Lower Bounds for the Graph Homomorphism Problem

The graph homomorphism problem (HOM) asks whether the vertices of a given $n$-vertex graph $G$ can be mapped to the vertices of a given $h$-vertex graph $H$ such that each edge of $G$ is mapped to an edge of $H$. The problem generalizes the graph coloring problem and at the same time can be viewed as a special case of the $2$-CSP problem. In this paper, we prove several lower bound for HOM under the Exponential Time Hypothesis (ETH) assumption. The main result is a lower bound $2^{\Omega\left( \frac{n \log h}{\log \log h}\right)}$. This rules out the existence of a single-exponential algorithm and shows that the trivial upper bound $2^{{\mathcal O}(n\log{h})}$ is almost asymptotically tight. We also investigate what properties of graphs $G$ and $H$ make it difficult to solve HOM$(G,H)$. An easy observation is that an ${\mathcal O}(h^n)$ upper bound can be improved to ${\mathcal O}(h^{\operatorname{vc}(G)})$ where $\operatorname{vc}(G)$ is the minimum size of a vertex cover of $G$. The second lower bound $h^{\Omega(\operatorname{vc}(G))}$ shows that the upper bound is asymptotically tight. As to the properties of the "right-hand side" graph $H$, it is known that HOM$(G,H)$ can be solved in time $(f(\Delta(H)))^n$ and $(f(\operatorname{tw}(H)))^n$ where $\Delta(H)$ is the maximum degree of $H$ and $\operatorname{tw}(H)$ is the treewidth of $H$. This gives single-exponential algorithms for graphs of bounded maximum degree or bounded treewidth. Since the chromatic number $\chi(H)$ does not exceed $\operatorname{tw}(H)$ and $\Delta(H)+1$, it is natural to ask whether similar upper bounds with respect to $\chi(H)$ can be obtained. We provide a negative answer to this question by establishing a lower bound $(f(\chi(H)))^n$ for any function $f$. We also observe that similar lower bounds can be obtained for locally injective homomorphisms.Comment: 19 page

Polynomial Kernels for Weighted Problems

Kernelization is a formalization of efficient preprocessing for NP-hard problems using the framework of parameterized complexity. Among open problems in kernelization it has been asked many times whether there are deterministic polynomial kernelizations for Subset Sum and Knapsack when parameterized by the number $n$ of items. We answer both questions affirmatively by using an algorithm for compressing numbers due to Frank and Tardos (Combinatorica 1987). This result had been first used by Marx and V\'egh (ICALP 2013) in the context of kernelization. We further illustrate its applicability by giving polynomial kernels also for weighted versions of several well-studied parameterized problems. Furthermore, when parameterized by the different item sizes we obtain a polynomial kernelization for Subset Sum and an exponential kernelization for Knapsack. Finally, we also obtain kernelization results for polynomial integer programs

What to expect when trading with the U.S.S.R.: The Problems Confronting The American Exporter

An examination of the ramifications of the 1972 Trade Agreement between the United States and the Soviet Union

The Coupled Electronic-Ionic Monte Carlo Simulation Method

Quantum Monte Carlo (QMC) methods such as Variational Monte Carlo, Diffusion Monte Carlo or Path Integral Monte Carlo are the most accurate and general methods for computing total electronic energies. We will review methods we have developed to perform QMC for the electrons coupled to a classical Monte Carlo simulation of the ions. In this method, one estimates the Born-Oppenheimer energy E(Z) where Z represents the ionic degrees of freedom. That estimate of the energy is used in a Metropolis simulation of the ionic degrees of freedom. Important aspects of this method are how to deal with the noise, which QMC method and which trial function to use, how to deal with generalized boundary conditions on the wave function so as to reduce the finite size effects. We discuss some advantages of the CEIMC method concerning how the quantum effects of the ionic degrees of freedom can be included and how the boundary conditions can be integrated over. Using these methods, we have performed simulations of liquid H2 and metallic H on a parallel computer.Comment: 27 pages, 10 figure

Momentum of an electromagnetic wave in dielectric media

Almost a hundred years ago, two different expressions were proposed for the energy--momentum tensor of an electromagnetic wave in a dielectric. Minkowski's tensor predicted an increase in the linear momentum of the wave on entering a dielectric medium, whereas Abraham's tensor predicted its decrease. Theoretical arguments were advanced in favour of both sides, and experiments proved incapable of distinguishing between the two. Yet more forms were proposed, each with their advocates who considered the form that they were proposing to be the one true tensor. This paper reviews the debate and its eventual conclusion: that no electromagnetic wave energy--momentum tensor is complete on its own. When the appropriate accompanying energy--momentum tensor for the material medium is also considered, experimental predictions of all the various proposed tensors will always be the same, and the preferred form is therefore effectively a matter of personal choice.Comment: 23 pages, 3 figures, RevTeX 4. Removed erroneous factor of mu/mu_0 from Eq.(44

Antitrust Enforcement Encourages Health Care Providers to Cooperate Procompetitively

A review of significant administrative and judicial rulings in antitrust law during 1993 shows that the antitrust laws should not impede innovative, cost-cutting cooperative arrangements among providers, so long as their actual or potential procompetitive benefits are not outweighed by their anticompetitive effects

Motivation and incentives of rural maternal and neonatal health care providers: a comparison of qualitative findings from Burkina Faso, Ghana and Tanzania.

In Burkina Faso, Ghana and Tanzania strong efforts are being made to improve the quality of maternal and neonatal health (MNH) care. However, progress is impeded by challenges, especially in the area of human resources. All three countries are striving not only to scale up the number of available health staff, but also to improve performance by raising skill levels and enhancing provider motivation. In-depth interviews were used to explore MNH provider views about motivation and incentives at primary care level in rural Burkina Faso, Ghana and Tanzania. Interviews were held with 25 MNH providers, 8 facility and district managers, and 2 policy-makers in each country. Across the three countries some differences were found in the reasons why people became health workers. Commitment to remaining a health worker was generally high. The readiness to remain at a rural facility was far less, although in all settings there were some providers that were willing to stay. In Burkina Faso it appeared to be particularly difficult to recruit female MNH providers to rural areas. There were indications that MNH providers in all the settings sometimes failed to treat their patients well. This was shown to be interlinked with differences in how the term 'motivation' was understood, and in the views held about remuneration and the status of rural health work. Job satisfaction was shown to be quite high, and was particularly linked to community appreciation. With some important exceptions, there was a strong level of agreement regarding the financial and non-financial incentives that were suggested by these providers, but there were clear country preferences as to whether incentives should be for individuals or teams. Understandings of the terms and concepts pertaining to motivation differed between the three countries. The findings from Burkina Faso underline the importance of gender-sensitive health workforce planning. The training that all levels of MNH providers receive in professional ethics, and the way this is reinforced in practice require closer attention. The differences in the findings across the three settings underscore the importance of in-depth country-level research to tailor the development of incentives schemes