Data Reduction for Graph Coloring Problems

This paper studies the kernelization complexity of graph coloring problems with respect to certain structural parameterizations of the input instances. We are interested in how well polynomial-time data reduction can provably shrink instances of coloring problems, in terms of the chosen parameter. It is well known that deciding 3-colorability is already NP-complete, hence parameterizing by the requested number of colors is not fruitful. Instead, we pick up on a research thread initiated by Cai (DAM, 2003) who studied coloring problems parameterized by the modification distance of the input graph to a graph class on which coloring is polynomial-time solvable; for example parameterizing by the number k of vertex-deletions needed to make the graph chordal. We obtain various upper and lower bounds for kernels of such parameterizations of q-Coloring, complementing Cai's study of the time complexity with respect to these parameters. Our results show that the existence of polynomial kernels for q-Coloring parameterized by the vertex-deletion distance to a graph class F is strongly related to the existence of a function f(q) which bounds the number of vertices which are needed to preserve the NO-answer to an instance of q-List-Coloring on F.Comment: Author-accepted manuscript of the article that will appear in the FCT 2011 special issue of Information & Computatio

Matter fields from a decaying background Lambda vacuum

We suggest an alternative framework for interpreting the current state of the visible universe. Our approach is based on a dynamical ``Cosmological Constant'' and the starting point is that a decaying vacuum produces matter. As we point out, such a dynamical Lambda is not incompatible with the general requirements of general relativity. By assuming inflation and big bang nucleosynthesis we can solve for the present fractional densities of matter Omega_{m,0} and vacuum Omega_{Lambda, 0} in terms of only one parameter which we call the vacuum domination crossing redshift, z_c. We put constraints on z_c to obtain a universe that is presently vacuum dominated and with characteristic densities consistent with observations. The model points to the possible existence of newly formed dark matter in the inter-cluster voids. We argue that some of this matter could be accreting onto clusters through the latter's long range gravitational potentials. If so, then cluster dark matter halos may not manifest clear cut-offs in their radial density profiles. Furthermore, if a substantial amount of this newly produced matter has already drained onto the clusters, then the CMB power spectrum may favor lower dark matter density values than is currently observed bound in the clusters. A final feature of our approach relates to the combined effect of the matter production by a decaying vacuum and the different rates at which matter and the vacuum will dilute with the scale factor. Such combination may create conditions for a universe in which the vacuum and matter densities dilute and evolve towards comparable amplitudes. In this sense the model offers a natural and conceptually simple explanation to the Coincidence Problem.Comment: 22 pages, 1 figure, accepted for publication in Int. J. Mod. Phys. Lett.

On the algorithmic complexity of twelve covering and independence parameters of graphs

The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs

A halo-independent lower bound on the dark matter capture rate in the Sun from a direct detection signal

We show that a positive signal in a dark matter (DM) direct detection experiment can be used to place a lower bound on the DM capture rate in the Sun, independent of the DM halo. For a given particle physics model and DM mass we obtain a lower bound on the capture rate independent of the local DM density, velocity distribution, galactic escape velocity, as well as the scattering cross section. We illustrate this lower bound on the capture rate by assuming that upcoming direct detection experiments will soon obtain a significant signal. When comparing the lower bound on the capture rate with limits on the high-energy neutrino flux from the Sun from neutrino telescopes, we can place upper limits on the branching fraction of DM annihilation channels leading to neutrinos. With current data from IceCube and Super-Kamiokande non-trivial limits can be obtained for spin-dependent interactions and direct annihilations into neutrinos. In some cases also annihilations into $\tau\tau$ or $b\bar b$ start getting constrained. For spin-independent interactions current constraints are weak, but they may become interesting for data from future neutrino telescopes.Comment: 27 pages, 8 figures. Added discussion on equilibrium. Added section 5.4 on form factor uncertainties. Updated figures with SK new limits. Published in JCA

TASI Lectures on Cosmological Perturbations

We present a self-contained summary of the theory of linear cosmological perturbations. We emphasize the effect of the six parameters of the minimal cosmological model, first, on the spectrum of Cosmic Microwave Background temperature anisotropies, and second, on the linear matter power spectrum. We briefly review at the end the possible impact of a few non-minimal dark matter and dark energy models.Comment: TASI 2013 lecture note

Online Learning with Feedback Graphs: Beyond Bandits

We study a general class of online learning problems where the feedback is specified by a graph. This class includes online prediction with expert advice and the multi-armed bandit problem, but also several learning problems where the online player does not necessarily observe his own loss. We analyze how the structure of the feedback graph controls the inherent difficulty of the induced $T$-round learning problem. Specifically, we show that any feedback graph belongs to one of three classes: strongly observable graphs, weakly observable graphs, and unobservable graphs. We prove that the first class induces learning problems with $\widetilde\Theta(\alpha^{1/2} T^{1/2})$ minimax regret, where $\alpha$ is the independence number of the underlying graph; the second class induces problems with $\widetilde\Theta(\delta^{1/3}T^{2/3})$ minimax regret, where $\delta$ is the domination number of a certain portion of the graph; and the third class induces problems with linear minimax regret. Our results subsume much of the previous work on learning with feedback graphs and reveal new connections to partial monitoring games. We also show how the regret is affected if the graphs are allowed to vary with time