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)

On approximability and LP formulations for multicut and feedback set problems

Graph cut algorithms are an important tool for solving optimization problems in a variety of areas in computer science. Of particular importance is the min $s$-$t$ cut problem and an efficient (polynomial time) algorithm for it. Unfortunately, efficient algorithms are not known for several other cut problems. Furthermore, the theory of NP-completeness rules out the existence of efficient algorithms for these problems if the $P\neq NP$ conjecture is true. For this reason, much of the focus has shifted to the design of approximation algorithms. Over the past 30 years significant progress has been made in understanding the approximability of various graph cut problems. In this thesis we further advance our understanding by closing some of the gaps in the known approximability results. Our results comprise of new approximation algorithms as well as new hardness of approximation bounds. For both of these, new linear programming (LP) formulations based on a labeling viewpoint play a crucial role. One of the problems we consider is a generalization of the min $s$-$t$ cut problem, known as the multicut problem. In a multicut instance, we are given an undirected or directed weighted supply graph and a set of pairs of vertices which can be encoded as a demand graph. The goal is to remove a minimum weight set of edges from the supply graph such that all the demand pairs are disconnected. We study the effect of the structure of the demand graph on the approximability of multicut. We prove several algorithmic and hardness results which unify previous results and also yield new results. Our algorithmic result generalizes the constant factor approximations known for the undirected and directed multiway cut problems to a much larger class of demand graphs. Our hardness result proves the optimality of the hitting-set LP for directed graphs. In addition to the results on multicut, we also prove results for multiway cut and another special case of multicut, called linear-3-cut. Our results exhibit tight approximability bounds in some cases and improve upon the existing bound in other cases. As a consequence, we also obtain tight approximation results for related problems. Another part of the thesis is focused on feedback set problems. In a subset feedback edge or vertex set instance, we are given an undirected edge or vertex weighted graph, and a set of terminals. The goal is to find a minimum weight set of edges or vertices which hit all of the cycles that contain some terminal vertex. There is a natural hitting-set LP which has an $\Omega(\log k)$ integrality gap for $k$ terminals. Constant factor approximation algorithms have been developed using combinatorial techniques. However, the factors are not tight, and the algorithms are sometimes complicated. Since most of the related problems admit optimal approximation algorithms using LP relaxations, lack of good LP relaxations was seen as a fundamental roadblock towards resolving the approximability of these problems. In this thesis we address this by developing new LP relaxations with constant integrality gaps for subset feedback edge and vertex set problems

Improving the integrality gap for multiway cut

In the multiway cut problem, we are given an undirected graph with non-negative edge weights and a collection of $k$ terminal nodes, and the goal is to partition the node set of the graph into $k$ non-empty parts each containing exactly one terminal so that the total weight of the edges crossing the partition is minimized. The multiway cut problem for $k\ge 3$ is APX-hard. For arbitrary $k$, the best-known approximation factor is $1.2965$ due to [Sharma and Vondr\'{a}k, 2014] while the best known inapproximability factor is $1.2$ due to [Angelidakis, Makarychev and Manurangsi, 2017]. In this work, we improve on the lower bound to $1.20016$ by constructing an integrality gap instance for the CKR relaxation. A technical challenge in improving the gap has been the lack of geometric tools to understand higher-dimensional simplices. Our instance is a non-trivial $3$-dimensional instance that overcomes this technical challenge. We analyze the gap of the instance by viewing it as a convex combination of $2$-dimensional instances and a uniform 3-dimensional instance. We believe that this technique could be exploited further to construct instances with larger integrality gap. One of the ingredients of our proof technique is a generalization of a result on \emph{Sperner admissible labelings} due to [Mirzakhani and Vondr\'{a}k, 2015] that might be of independent combinatorial interest.Comment: 28 page

Spectrally Robust Graph Isomorphism

We initiate the study of spectral generalizations of the graph isomorphism problem. b) The Spectral Graph Dominance (SGD) problem: On input of two graphs G and H does there exist a permutation pi such that G preceq pi(H)? c) The Spectrally Robust Graph Isomorphism (kappa-SRGI) problem: On input of two graphs G and H, find the smallest number kappa over all permutations pi such that pi(H) preceq G preceq kappa c pi(H) for some c. SRGI is a natural formulation of the network alignment problem that has various applications, most notably in computational biology. G preceq c H means that for all vectors x we have x^T L_G x <= c x^T L_H x, where L_G is the Laplacian G. We prove NP-hardness for SGD. We also present a kappa^3-approximation algorithm for SRGI for the case when both G and H are bounded-degree trees. The algorithm runs in polynomial time when kappa is a constant

Cyclodextrin-complexed curcumin exhibits anti-inflammatory and antiproliferative activities superior to those of curcumin through higher cellular uptake (vol 80, vol 1021, 2010)

Retraction notice to “Cyclodextrin-complexed curcumin exhibits anti-inflammatory and antiproliferative activities superior to those of curcumin through higher cellular uptake” [Biochem. Pharmacol. 80 (2010) 1021–1032].Non peer reviewe

Effect of Convective Drying on Texture, Rehydration, Microstructure and Drying Behavior of Yam (Dioscorea pentaphylla) Slices

Drying is a critical primary processing technique in enhancing and maintaining the quality and storability of Dioscorea pentaphylla. The present work investigated the effect of forced convective drying at three drying temperatures (50, 60, and 70℃). Ten drying and four-color kinetics models were used to fit the drying data to study the drying behavior and the effect of temperature and time on color change. Moisture diffusivity increased with hot air temperature (4.88526 × 10−10– 8.8069×10−10 m2/s). For Dioscorea pentaphylla slices, 27.04 (kJ/mol) of activation energy was found. Hii and others model gives the superior fitting for all the drying temperatures followed by logarithmic and Avhad and Marchetti model. Color kinetics was evaluated using L, a, and b values at a specified time during whole drying process. Temperature and time influenced the Lightness (L), yellowness (b), a value, chroma, hue, and browning index (BI). Dried slices from 70℃ showed more color change, whereas those from 50℃ had a medium-light brown. The modified color model is best fitted with high R2 and lower chi-square. Potassium metabisulfite (K2S2O5) pre-treatment and boiling significantly affected the drying time and final color of slices. The study reveals that drying at 50℃ exhibits better color retention and could be effectively used to dry Dioscorea pentaphylla. Dried Dioscorea pentaphylla can be utilized in both food and pharmaceutical industries for several applications for formulations food products and health supplements