How to Hide a Clique?

In the well known planted clique problem, a clique (or alternatively, an independent set) of size k is planted at random in an Erdos-Renyi random G(n, p) graph, and the goal is to design an algorithm that finds the maximum clique (or independent set) in the resulting graph. We introduce a variation on this problem, where instead of planting the clique at random, the clique is planted by an adversary who attempts to make it difficult to find the maximum clique in the resulting graph. We show that for the standard setting of the parameters of the problem, namely, a clique of size k = ?n planted in a random G(n, 1/2) graph, the known polynomial time algorithms can be extended (in a non-trivial way) to work also in the adversarial setting. In contrast, we show that for other natural settings of the parameters, such as planting an independent set of size k = n/2 in a G(n, p) graph with p = n^{-1/2}, there is no polynomial time algorithm that finds an independent set of size k, unless NP has randomized polynomial time algorithms

An algorithmic framework for colouring locally sparse graphs

We develop an algorithmic framework for graph colouring that reduces the problem to verifying a local probabilistic property of the independent sets. With this we give, for any fixed $k\ge 3$ and $\varepsilon>0$, a randomised polynomial-time algorithm for colouring graphs of maximum degree $\Delta$ in which each vertex is contained in at most $t$ copies of a cycle of length $k$, where $1/2\le t\le \Delta^\frac{2\varepsilon}{1+2\varepsilon}/(\log\Delta)^2$, with $\lfloor(1+\varepsilon)\Delta/\log(\Delta/\sqrt t)\rfloor$ colours. This generalises and improves upon several notable results including those of Kim (1995) and Alon, Krivelevich and Sudakov (1999), and more recent ones of Molloy (2019) and Achlioptas, Iliopoulos and Sinclair (2019). This bound on the chromatic number is tight up to an asymptotic factor $2$ and it coincides with a famous algorithmic barrier to colouring random graphs.Comment: 23 page

Subgraph densities in a surface

Given a fixed graph $H$ that embeds in a surface $\Sigma$, what is the maximum number of copies of $H$ in an $n$-vertex graph $G$ that embeds in $\Sigma$? We show that the answer is $\Theta(n^{f(H)})$, where $f(H)$ is a graph invariant called the `flap-number' of $H$, which is independent of $\Sigma$. This simultaneously answers two open problems posed by Eppstein (1993). When $H$ is a complete graph we give more precise answers.Comment: v4: referee's comments implemented. v3: proof of the main theorem fully rewritten, fixes a serious error in the previous version found by Kevin Hendre

Quantum Apices: Identifying Limits of Entanglement, Nonlocality, & Contextuality

This work develops analytic methods to quantitatively demarcate quantum reality from its subset of classical phenomenon, as well as from the superset of general probabilistic theories. Regarding quantum nonlocality, we discuss how to determine the quantum limit of Bell-type linear inequalities. In contrast to semidefinite programming approaches, our method allows for the consideration of inequalities with abstract weights, by means of leveraging the Hermiticity of quantum states. Recognizing that classical correlations correspond to measurements made on separable states, we also introduce a practical method for obtaining sufficient separability criteria. We specifically vet the candidacy of driven and undriven superradiance as schema for entanglement generation. We conclude by reviewing current approaches to quantum contextuality, emphasizing the operational distinction between nonlocal and contextual quantum statistics. We utilize our abstractly-weighted linear quantum bounds to explicitly demonstrate a set of conditional probability distributions which are simultaneously compatible with quantum contextuality while being incompatible with quantum nonlocality. It is noted that this novel statistical regime implies an experimentally-testable target for the Consistent Histories theory of quantum gravity.Comment: Doctoral Thesis for the University of Connecticu

Representations of Partition Problems and the Method of Moments

The thesis follows two main goals. The first is to formulate, explain and link representations of partitions that can be used to model partition problems. The second is to use the method of moments, an approach from polynomial optimization, to bound the global optima of the corresponding partition problems by constructing convex relaxations of these representations. For problems like Euclidean k-clustering, this is a stark contrast to their usual treatment, which mostly involves heuristics that are content with local optima. Since the method of moments results in a convex approach, the focus lies on finding and exploiting representations that lack a non-trivial symmetry-invariant solution space in order to be able to round the relaxations to feasible solutions. The representations considered in the thesis are assignment matrices, partition matrices, projection matrices and simplicial covers for a generalized version of Euclidean k-clustering. Connections and transformations between the matrix classes are established and compared to the literature, and it is explicitly shown how partition matrices arise naturally from assignment matrices through the method of moments. Using projection matrices, we are able to give a new formulation of the colouring number, and the resulting relaxations from the method of moments are compared to the Lovász theta number. We characterize under which circumstances the relaxations agree and explain when they do not, indicating our first main result that in this case, relaxing binary matrix entries yields better results than relaxing binary eigenvalues. The final part of the thesis is devoted to what we call the affine Euclidean k-clustering problem, which is a more general version of the Euclidean k-clustering problem. As our second main result of the thesis, we introduce a new method for this challenging problem, utilizing simplicial covers of the feasible region to formulate unique representations of the optimal solutions of the underlying problem. In contrast to applying the method of moments directly, applying it to our formulation yields a slower growth in size, better parallelizability and enables us to recover information that can be used for rounding, which is not possible for the standard formulation due to symmetry