The Lazy Bureaucrat Scheduling Problem

We introduce a new class of scheduling problems in which the optimization is performed by the worker (single ``machine'') who performs the tasks. A typical worker's objective is to minimize the amount of work he does (he is ``lazy''), or more generally, to schedule as inefficiently (in some sense) as possible. The worker is subject to the constraint that he must be busy when there is work that he can do; we make this notion precise both in the preemptive and nonpreemptive settings. The resulting class of ``perverse'' scheduling problems, which we denote ``Lazy Bureaucrat Problems,'' gives rise to a rich set of new questions that explore the distinction between maximization and minimization in computing optimal schedules.Comment: 19 pages, 2 figures, Latex. To appear, Information and Computatio

On the Extended TSP Problem

We initiate the theoretical study of Ext-TSP, a problem that originates in the area of profile-guided binary optimization. Given a graph $G=(V, E)$ with positive edge weights $w: E \rightarrow R^+$, and a non-increasing discount function $f(\cdot)$ such that $f(1) = 1$ and $f(i) = 0$ for $i > k$, for some parameter $k$ that is part of the problem definition. The problem is to sequence the vertices $V$ so as to maximize $\sum_{(u, v) \in E} f(|d_u - d_v|)\cdot w(u,v)$, where $d_v \in \{1, \ldots, |V| \}$ is the position of vertex~$v$ in the sequence. We show that \prob{Ext-TSP} is APX-hard to approximate in general and we give a $(k+1)$-approximation algorithm for general graphs and a PTAS for some sparse graph classes such as planar or treewidth-bounded graphs. Interestingly, the problem remains challenging even on very simple graph classes; indeed, there is no exact $n^{o(k)}$ time algorithm for trees unless the ETH fails. We complement this negative result with an exact $n^{O(k)}$ time algorithm for trees.Comment: 17 page

Vertex operators for cosmic strings

Superstring theory posits that as complicated as nature may seem to the naive observer, the variety of observed phenomena may be explained by postulating that at the fundamental scale, matter is composed of lines of energy, namely strings. These oscillating lines would be elementary and would hence have no substructure. They are expected to be incredibly tiny, their line-like structure would become noticeable at scales close to the string scale (which may lie anywhere from the TeV scale all the way up to the Planck scale) and would appear to be point-like to the macroscopic observer. Internal consistency then also requires the presence of higher dimensional objects, namely D-branes, all of which conspire and combine in such a way so as to give rise to the observable Universe. Advances in cosmology suggest the early universe was much hotter and denser than is the Universe at present, that the Universe has expanded and continues to expand (exponentially in fact) at present. This in turn has led a number of theorists to point out the remarkable possibility that some of these strings or D-branes were also stretched with the expansion. The resulting macroscopic strings, the so-called cosmic strings, would potentially stretch across the entire Universe. Cosmic strings make their presence manifest by oscillating, scattering off other structures, by decaying, producing gravitational waves and so on, and this in turn hints at the available handles that may be used to observe them. Before we can hope to observe cosmic strings however, the first step is then clearly to understand these properties which determine their evolution. A number of approximate (classical) descriptions of cosmic strings have been constructed to date, but approximations break down, especially when potentially interesting things happen (e.g. close to cusps, i.e. points on the string that reach the speed of light) and can obscure the physics. Thankfully, one can go beyond these approximations: all properties of cosmic strings can be concisely and accurately contained or encoded in a single object, the so-called fundamental cosmic string vertex operator. In the present thesis I construct precisely this, covariant vertex operators for general cosmic strings and this is the first such construction. Cosmic strings, being macroscopic, are likely to exhibit classical behaviour in which case they would most accurately be described by a string theory analogue of the well known harmonic oscillator coherent states. By minimally extending the standard definition of coherent states, so as to include the string theory requirements, I go on to construct both open and closed covariant coherent state vertex operators. The naive construction of the latter requires the existence of a lightlike compactification of spacetime. When the lightlike winding states in the underlying Hilbert space are projected out, the resulting vertex operators have a classical interpretation and can consistently propagate in noncompact spacetime. Using the DDF map I identify explicitly the corresponding general lightcone gauge classical solutions around which the exact macroscopic quantum states are fluctuating. We go on to show that both the covariant gauge coherent vertex operators, the corresponding lightcone gauge coherent states and the classical solutions all share the same mass and angular momenta, which leads us to conjecture that the covariant and lightcone gauge states are different manifestations of the same state and share identical interactions. Apart from the coherent state vertices I also present a complete set of covariant mass eigenstate vertex operators and these may also be relevant in cosmic string evolution. Finally, I also present the first amplitude computation with the coherent states, the graviton emission amplitude (including the effects of gravitational backreaction) for a simple class of cosmic string loops. As a byproduct of the above, I find that the fundamental building blocks of arbitrarily massive covariant string states are given by elementary Schur polynomials (equivalently complete Bell polynomials). This construction enables one to address the aforementioned questions concerning the properties of cosmic strings, their cosmological signatures, and may lead to the first observations of such objects in the sky. This in turn would be a remarkable way of verifying Superstring theory as the framework underlying the structure of our Universe

Planck 2015 results: XIII. Cosmological parameters

We present results based on full-mission Planck observations of temperature and polarization anisotropies of the CMB. These data are consistent with the six-parameter inflationary LCDM cosmology. From the Planck temperature and lensing data, for this cosmology we find a Hubble constant, H0= (67.8 +/- 0.9) km/s/Mpc, a matter density parameter Omega_m = 0.308 +/- 0.012 and a scalar spectral index with n_s = 0.968 +/- 0.006. (We quote 68% errors on measured parameters and 95% limits on other parameters.) Combined with Planck temperature and lensing data, Planck LFI polarization measurements lead to a reionization optical depth of tau = 0.066 +/- 0.016. Combining Planck with other astrophysical data we find N_ eff = 3.15 +/- 0.23 for the effective number of relativistic degrees of freedom and the sum of neutrino masses is constrained to < 0.23 eV. Spatial curvature is found to be |Omega_K| < 0.005. For LCDM we find a limit on the tensor-to-scalar ratio of r <0.11 consistent with the B-mode constraints from an analysis of BICEP2, Keck Array, and Planck (BKP) data. Adding the BKP data leads to a tighter constraint of r < 0.09. We find no evidence for isocurvature perturbations or cosmic defects. The equation of state of dark energy is constrained to w = -1.006 +/- 0.045. Standard big bang nucleosynthesis predictions for the Planck LCDM cosmology are in excellent agreement with observations. We investigate annihilating dark matter and deviations from standard recombination, finding no evidence for new physics. The Planck results for base LCDM are in agreement with BAO data and with the JLA SNe sample. However the amplitude of the fluctuations is found to be higher than inferred from rich cluster counts and weak gravitational lensing. Apart from these tensions, the base LCDM cosmology provides an excellent description of the Planck CMB observations and many other astrophysical data sets