QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Clique on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show the rather surprising structural result that a disjoint union of cycles is the complement of a disk graph if and only if at most one of those cycles is of odd length. From that, we derive the first QPTAS and subexponential algorithm running in time 2^{O~(n^{2/3})} for Maximum Clique on disk graphs. In stark contrast, Maximum Clique on intersection graphs of filled ellipses or filled triangles is unlikely to have such algorithms, even when the ellipses are close to unit disks. Indeed, we show that there is a constant ratio of approximation which cannot be attained even in time 2^{n^{1-epsilon}}, unless the Exponential Time Hypothesis fails

Finding list homomorphisms from bounded-treewidth graphs to reflexive graphs: A complete complexity characterization

In the list homomorphism problem, the input consists of two graphs G and H, together with a list L(v) subseteq V(H) for every vertex v in V(G). The task is to find a homomorphism phi:V(G) -> V(H) respecting the lists, that is, we have that phi(v) in L(v) for every v in V(H) and if u and v are adjacent in G, then phi(u) and phi(v) are adjacent in H. If H is a fixed graph, then the problem is denoted LHom(H). We consider the reflexive version of the problem, where we assume that every vertex in H has a self-loop. If is known that reflexive LHom(H) is polynomial-time solvable if H is an interval graph and it is NP-complete otherwise [Feder and Hell, JCTB 1998]. We explore the complexity of the problem parameterized by the treewidth tw(G) of the input graph G. If a tree decomposition of G of width tw(G) is given in the input, then the problem can be solved in time |V(H)|^{tw(G)} n^{O(1)} by naive dynamic programming. Our main result completely reveals when and by exactly how much this naive algorithm can be improved. We introduce a simple combinatorial invariant i^*(H), which is based on the existence of decompositions and incomparable sets, and show that this number should appear as the base of the exponent in the best possible running time. Specifically, we prove for every fixed non-interval graph H that * If a tree decomposition of width tw(G) is given in the input, then the problem can be solved in time i^*(H)^{tw(G)} n^{O(1)}. * Assuming the Strong Exponential-Time Hypothesis (SETH), the probem cannot be solved in time (i^*(H)-epsilon)^{tw(G)} n^{O(1)} for any epsilon>0. Thus by matching upper and lower bounds, our result exactly characterizes for every fixed H the complexity of reflexive LHom(H) parameterized by treewidth

Noncanonical quantum optics

Modification of the right-hand-side of canonical commutation relations (CCR) naturally occurs if one considers a harmonic oscillator with indefinite frequency. Quantization of electromagnetic field by means of such a non-CCR algebra naturally removes the infinite energy of vacuum but still results in a theory which is very similar to quantum electrodynamics. An analysis of perturbation theory shows that the non-canonical theory has an automatically built-in cut-off but requires charge/mass renormalization already at the nonrelativistic level. A simple rule allowing to compare perturbative predictions of canonical and non-canonical theories is given. The notion of a unique vacuum state is replaced by a set of different vacua. Multi-photon states are defined in the standard way but depend on the choice of vacuum. Making a simplified choice of the vacuum state we estimate corrections to atomic lifetimes, probabilities of multiphoton spontaneous and stimulated emission, and the Planck law. The results are practically identical to the standard ones. Two different candidates for a free-field Hamiltonian are compared.Comment: Completely rewritten version of quant-ph/0002003v2. There are overlaps between the papers, but sections on perturbative calculations show the same problem from different sides, therefore quant-ph/0002003v2 is not replace

Rate limit for photoassociation of a Bose-Einstein condensate

We simulate numerically the photodissociation of molecules into noncondensate atom pairs that accompanies photoassociation of an atomic Bose-Einstein condensate into a molecular condensate. Such rogue photodissociation sets a limit on the achievable rate of photoassociation. Given the atom density \rho and mass m, the limit is approximately 6\hbar\rho^{2/3}/m. At low temperatures this is a more stringent restriction than the unitary limit of scattering theory.Comment: 5 pgs, 18 refs., 3 figs., submitted to Phys. Rev. Let

EPTAS and Subexponential Algorithm for Maximum Clique on Disk and Unit Ball Graphs

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Cliqe on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics ’90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time 2O˜(n2/3) for Maximum Cliqe on disk and unit ball graphs. We then obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. This, in combination with our structural results, yields a randomized EPTAS for Max Cliqe on disk and unit ball graphs. Max Cliqe on unit ball graphs is equivalent to finding, given a collection of points in R3, a maximum subset of points with diameter at most some fixed value. In stark contrast, Maximum Cliqe on ball graphs and unit 4-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have such algorithms. Indeed, we show that, for all those problems, there is a constant ratio of approximation which cannot be attained even in time 2n1−ε, unless the Exponential Time Hypothesis fails

Elastic scattering losses in the four-wave mixing of Bose Einstein Condensates

We introduce a classical stochastic field method that accounts for the quantum fluctuations responsible for spontaneous initiation of various atom optics processes. We assume a delta-correlated Gaussian noise in all initially empty modes of atomic field. Its strength is determined by comparison with the analytical results for two colliding condensates in the low loss limit. Our method is applied to the atomic four wave mixing experiment performed at MIT [Vogels {\it et. al.}, Phys. Rev. Lett. {\bf 89}, 020401, (2002)], for the first time reproducing experimental data

Probing the classical field approximation - thermodynamics and decaying vortices

We review our version of the classical field approximation to the dynamics of a finite temperature Bose gas. In the case of a periodic box potential, we investigate the role of the high momentum cut-off, essential in the method. In particular, we show that the cut-off going to infinity limit decribes the particle number going to infinity with the scattering length going to zero. In this weak interaction limit, the relative population of the condensate tends to unity. We also show that the cross-over energy, at which the probability distribution of the condensate occupation changes its character, grows with a growing scattering length. In the more physical case of the condensate in the harmonic trap we investigate the dissipative dynamics of a vortex. We compare the decay time and the velocities of the vortex with the available analytic estimates.Comment: 7 pages, 8 eps figures, submitted to J. Optics B for the proceedings of the "Atom Optics and Interferometry" Lunteren 2002 worksho

Qubit Disentanglement and Decoherence via Dephasing

We consider whether quantum coherence in the form of mutual entanglement between a pair of qubits is susceptible to decay that may be more rapid than the decay of the coherence of either qubit individually. An instance of potential importance for solid state quantum computing arises if embedded qubits (spins, quantum dots, Cooper pair boxes, etc.) are exposed to global and local noise at the same time. Here we allow separate phase-noisy channels to affect local and non-local measures of system coherence. We find that the time for decay of the qubit entanglement can be significantly shorter than the time for local dephasing of the individual qubits.Comment: REVTeX, 9 pages, 1 figure, v2 with minor changes, reference adde

Science of Extreme Light Infrastructure

The infrastructure of Extreme Light Infrastructure (ELI) provides an unprecedented opportunity for a broad range of frontier science. Its highest ever intensity of lasers, as well as high fluence, high power, and/or ultrafast optical characteristics carve out new territories of discovery, ranging from attosecond science to photonuclear science, laser acceleration and associated beams, and high field science (Four Pillars of ELI). Its applications span from medicine, biology, engineering, energy, chemistry, physics, and fundamental understanding of the Universe. The relativistic optics that intense lasers have begun exploring may be extended into a new regime of ultra‐relativistic regime, where even protons fly relativistically in the optical fields. ELI provides the highest intensity to date such that photon fields begin to feel even the texture of vacuum. This is a singular appeal of ELI with its relatively modest infrastructure (compared to the contemporary largest scientific infrastructures), yet provides an exceptional avenue along which the 21st Century science and society need to answer the toughest questions. The intensity frontier simultaneously brings in the energy horizon (TeV and PeV) as well as temporal frontier (attoseconds and zeptoseconds). It also turns over optics of atoms and molecules into that of nuclei with the ability to produce monoenergetic collimated γ‐ray photons. As such, the ELI concept acutely demands an effort to encompass and integrate its Four Pillars