Koyel Mukherjee — Page 2

arXiv2016

LP Rounding and Combinatorial Algorithms for Minimizing Active and Busy Time

We consider fundamental scheduling problems motivated by energy issues. In this framework, we are given a set of jobs, each with a release time, deadline and required processing length. The jobs need to be scheduled on a machine so that at most g jobs are active at any given time. The duration for which a machine is active (i.e., "on") is referred to as its active time. The goal is to find a feasible schedule for all jobs, minimizing the total active time. When preemption is allowed at integer time points, we show that a minimal feasible schedule already yields a 3-approximation (and this bound is tight) and we further improve this to a 2-approximation via LP rounding techniques. Our second contribution is for the non-preemptive version of this problem. However, since even asking if a feasible schedule on one machine exists is NP-hard, we allow for an unbounded number of virtual machines, each having capacity of g. This problem is known as the busy time problem in the literature and a 4-approximation is known for this problem. We develop a new combinatorial algorithm that gives a 3-approximation. Furthermore, we consider the preemptive busy time problem, giving a simple and exact greedy algorithm when unbounded parallelism is allowed, i.e., g is unbounded. For arbitrary g, this yields an algorithm that is 2-approximate.

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arXiv2013

A Game-Theoretic Model Motivated by the DARPA Network Challenge

In this paper we propose a game-theoretic model to analyze events similar to the 2009 \emph{DARPA Network Challenge}, which was organized by the Defense Advanced Research Projects Agency (DARPA) for exploring the roles that the Internet and social networks play in incentivizing wide-area collaborations. The challenge was to form a group that would be the first to find the locations of ten moored weather balloons across the United States. We consider a model in which $N$ people (who can form groups) are located in some topology with a fixed coverage volume around each person's geographical location. We consider various topologies where the players can be located such as the Euclidean $d$-dimension space and the vertices of a graph. A balloon is placed in the space and a group wins if it is the first one to report the location of the balloon. A larger team has a higher probability of finding the balloon, but we assume that the prize money is divided equally among the team members. Hence there is a competing tension to keep teams as small as possible. \emph{Risk aversion} is the reluctance of a person to accept a bargain with an uncertain payoff rather than another bargain with a more certain, but possibly lower, expected payoff. In our model we consider the \emph{isoelastic} utility function derived from the Arrow-Pratt measure of relative risk aversion. The main aim is to analyze the structures of the groups in Nash equilibria for our model. For the $d$-dimensional Euclidean space ($d\geq 1$) and the class of bounded degree regular graphs we show that in any Nash Equilibrium the \emph{richest} group (having maximum expected utility per person) covers a constant fraction of the total volume.

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