Approximation of non-boolean 2CSP

We develop a polynomial time $\Omega\left ( \frac 1R \log R \right)$ approximate algorithm for Max 2CSP-$R$, the problem where we are given a collection of constraints, each involving two variables, where each variable ranges over a set of size $R$, and we want to find an assignment to the variables that maximizes the number of satisfied constraints. Assuming the Unique Games Conjecture, this is the best possible approximation up to constant factors. Previously, a $1/R$-approximate algorithm was known, based on linear programming. Our algorithm is based on semidefinite programming (SDP) and on a novel rounding technique. The SDP that we use has an almost-matching integrality gap

How to Play Unique Games against a Semi-Random Adversary

In this paper, we study the average case complexity of the Unique Games problem. We propose a natural semi-random model, in which a unique game instance is generated in several steps. First an adversary selects a completely satisfiable instance of Unique Games, then she chooses an epsilon-fraction of all edges, and finally replaces ("corrupts") the constraints corresponding to these edges with new constraints. If all steps are adversarial, the adversary can obtain any (1-epsilon) satisfiable instance, so then the problem is as hard as in the worst case. In our semi-random model, one of the steps is random, and all other steps are adversarial. We show that known algorithms for unique games (in particular, all algorithms that use the standard SDP relaxation) fail to solve semi-random instances of Unique Games. We present an algorithm that with high probability finds a solution satisfying a (1-delta) fraction of all constraints in semi-random instances (we require that the average degree of the graph is Omega(log k). To this end, we consider a new non-standard SDP program for Unique Games, which is not a relaxation for the problem, and show how to analyze it. We present a new rounding scheme that simultaneously uses SDP and LP solutions, which we believe is of independent interest. Our result holds only for epsilon less than some absolute constant. We prove that if epsilon > 1/2, then the problem is hard in one of the models, the result assumes the 2-to-2 conjecture. Finally, we study semi-random instances of Unique Games that are at most (1-epsilon) satisfiable. We present an algorithm that with high probability, distinguishes between the case when the instance is a semi-random instance and the case when the instance is an (arbitrary) (1-delta) satisfiable instance if epsilon > c delta

Collaborative Learning of Stochastic Bandits over a Social Network

We consider a collaborative online learning paradigm, wherein a group of agents connected through a social network are engaged in playing a stochastic multi-armed bandit game. Each time an agent takes an action, the corresponding reward is instantaneously observed by the agent, as well as its neighbours in the social network. We perform a regret analysis of various policies in this collaborative learning setting. A key finding of this paper is that natural extensions of widely-studied single agent learning policies to the network setting need not perform well in terms of regret. In particular, we identify a class of non-altruistic and individually consistent policies, and argue by deriving regret lower bounds that they are liable to suffer a large regret in the networked setting. We also show that the learning performance can be substantially improved if the agents exploit the structure of the network, and develop a simple learning algorithm based on dominating sets of the network. Specifically, we first consider a star network, which is a common motif in hierarchical social networks, and show analytically that the hub agent can be used as an information sink to expedite learning and improve the overall regret. We also derive networkwide regret bounds for the algorithm applied to general networks. We conduct numerical experiments on a variety of networks to corroborate our analytical results.Comment: 14 Pages, 6 Figure

On the Expansion of Group-Based Lifts

A $k$-lift of an $n$-vertex base graph $G$ is a graph $H$ on $n\times k$ vertices, where each vertex $v$ of $G$ is replaced by $k$ vertices $v_1,\cdots{},v_k$ and each edge $(u,v)$ in $G$ is replaced by a matching representing a bijection $\pi_{uv}$ so that the edges of $H$ are of the form $(u_i,v_{\pi_{uv}(i)})$. Lifts have been studied as a means to efficiently construct expanders. In this work, we study lifts obtained from groups and group actions. We derive the spectrum of such lifts via the representation theory principles of the underlying group. Our main results are: (1) There is a constant $c_1$ such that for every $k\geq 2^{c_1nd}$, there does not exist an abelian $k$-lift $H$ of any $n$-vertex $d$-regular base graph with $H$ being almost Ramanujan (nontrivial eigenvalues of the adjacency matrix at most $O(\sqrt{d})$ in magnitude). This can be viewed as an analogue of the well-known no-expansion result for abelian Cayley graphs. (2) A uniform random lift in a cyclic group of order $k$ of any $n$-vertex $d$-regular base graph $G$, with the nontrivial eigenvalues of the adjacency matrix of $G$ bounded by $\lambda$ in magnitude, has the new nontrivial eigenvalues also bounded by $\lambda+O(\sqrt{d})$ in magnitude with probability $1-ke^{-\Omega(n/d^2)}$. In particular, there is a constant $c_2$ such that for every $k\leq 2^{c_2n/d^2}$, there exists a lift $H$ of every Ramanujan graph in a cyclic group of order $k$ with $H$ being almost Ramanujan. We use this to design a quasi-polynomial time algorithm to construct almost Ramanujan expanders deterministically. The existence of expanding lifts in cyclic groups of order $k=2^{O(n/d^2)}$ can be viewed as a lower bound on the order $k_0$ of the largest abelian group that produces expanding lifts. Our results show that the lower bound matches the upper bound for $k_0$ (upto $d^3$ in the exponent)

Scalar dissipation rate based flamelet modelling of turbulent premixed flames

Lean premixed combustion has potential for reducing emissions from combustion devices without compromising fuel efficiency, but it is prone to instabilities which presents design difficulties. From emissions point of view reliable predictions of species formation rates in the flame zone are required while from the point of view of thermo--acoustics the prediction of spatial variation of heat release rate is crucial; both tasks are challenging but imperative in CFD based design of combustion systems. In this thesis a computational model for turbulent premixed combustion is proposed in the RANS framework and its predictive ability is studied. The model is based on the flamelet concept and employs strained laminar flamelets in reactant--to--product opposed flow configuration. The flamelets are parametrised by scalar dissipation rate of progress variable which is a suitable quantity to describe the flamelet structure since it is governed by convection--diffusion--reaction balance and represents the flame front dynamics. This paramaterisation is new. The mean reaction rate and mean species concentrations are obtained by integrating the corresponding flamelets quantity weighted by the joint pdf of the progress variable and its dissipation rate. The marginal pdf of the progress variable is obtained using $\beta$--pdf and the pdf of the conditional dissipation rate is presumed to be log--normal. The conditional mean dissipation rate is obtained from unconditional mean dissipation rate which is a modelling parameter. An algebraic model for the unconditional mean scalar dissipation rate is proposed based on the relevant physics of reactive scalar mixing in turbulent premixed flames. This algebraic model is validated directly using DNS data. An indirect validation is performed by deriving a turbulent flame speed expression using the Kolmogorov--Petrovskii--Piskunov analysis and comparing its predictions with experimental data from a wide range of flame and flow conditions. The mean reaction rate closure of the strained flamelets model is assessed using RANS calculations of statistically planar one--dimensional flames in corrugated flamelets and thin reaction zones regimes. The flame speeds predicted by this closure were close to experimental data in both the regimes. On the other hand, an unstrained flamelets closure predicts flame speed close to the experimental data in the corrugated flamelets regime, but overpredicts in the thin reaction zones regime indicating an overprediction of the mean reaction rate. The overall predictive ability of the strained flamelets model is assessed via calculations of laboratory flames of two different configurations: a rod stabilised V--flame and pilot stabilised Bunsen flames. For the V--flame, whose conditions correspond to the corrugated flamelets regime, the strained and unstrained flamelets models yield similar predictions which are in good agreement with experimental measurements. For the Bunsen flames which are in the thin reaction zones regime, the unstrained flamelet model predicts a smaller flame brush while the predictions of the strained flamelets model are in good agreement with the experimental data. The major and minor species concentrations are also reasonably well predicted by the strained flamelets model, although the minor species predictions seem sensitive to the product stream composition of the laminar flamelets. The fluid dynamics induced attenuation of the reaction rate is captured by the strained flamelets model enabling it to give better predictions than the unstrained flamelets model in the thin reaction zones regime. The planar flames and laboratory flames calculations illustrate the importance of appropriately accounting for fluid dynamic effects on flamelet structure and the scalar dissipation rate based strained flamelet model seems promising in this respect. Furthermore, this model seems to have a wide range of applicability with a fixed set of model parameters.This doctoral work was supported by a scholarship awarded by the Nehru Cambridge Trust of Cambridge Commonwealth Trusts

Design Method Development for the Design of Traction Systems

The objective of this research is to develop a design method for rapid exploration of traction concepts primarily for off-road vehicles. Different approaches available to achieve this objective are discussed and compared, such as computational, analytical, and physical methods. Computational approaches are based on simulations performed using Finite Element Method (FEM), Discrete Element Method (DEM), and combined Finite Element-Discrete Element (FE-DE) methods. Analytical approaches are based on closed form mathematical models developed by previous researchers based on the theory of plasticity. Physical approaches include fabrication and testing of prototypes at different levels of abstraction. This thesis compares these different approaches to design with respect to design process requirements of (1) timeliness, (2) cost, (3) required expertise, (4) accuracy of results, (5) flexibility to adapt to new designs and (6) stage of design process. This comparison is done both at a theoretical level and at an implemented level where each of the strategies are used to try and delineate between different classes of traction concepts. It is proposed that the physical prototyping approach should be the preferred approach with respect to these criteria. A new structured design approach is developed based on these findings to employ the different modeling schemes at stages of the design process that are most appropriate based on the technological maturity of this specific application domain