Varsha Dani

Allocating indivisible goods

The problem of allocating divisible goods has enjoyed a lot of attention in both mathematics (e.g. the cake-cutting problem) and economics (e.g. market equilibria). On the other hand, the natural requirement of indivisible goods has been somewhat neglected, perhaps because of its more complicated nature. In this work we study a fairness criterion, called the Max-Min Fairness problem, for k players who want to allocate among themselves m indivisible goods. Each player has a specified valuation function on the subsets of the goods and the goal is to split the goods between the players so as to maximize the minimum valuation. Viewing the problem from a game theoretic perspective, we show that for two players and additive valuations the expected minimum of the (randomized) cut-and-choose mechanism is a 1/2-approximation of the optimum. To complement this result we show that no truthful mechanism can compute the exact optimum.We also consider the algorithmic perspective when the (true) additive valuation functions are part of the input. We present a simple 1/(m - k + 1) approximation algorithm which allocates to every player at least 1/k fraction of the value of all but the k - 1 heaviest items. We also give an algorithm with additive error against the fractional optimum bounded by the value of the largest item. The two approximation algorithms are incomparable in the sense that there exist instances when one outperforms the other.

Error limiting reductions between classification tasks

We introduce a reduction-based model for analyzing supervised learning tasks. We use this model to devise a new reduction from multi-class cost-sensitive classification to binary classification with the following guarantee: If the learned binary classifier has error rate at most ε then the cost-sensitive classifier has cost at most 2ε times the expected sum of costs of all possible lables. Since cost-sensitive classification can embed any bounded loss finite choice supervised learning task, this result shows that any such task can be solved using a binary classification oracle. Finally, we present experimental results showing that our new reduction outperforms existing algorithms for multi-class cost-sensitive learning.

Brief announcement: breaking the O(nm) bit barrier, secure multiparty computation with a static adversary

We describe scalable algorithms for secure multiparty computation (SMPC). We assume a synchronous message passing communication model, but we do not assume the existence of a broadcast channel. Our main result holds for the case where there are <i>n</i> players, of which a 1/3-ε fraction are controlled by an adversary, for ε any positive constant. We describe an SMPC algorithm for this model that requires each player to send Õ(⁄<i>n</i>+<i>m</i><i>n</i> + √<i>n</i>) messages and perform Õ(⁄<i>n</i>+<i>m</i><i>n</i> + √<i>n</i>) computations to compute any function <i>f</i>, where <i>m</i> is the size of a circuit to compute <i>f</i>. We also consider a model where all players are rational. In this model, we describe a Nash equilibrium protocol that solves SMPC and requires each player to send Õ(⁄<i>n</i>+<i>m</i><i>n</i>) messages and perform Õ(⁄<i>n</i>+<i>mn</i>) computations. These results significantly improve over past results for SMPC which require each player to send a number of bits and perform a number of computations that is Θ(<i>n, m</i>)

Quorums Quicken Queries: Efficient Asynchronous Secure Multiparty Computation

We describe an asynchronous algorithm to solve secure multiparty computation MPC over n players, when strictly less than a

Independent sets in random graphs from the weighted second moment method

{1}\over{8}

Resource-Competitive Algorithms

fraction of the players are controlled by a static adversary. For any function f over a field that can be computed by a circuit with m gates, our algorithm requires each player to send a number of field elements and perform an amount of computation that is

Scalable rational secret sharing

\tilde{O}\frac{m}{n} + \sqrt n

Tight Bounds on the Threshold for Permuted k-Colorability

. This significantly improves over traditional algorithms, which require each player to both send a number of messages and perform computation that is Ωnm. Additionaly, we define the threshold counting problem and present a distributed algorithm to solve it in the asynchronous communication model. Our algorithm is load balanced, with computation, communication and latency complexity of Ologn, and may be of independent interest to other applications with a load balancing goal in mind.

Interactive Communication with Unknown Noise Rate

We prove new lower bounds on the likely size of a maximum independent set in a random graph with a given average degree. Our method is a weighted version of the second moment method, where we give each independent set a weight based on the total degree of its vertices.

The Energy Complexity of Broadcast

The point of adversarial analysis is to model the worst-case performance of an algorithm. Unfortunately, this analysis may not always reect performance in practice because the adversarial assumption can be overly pessimistic. In such cases, several techniques have been developed to provide a more refined understanding of how an algorithm performs e.g., competitive analysis, parameterized analysis, and the theory of approximation algorithms. Here, we describe an analogous technique called resource competitiveness, tailored for distributed systems. Often there is an operational cost for adversarial behavior arising from bandwidth usage, computational power, energy limitations, etc. Modeling this cost provides some notion of how much disruption the adversary can inict on the system. In parameterizing by this cost, we can design an algorithm with the following guarantee: if the adversary pays T, then the additional cost of the algorithm is some function of T. Resource competitiveness yields results pertaining to secure, fault tolerant, and efficient distributed computation. We summarize these results and highlight future challenges where we expect this algorithmic tool to provide new insights.