1,486 research outputs found
Algorithms as Mechanisms: The Price of Anarchy of Relax-and-Round
Many algorithms that are originally designed without explicitly considering
incentive properties are later combined with simple pricing rules and used as
mechanisms. The resulting mechanisms are often natural and simple to
understand. But how good are these algorithms as mechanisms? Truthful reporting
of valuations is typically not a dominant strategy (certainly not with a
pay-your-bid, first-price rule, but it is likely not a good strategy even with
a critical value, or second-price style rule either). Our goal is to show that
a wide class of approximation algorithms yields this way mechanisms with low
Price of Anarchy.
The seminal result of Lucier and Borodin [SODA 2010] shows that combining a
greedy algorithm that is an -approximation algorithm with a
pay-your-bid payment rule yields a mechanism whose Price of Anarchy is
. In this paper we significantly extend the class of algorithms for
which such a result is available by showing that this close connection between
approximation ratio on the one hand and Price of Anarchy on the other also
holds for the design principle of relaxation and rounding provided that the
relaxation is smooth and the rounding is oblivious.
We demonstrate the far-reaching consequences of our result by showing its
implications for sparse packing integer programs, such as multi-unit auctions
and generalized matching, for the maximum traveling salesman problem, for
combinatorial auctions, and for single source unsplittable flow problems. In
all these problems our approach leads to novel simple, near-optimal mechanisms
whose Price of Anarchy either matches or beats the performance guarantees of
known mechanisms.Comment: Extended abstract appeared in Proc. of 16th ACM Conference on
Economics and Computation (EC'15
Balancing Relevance and Diversity in Online Bipartite Matching via Submodularity
In bipartite matching problems, vertices on one side of a bipartite graph are
paired with those on the other. In its online variant, one side of the graph is
available offline, while the vertices on the other side arrive online. When a
vertex arrives, an irrevocable and immediate decision should be made by the
algorithm; either match it to an available vertex or drop it. Examples of such
problems include matching workers to firms, advertisers to keywords, organs to
patients, and so on. Much of the literature focuses on maximizing the total
relevance---modeled via total weight---of the matching. However, in many
real-world problems, it is also important to consider contributions of
diversity: hiring a diverse pool of candidates, displaying a relevant but
diverse set of ads, and so on. In this paper, we propose the Online Submodular
Bipartite Matching (\osbm) problem, where the goal is to maximize a submodular
function over the set of matched edges. This objective is general enough to
capture the notion of both diversity (\emph{e.g.,} a weighted coverage
function) and relevance (\emph{e.g.,} the traditional linear function)---as
well as many other natural objective functions occurring in practice
(\emph{e.g.,} limited total budget in advertising settings). We propose novel
algorithms that have provable guarantees and are essentially optimal when
restricted to various special cases. We also run experiments on real-world and
synthetic datasets to validate our algorithms.Comment: To appear in AAAI 201
Streaming Algorithms for Submodular Function Maximization
We consider the problem of maximizing a nonnegative submodular set function
subject to a -matchoid
constraint in the single-pass streaming setting. Previous work in this context
has considered streaming algorithms for modular functions and monotone
submodular functions. The main result is for submodular functions that are {\em
non-monotone}. We describe deterministic and randomized algorithms that obtain
a -approximation using -space, where is
an upper bound on the cardinality of the desired set. The model assumes value
oracle access to and membership oracles for the matroids defining the
-matchoid constraint.Comment: 29 pages, 7 figures, extended abstract to appear in ICALP 201
A Bridge between Liquid and Social Welfare in Combinatorial Auctions with Submodular Bidders
We study incentive compatible mechanisms for Combinatorial Auctions where the
bidders have submodular (or XOS) valuations and are budget-constrained. Our
objective is to maximize the \emph{liquid welfare}, a notion of efficiency for
budget-constrained bidders introduced by Dobzinski and Paes Leme (2014). We
show that some of the known truthful mechanisms that best-approximate the
social welfare for Combinatorial Auctions with submodular bidders through
demand query oracles can be adapted, so that they retain truthfulness and
achieve asymptotically the same approximation guarantees for the liquid
welfare. More specifically, for the problem of optimizing the liquid welfare in
Combinatorial Auctions with submodular bidders, we obtain a universally
truthful randomized -approximate mechanism, where is the number
of items, by adapting the mechanism of Krysta and V\"ocking (2012).
Additionally, motivated by large market assumptions often used in mechanism
design, we introduce a notion of competitive markets and show that in such
markets, liquid welfare can be approximated within a constant factor by a
randomized universally truthful mechanism. Finally, in the Bayesian setting, we
obtain a truthful -approximate mechanism for the case where bidder
valuations are generated as independent samples from a known distribution, by
adapting the results of Feldman, Gravin and Lucier (2014).Comment: AAAI-1
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