19 research outputs found
Submodular Welfare Maximization
An overview of different variants of the submodular welfare maximization
problem in combinatorial auctions. In particular, I studied the existing
algorithmic and game theoretic results for submodular welfare maximization
problem and its applications in other areas such as social networks
A Truthful Mechanism for the Generalized Assignment Problem
We propose a truthful-in-expectation, -approximation mechanism for a
strategic variant of the generalized assignment problem (GAP). In GAP, a set of
items has to be optimally assigned to a set of bins without exceeding the
capacity of any singular bin. In the strategic variant of the problem we study,
values for assigning items to bins are the private information of bidders and
the mechanism should provide bidders with incentives to truthfully report their
values. The approximation ratio of the mechanism is a significant improvement
over the approximation ratio of the existing truthful mechanism for GAP.
The proposed mechanism comprises a novel convex optimization program as the
allocation rule as well as an appropriate payment rule. To implement the convex
program in polynomial time, we propose a fractional local search algorithm
which approximates the optimal solution within an arbitrarily small error
leading to an approximately truthful-in-expectation mechanism. The presented
algorithm improves upon the existing optimization algorithms for GAP in terms
of simplicity and runtime while the approximation ratio closely matches the
best approximation ratio given for GAP when all inputs are publicly known.Comment: 18 pages, Earlier version accepted at WINE 201
Locally Adaptive Optimization: Adaptive Seeding for Monotone Submodular Functions
The Adaptive Seeding problem is an algorithmic challenge motivated by
influence maximization in social networks: One seeks to select among certain
accessible nodes in a network, and then select, adaptively, among neighbors of
those nodes as they become accessible in order to maximize a global objective
function. More generally, adaptive seeding is a stochastic optimization
framework where the choices in the first stage affect the realizations in the
second stage, over which we aim to optimize.
Our main result is a -approximation for the adaptive seeding
problem for any monotone submodular function. While adaptive policies are often
approximated via non-adaptive policies, our algorithm is based on a novel
method we call \emph{locally-adaptive} policies. These policies combine a
non-adaptive global structure, with local adaptive optimizations. This method
enables the -approximation for general monotone submodular functions
and circumvents some of the impossibilities associated with non-adaptive
policies.
We also introduce a fundamental problem in submodular optimization that may
be of independent interest: given a ground set of elements where every element
appears with some small probability, find a set of expected size at most
that has the highest expected value over the realization of the elements. We
show a surprising result: there are classes of monotone submodular functions
(including coverage) that can be approximated almost optimally as the
probability vanishes. For general monotone submodular functions we show via a
reduction from \textsc{Planted-Clique} that approximations for this problem are
not likely to be obtainable. This optimization problem is an important tool for
adaptive seeding via non-adaptive policies, and its hardness motivates the
introduction of \emph{locally-adaptive} policies we use in the main result
Reallocation Mechanisms
We consider reallocation problems in settings where the initial endowment of
each agent consists of a subset of the resources. The private information of
the players is their value for every possible subset of the resources. The goal
is to redistribute resources among agents to maximize efficiency. Monetary
transfers are allowed, but participation is voluntary.
We develop incentive-compatible, individually-rational and budget balanced
mechanisms for several classic settings, including bilateral trade, partnership
dissolving, Arrow-Debreu markets, and combinatorial exchanges. All our
mechanisms (except one) provide a constant approximation to the optimal
efficiency in these settings, even in ones where the preferences of the agents
are complex multi-parameter functions
The Limitations of Optimization from Samples
In this paper we consider the following question: can we optimize objective
functions from the training data we use to learn them? We formalize this
question through a novel framework we call optimization from samples (OPS). In
OPS, we are given sampled values of a function drawn from some distribution and
the objective is to optimize the function under some constraint.
While there are interesting classes of functions that can be optimized from
samples, our main result is an impossibility. We show that there are classes of
functions which are statistically learnable and optimizable, but for which no
reasonable approximation for optimization from samples is achievable. In
particular, our main result shows that there is no constant factor
approximation for maximizing coverage functions under a cardinality constraint
using polynomially-many samples drawn from any distribution.
We also show tight approximation guarantees for maximization under a
cardinality constraint of several interesting classes of functions including
unit-demand, additive, and general monotone submodular functions, as well as a
constant factor approximation for monotone submodular functions with bounded
curvature
Combinatorial Auctions via Posted Prices
We study anonymous posted price mechanisms for combinatorial auctions in a
Bayesian framework. In a posted price mechanism, item prices are posted, then
the consumers approach the seller sequentially in an arbitrary order, each
purchasing her favorite bundle from among the unsold items at the posted
prices. These mechanisms are simple, transparent and trivially dominant
strategy incentive compatible (DSIC).
We show that when agent preferences are fractionally subadditive (which
includes all submodular functions), there always exist prices that, in
expectation, obtain at least half of the optimal welfare. Our result is
constructive: given black-box access to a combinatorial auction algorithm A,
sample access to the prior distribution, and appropriate query access to the
sampled valuations, one can compute, in polytime, prices that guarantee at
least half of the expected welfare of A. As a corollary, we obtain the first
polytime (in n and m) constant-factor DSIC mechanism for Bayesian submodular
combinatorial auctions, given access to demand query oracles. Our results also
extend to valuations with complements, where the approximation factor degrades
linearly with the level of complementarity
Learning Coverage Functions and Private Release of Marginals
We study the problem of approximating and learning coverage functions. A
function is a coverage function, if
there exists a universe with non-negative weights for each
and subsets of such that . Alternatively, coverage functions can be described
as non-negative linear combinations of monotone disjunctions. They are a
natural subclass of submodular functions and arise in a number of applications.
We give an algorithm that for any , given random and uniform
examples of an unknown coverage function , finds a function that
approximates within factor on all but -fraction of the
points in time . This is the first fully-polynomial
algorithm for learning an interesting class of functions in the demanding PMAC
model of Balcan and Harvey (2011). Our algorithms are based on several new
structural properties of coverage functions. Using the results in (Feldman and
Kothari, 2014), we also show that coverage functions are learnable agnostically
with excess -error over all product and symmetric
distributions in time . In contrast, we show that,
without assumptions on the distribution, learning coverage functions is at
least as hard as learning polynomial-size disjoint DNF formulas, a class of
functions for which the best known algorithm runs in time
(Klivans and Servedio, 2004).
As an application of our learning results, we give simple
differentially-private algorithms for releasing monotone conjunction counting
queries with low average error. In particular, for any , we obtain
private release of -way marginals with average error in time