Minimizing Flow-Time on Unrelated Machines

We consider some flow-time minimization problems in the unrelated machines setting. In this setting, there is a set of $m$ machines and a set of $n$ jobs, and each job $j$ has a machine dependent processing time of $p_{ij}$ on machine $i$. The flow-time of a job is the total time the job spends in the system (completion time minus its arrival time), and is one of the most natural quality of service measure. We show the following two results: an $O(\min(\log^2 n,\log n \log P))$ approximation algorithm for minimizing the total-flow time, and an $O(\log n)$ approximation for minimizing the maximum flow-time. Here $P$ is the ratio of maximum to minimum job size. These are the first known poly-logarithmic guarantees for both the problems.Comment: The new version fixes some typos in the previous version. The paper is accepted for publication in STOC 201

The Geometry of Scheduling

We consider the following general scheduling problem: The input consists of n jobs, each with an arbitrary release time, size, and a monotone function specifying the cost incurred when the job is completed at a particular time. The objective is to find a preemptive schedule of minimum aggregate cost. This problem formulation is general enough to include many natural scheduling objectives, such as weighted flow, weighted tardiness, and sum of flow squared. Our main result is a randomized polynomial-time algorithm with an approximation ratio O(log log nP), where P is the maximum job size. We also give an O(1) approximation in the special case when all jobs have identical release times. The main idea is to reduce this scheduling problem to a particular geometric set-cover problem which is then solved using the local ratio technique and Varadarajan's quasi-uniform sampling technique. This general algorithmic approach improves the best known approximation ratios by at least an exponential factor (and much more in some cases) for essentially all of the nontrivial common special cases of this problem. Our geometric interpretation of scheduling may be of independent interest.Comment: Conference version in FOCS 201

Semidefinite optimization in discrepancy theory

Recently, there have been several new developments in discrepancy theory based on connections to semidefinite programming. This connection has been useful in several ways. It gives efficient polynomial time algorithms for several problems for which only non-constructive results were previously known. It also leads to several new structural results in discrepancy itself, such as tightness of the so-called determinant lower bound, improved bounds on the discrepancy of the union of set systems and so on. We will give a brief survey of these results, focussing on the main ideas and the techniques involved

Weighted k-Server Bounds via Combinatorial Dichotomies

The weighted $k$-server problem is a natural generalization of the $k$-server problem where each server has a different weight. We consider the problem on uniform metrics, which corresponds to a natural generalization of paging. Our main result is a doubly exponential lower bound on the competitive ratio of any deterministic online algorithm, that essentially matches the known upper bounds for the problem and closes a large and long-standing gap. The lower bound is based on relating the weighted $k$-server problem to a certain combinatorial problem and proving a Ramsey-theoretic lower bound for it. This combinatorial connection also reveals several structural properties of low cost feasible solutions to serve a sequence of requests. We use this to show that the generalized Work Function Algorithm achieves an almost optimum competitive ratio, and to obtain new refined upper bounds on the competitive ratio for the case of $d$ different weight classes.Comment: accepted to FOCS'1

Synthesis, Characterization and Spectral Studies of Novel Eu3+ Based Phosphors and its Solid Solutions

Tungstate and molybdate based metal oxides with substituted Eu3+ based phosphors are promising red emitting phosphor materials for white LEDs. In the present research, a series of novel red phosphors including La2GdBWO9 as a host and doped La2Gd0.9Eu0.1BW1-xMoxO9 (x = 0 – 0.4, insteps of 0.1) samples were prepared by the conventional solid state reaction. The phase formation and transformation of the synthesized compositions was investigated by powder X-ray diffraction, DRS studies and photoluminescence properties were studied as a function of the Mo/W ratio. All the compositions show broad charge transfer (CT) band due to CT from O to W/Mo and red emission due to Eu3+ ions. Eu3+ substituted compositions show high red emission intensity under NUV/blue ray excitation as well as the emission of the phosphor shows very good CIE (Commission Internationale de l’Eclairage) chromaticity coordinates (x = 0.6647, y = 0.3349). Band gap of La2Gd0.9Eu0.1BW1-xMoxO9 (x = 0 – 0.3, insteps of 0.1) becomes narrower than that of host La2GdBWO9 by introducing Mo6+. The obtained results are reveals that the investigated phosphors are potential phosphors for LEDs

New Notions and Constructions of Sparsification for Graphs and Hypergraphs

A sparsifier of a graph $G$ (Bencz\'ur and Karger; Spielman and Teng) is a sparse weighted subgraph $\tilde G$ that approximately retains the cut structure of $G$. For general graphs, non-trivial sparsification is possible only by using weighted graphs in which different edges have different weights. Even for graphs that admit unweighted sparsifiers, there are no known polynomial time algorithms that find such unweighted sparsifiers. We study a weaker notion of sparsification suggested by Oveis Gharan, in which the number of edges in each cut $(S,\bar S)$ is not approximated within a multiplicative factor $(1+\epsilon)$, but is, instead, approximated up to an additive term bounded by $\epsilon$ times $d\cdot |S| + \text{vol}(S)$, where $d$ is the average degree, and $\text{vol}(S)$ is the sum of the degrees of the vertices in $S$. We provide a probabilistic polynomial time construction of such sparsifiers for every graph, and our sparsifiers have a near-optimal number of edges $O(\epsilon^{-2} n {\rm polylog}(1/\epsilon))$. We also provide a deterministic polynomial time construction that constructs sparsifiers with a weaker property having the optimal number of edges $O(\epsilon^{-2} n)$. Our constructions also satisfy a spectral version of the ``additive sparsification'' property. Our construction of ``additive sparsifiers'' with $O_\epsilon (n)$ edges also works for hypergraphs, and provides the first non-trivial notion of sparsification for hypergraphs achievable with $O(n)$ hyperedges when $\epsilon$ and the rank $r$ of the hyperedges are constant. Finally, we provide a new construction of spectral hypergraph sparsifiers, according to the standard definition, with ${\rm poly}(\epsilon^{-1},r)\cdot n\log n$ hyperedges, improving over the previous spectral construction (Soma and Yoshida) that used $\tilde O(n^3)$ hyperedges even for constant $r$ and $\epsilon$.Comment: 31 page

An Algorithm for Koml\'os Conjecture Matching Banaszczyk's bound

We consider the problem of finding a low discrepancy coloring for sparse set systems where each element lies in at most t sets. We give an efficient algorithm that finds a coloring with discrepancy O((t log n)^{1/2}), matching the best known non-constructive bound for the problem due to Banaszczyk. The previous algorithms only achieved an O(t^{1/2} log n) bound. The result also extends to the more general Koml\'{o}s setting and gives an algorithmic O(log^{1/2} n) bound