and cost-sharing design with low PoA. We exploit this connection for both problems and we enforce approximate solutions of the stochastic problem, as Bayesian Nash equilibria, with the same guarantees on the PoA. More interestingly, we show how to obtain the same bounds on the PoA, by using anonymous posted prices which are desirable because they are easy to implement and, as we show, induce dominant strategies for the players

Generating Converging Eigenenergy Bounds for the Discrete States of the -ix^3 Non-Hermitian Potential

Recent investigations by Bender and Boettcher (Phys. Rev. Lett 80, 5243 (1998)) and Mezincescu (J. Phys. A. 33, 4911 (2000)) have argued that the discrete spectrum of the non-hermitian potential $V(x) = -ix^3$ should be real. We give further evidence for this through a novel formulation which transforms the general one dimensional Schrodinger equation (with complex potential) into a fourth order linear differential equation for $|\Psi(x)|^2$. This permits the application of the Eigenvalue Moment Method, developed by Handy, Bessis, and coworkers (Phys. Rev. Lett. 55, 931 (1985);60, 253 (1988a,b)), yielding rapidly converging lower and upper bounds to the low lying discrete state energies. We adapt this formalism to the pure imaginary cubic potential, generating tight bounds for the first five discrete state energy levels.Comment: Work to appear in J. Phys. A: Math & Ge

Generating Converging Bounds to the (Complex) Discrete States of the P2+iX3+iαXP^2 + iX^3 + i\alpha X Hamiltonian

The Eigenvalue Moment Method (EMM), Handy (2001), Handy and Wang (2001)) is applied to the $H_\alpha \equiv P^2 + iX^3 + i\alpha X$ Hamiltonian, enabling the algebraic/numerical generation of converging bounds to the complex energies of the $L^2$ states, as argued (through asymptotic methods) by Delabaere and Trinh (J. Phys. A: Math. Gen. {\bf 33} 8771 (2000)).Comment: Submitted to J. Phys.

The school bus routing problem: An analysis and algorithm

In this paper we analyse a flexible real world-based model for designing school bus transit systems and note a number of parallels between this and other well-known combinatorial optimisation problems including the vehicle routing problem, the set covering problem, and one-dimensional bin packing. We then describe an iterated local search algorithm for this problem and demonstrate the sort of solutions that we can expect with different types of problem instance

Extension of a Spectral Bounding Method to Complex Rotated Hamiltonians, with Application to p2−ix3p^2-ix^3

We show that a recently developed method for generating bounds for the discrete energy states of the non-hermitian $-ix^3$ potential (Handy 2001) is applicable to complex rotated versions of the Hamiltonian. This has important implications for extension of the method in the analysis of resonant states, Regge poles, and general bound states in the complex plane (Bender and Boettcher (1998)).Comment: Submitted to J. Phys.

Overcoming controllability problems in distributed testing from an input output transition system

This is the Pre-print version of the Article. The official published version can be accessed from the link below - Copyright @ 2012 Springer VerlagThis paper concerns the testing of a system with physically distributed interfaces, called ports, at which it interacts with its environment. We place a tester at each port and the tester at port p observes events at p only. This can lead to controllability problems, where the observations made by the tester at a port p are not sufficient for it to be able to know when to send an input. It is known that there are test objectives, such as executing a particular transition, that cannot be achieved if we restrict attention to test cases that have no controllability problems. This has led to interest in schemes where the testers at the individual ports send coordination messages to one another through an external communications network in order to overcome controllability problems. However, such approaches have largely been studied in the context of testing from a deterministic finite state machine. This paper investigates the use of coordination messages to overcome controllability problems when testing from an input output transition system and gives an algorithm for introducing sufficient messages. It also proves that the problem of minimising the number of coordination messages used is NP-hard