Evasiveness and the Distribution of Prime Numbers

We confirm the eventual evasiveness of several classes of monotone graph properties under widely accepted number theoretic hypotheses. In particular we show that Chowla's conjecture on Dirichlet primes implies that (a) for any graph $H$, "forbidden subgraph $H$" is eventually evasive and (b) all nontrivial monotone properties of graphs with $\le n^{3/2-\epsilon}$ edges are eventually evasive. ($n$ is the number of vertices.) While Chowla's conjecture is not known to follow from the Extended Riemann Hypothesis (ERH, the Riemann Hypothesis for Dirichlet's $L$ functions), we show (b) with the bound $O(n^{5/4-\epsilon})$ under ERH. We also prove unconditional results: (a$'$) for any graph $H$, the query complexity of "forbidden subgraph $H$" is $\binom{n}{2} - O(1)$; (b$'$) for some constant $c>0$, all nontrivial monotone properties of graphs with $\le cn\log n+O(1)$ edges are eventually evasive. Even these weaker, unconditional results rely on deep results from number theory such as Vinogradov's theorem on the Goldbach conjecture. Our technical contribution consists in connecting the topological framework of Kahn, Saks, and Sturtevant (1984), as further developed by Chakrabarti, Khot, and Shi (2002), with a deeper analysis of the orbital structure of permutation groups and their connection to the distribution of prime numbers. Our unconditional results include stronger versions and generalizations of some result of Chakrabarti et al.Comment: 12 pages (conference version for STACS 2010

A sharpened version of the aanderaa-rosenberg conjecture

Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe

Quantum query complexity of minor-closed graph properties

We study the quantum query complexity of minor-closed graph properties, which include such problems as determining whether an $n$-vertex graph is planar, is a forest, or does not contain a path of a given length. We show that most minor-closed properties---those that cannot be characterized by a finite set of forbidden subgraphs---have quantum query complexity \Theta(n^{3/2}). To establish this, we prove an adversary lower bound using a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. On the other hand, we show that minor-closed properties (and more generally, sparse graph properties) that can be characterized by finitely many forbidden subgraphs can be solved strictly faster, in o(n^{3/2}) queries. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.Comment: v1: 25 pages, 2 figures. v2: 26 page

HYDROGRAPHIC OBSERVATIONS OF OXYGEN AND RELATED PHYSICAL VARIABLES IN THE NORTH SEA AND WESTERN ROSSSEA POLYNYA investigations using seagliders, historical observations and numerical modelling

Shelf seas are one of the most ecologically and economically important ecosystems of the planet. Dissolved oxygen in particular is of critical importance to maintaining a healthy and stable biological community. This work investigates the physical, chemical and biological drivers of summer oxygen variability in the North Sea (Europe) and Ross Sea polynya (Antarctica). In particular, this work also focuses on the use of new autonomous underwater vehicles, Seagliders, for oceanographic observations of fine scale (a few metres) to basin-wide features (hundreds of kilometres). Two hydrographic surveys in 2010 and 2011 and an analysis of historical data dating back to 1902 revealed low dissolved oxygen in the bottom mixed layer of the central North Sea.We deployed a Seaglider in a region of known low oxygen during August 2011 to investigate the processes regulating supply and consumption of dissolved oxygen below the pycnocline. Historical data highlighted an increase in seasonal oxygen depletion and a warming over the past 20 years. Regions showing sub-saturation oxygen concentrations were identified in the central and northern North Sea post-1990 where previously no depletion was identified. Low dissolved oxygen was apparent in regions characterised by low advection, high stratification, elevated organic matter production from the spring bloom and a deep chlorophyll maximum. The constant consumption of oxygen for the remineralisation of the matter exported below the thermocline exceeded the supply from horizontal advection or vertical diffusion. The Seaglider identified cross-pycnocline mixing features responsible for reoxygenation of the bottom mixed layer not currently resolved by models of the North Sea. Using the data, we were also able to constrain the relative importance of different sources of organic matter leading to oxygen consumption. iii From November 2010 to February 2011, two Seagliders were deployed in the Ross polynya to observe the initiation and evolution of the spring bloom. Seagliders were a novel and effective tool to bypass the sampling difficulties caused by the presence of ice and the remoteness of the region, in particular they were able to obtain data in the polynya before access was possible by oceanographic vessels. Seagliders were able to survey the region at a fraction of the cost and inconvenience of traditional ship surveys and moorings. We present observations of a large phytoplankton bloom in the Ross Sea polynya, export of organic matter and related fluctuations in dissolved oxygen concentrations. The bloom was found to be widespread and unrelated to the presence of Ross Bank. Increased fluorescence was identified through the use of satellite ocean colour data and is likely related to the intrusion of modified circumpolar deep water. In parallel, changes in dissolved oxygen concentration are quantified and highlight the importance of a deep chlorophyll maximum as a driver of primary production in the Ross Sea polynya. Both the variability of the biological features and the inherent difficulties in observing these features using other means are highlighted by the analysis of Seaglider data. The Seaglider proved to be an excellent tool for monitoring shelf sea processes despite challenges to Seaglider deployments posed by the ice presence, high tidal velocities, shallow bathymetry and lack of accurate means of calibration. Data collected show great potential for improving biogeochemical models by providing means to obtain novel oceanographic observations along and across a range of scales

Quantum Query Complexity of Subgraph Isomorphism and Homomorphism

Let $H$ be a fixed graph on $n$ vertices. Let $f_H(G) = 1$ iff the input graph $G$ on $n$ vertices contains $H$ as a (not necessarily induced) subgraph. Let $\alpha_H$ denote the cardinality of a maximum independent set of $H$. In this paper we show: $$Q(f_H) = \Omega\left(\sqrt{\alpha_H \cdot n}\right),$$ where $Q(f_H)$ denotes the quantum query complexity of $f_H$. As a consequence we obtain a lower bounds for $Q(f_H)$ in terms of several other parameters of $H$ such as the average degree, minimum vertex cover, chromatic number, and the critical probability. We also use the above bound to show that $Q(f_H) = \Omega(n^{3/4})$ for any $H$, improving on the previously best known bound of $\Omega(n^{2/3})$. Until very recently, it was believed that the quantum query complexity is at least square root of the randomized one. Our $\Omega(n^{3/4})$ bound for $Q(f_H)$ matches the square root of the current best known bound for the randomized query complexity of $f_H$, which is $\Omega(n^{3/2})$ due to Gr\"oger. Interestingly, the randomized bound of $\Omega(\alpha_H \cdot n)$ for $f_H$ still remains open. We also study the Subgraph Homomorphism Problem, denoted by $f_{[H]}$, and show that $Q(f_{[H]}) = \Omega(n)$. Finally we extend our results to the $3$-uniform hypergraphs. In particular, we show an $\Omega(n^{4/5})$ bound for quantum query complexity of the Subgraph Isomorphism, improving on the previously known $\Omega(n^{3/4})$ bound. For the Subgraph Homomorphism, we obtain an $\Omega(n^{3/2})$ bound for the same.Comment: 16 pages, 2 figure

Evasiveness of Graph Properties and Topological Fixed-Point Theorems

Many graph properties (e.g., connectedness, containing a complete subgraph) are known to be difficult to check. In a decision-tree model, the cost of an algorithm is measured by the number of edges in the graph that it queries. R. Karp conjectured in the early 1970s that all monotone graph properties are evasive -- that is, any algorithm which computes a monotone graph property must check all edges in the worst case. This conjecture is unproven, but a lot of progress has been made. Starting with the work of Kahn, Saks, and Sturtevant in 1984, topological methods have been applied to prove partial results on the Karp conjecture. This text is a tutorial on these topological methods. I give a fully self-contained account of the central proofs from the paper of Kahn, Saks, and Sturtevant, with no prior knowledge of topology assumed. I also briefly survey some of the more recent results on evasiveness.Comment: Book version, 92 page