Deeparnab Chakrabarty

On Allocating Goods to Maximize Fairness

We consider the Max-Min Allocation problem: given a set of m agents and a set of n items, where agent A has utility u(A, i) for item i, our goal is to allocate items to agents so as to maximize fairness. Specifically, the utility of an agent is the sum of its utilities for the items it receives, and we seek to maximize the minimum utility of any agent. While this problem has received much attention recently, its approximability has not been well-understood thus far. The best known approximation algorithm achieves a roughly O(\sqrt m}-approximation, and in contrast, the best known hardness of approximation stands at 2. Our main result is an algorithm that achieves a \tilde{O}(n^{\eps})-approximation in time n^{O(1/\eps)} for any \eps=\Omega(log log n/log n). In particular, we obtain a poly-logarithmic approximation in quasi-polynomial time, and for every constant \eps ≫ 0, we obtain an n^{\eps}-approximation in polynomial time. Our algorithm also yields a quasi-polynomial time m^{\eps}-approximation algorithm for any constant \eps ≫ 0. An interesting technical aspect of our algorithm is that we use as a building block a linear program whose integrality gap is \Omega(\sqrt m). We bypass this obstacle by iteratively using the solutions produced by the LP to construct new instances with significantly smaller integrality gaps, eventually obtaining the desired approximation. We also investigate a special case of the problem, where every item has a non-zero utility for at most two agents. This problem is hard to approximate to within any factor better than 2. We give a factor 2-approximation algorithm.

On the Approximability of Budgeted Allocations and Improved Lower Bounds for Submodular Welfare Maximization and GAP

In this paper we consider the following maximum budgeted allocation (MBA) problem: Given a set of m indivisible items and n agents; each agent i willing to pay bij on item j and with a maximum budget of Bi, the goal is to allocate items to agents to maximize revenue. The problem naturally arises as auctioneer revenue maximization in budget-constrained auctions and as winner determination problem in combinatorial auctions when utilities of agents are budgeted-additive.We give a 3/4-approximation algorithm for MBA improving upon the previous best of sime0.632[2, 10]. Our techniques are based on a natural LP relaxation of MBA and our factor is optimal in the sense that it matches the integrality gap of the LP.We prove it is NP-hard to approximate MBA to any factor better than 15/16, previously only NP-hardness was known [21, 17]. Our result also implies NP- hardness of approximating maximum submodular welfare with demand oracle to a factor better than 15/16, improving upon the best known hardness of 275/276[10].Our hardness techniques can be modified to prove that it is NP-hard to approximate the Generalized Assignment Problem (GAP) to any factor better than 10/11. This improves upon the 422/423 hardness of [7, 9].We use iterative rounding on a natural LP relaxation of MBA to obtain the 3/4-approximation. We also give a (3/4 - epsiv) -factor algorithm based on the primal-dual schema which runs in O(nm) time, for any constant epsiv > 0.

An

A Boolean function

o(n)

f:\{0,1\}^n \mapsto \{0,1\}

Monotonicity Tester for Boolean Functions over the Hypercube

is said to be

On column-restricted and priority covering integer programs

\varepsilon

Approximability of Sparse Integer Programs

-far from monotone if

Fairness and optimality in congestion games

f

New results on rationality and strongly polynomial time solvability in eisenberg-gale markets

needs to be modified in at least

Approximability of the Firefighter Problem: Computing Cuts over Time

\varepsilon