437 research outputs found
The Price of Information in Combinatorial Optimization
Consider a network design application where we wish to lay down a
minimum-cost spanning tree in a given graph; however, we only have stochastic
information about the edge costs. To learn the precise cost of any edge, we
have to conduct a study that incurs a price. Our goal is to find a spanning
tree while minimizing the disutility, which is the sum of the tree cost and the
total price that we spend on the studies. In a different application, each edge
gives a stochastic reward value. Our goal is to find a spanning tree while
maximizing the utility, which is the tree reward minus the prices that we pay.
Situations such as the above two often arise in practice where we wish to
find a good solution to an optimization problem, but we start with only some
partial knowledge about the parameters of the problem. The missing information
can be found only after paying a probing price, which we call the price of
information. What strategy should we adopt to optimize our expected
utility/disutility?
A classical example of the above setting is Weitzman's "Pandora's box"
problem where we are given probability distributions on values of
independent random variables. The goal is to choose a single variable with a
large value, but we can find the actual outcomes only after paying a price. Our
work is a generalization of this model to other combinatorial optimization
problems such as matching, set cover, facility location, and prize-collecting
Steiner tree. We give a technique that reduces such problems to their non-price
counterparts, and use it to design exact/approximation algorithms to optimize
our utility/disutility. Our techniques extend to situations where there are
additional constraints on what parameters can be probed or when we can
simultaneously probe a subset of the parameters.Comment: SODA 201
Ignorance is Almost Bliss: Near-Optimal Stochastic Matching With Few Queries
The stochastic matching problem deals with finding a maximum matching in a
graph whose edges are unknown but can be accessed via queries. This is a
special case of stochastic -set packing, where the problem is to find a
maximum packing of sets, each of which exists with some probability. In this
paper, we provide edge and set query algorithms for these two problems,
respectively, that provably achieve some fraction of the omniscient optimal
solution.
Our main theoretical result for the stochastic matching (i.e., -set
packing) problem is the design of an \emph{adaptive} algorithm that queries
only a constant number of edges per vertex and achieves a
fraction of the omniscient optimal solution, for an arbitrarily small
. Moreover, this adaptive algorithm performs the queries in only a
constant number of rounds. We complement this result with a \emph{non-adaptive}
(i.e., one round of queries) algorithm that achieves a
fraction of the omniscient optimum. We also extend both our results to
stochastic -set packing by designing an adaptive algorithm that achieves a
fraction of the omniscient optimal solution, again
with only queries per element. This guarantee is close to the best known
polynomial-time approximation ratio of for the
\emph{deterministic} -set packing problem [Furer and Yu, 2013]
We empirically explore the application of (adaptations of) these algorithms
to the kidney exchange problem, where patients with end-stage renal failure
swap willing but incompatible donors. We show on both generated data and on
real data from the first 169 match runs of the UNOS nationwide kidney exchange
that even a very small number of non-adaptive edge queries per vertex results
in large gains in expected successful matches
Algorithms to Approximate Column-Sparse Packing Problems
Column-sparse packing problems arise in several contexts in both
deterministic and stochastic discrete optimization. We present two unifying
ideas, (non-uniform) attenuation and multiple-chance algorithms, to obtain
improved approximation algorithms for some well-known families of such
problems. As three main examples, we attain the integrality gap, up to
lower-order terms, for known LP relaxations for k-column sparse packing integer
programs (Bansal et al., Theory of Computing, 2012) and stochastic k-set
packing (Bansal et al., Algorithmica, 2012), and go "half the remaining
distance" to optimal for a major integrality-gap conjecture of Furedi, Kahn and
Seymour on hypergraph matching (Combinatorica, 1993).Comment: Extended abstract appeared in SODA 2018. Full version in ACM
Transactions of Algorithm
Almost Optimal Stochastic Weighted Matching With Few Queries
We consider the {\em stochastic matching} problem. An edge-weighted general
(i.e., not necessarily bipartite) graph is given in the input, where
each edge in is {\em realized} independently with probability ; the
realization is initially unknown, however, we are able to {\em query} the edges
to determine whether they are realized. The goal is to query only a small
number of edges to find a {\em realized matching} that is sufficiently close to
the maximum matching among all realized edges. This problem has received a
considerable attention during the past decade due to its numerous real-world
applications in kidney-exchange, matchmaking services, online labor markets,
and advertisements.
Our main result is an {\em adaptive} algorithm that for any arbitrarily small
, finds a -approximation in expectation, by
querying only edges per vertex. We further show that our approach leads
to a -approximate {\em non-adaptive} algorithm that also
queries only edges per vertex. Prior to our work, no nontrivial
approximation was known for weighted graphs using a constant per-vertex budget.
The state-of-the-art adaptive (resp. non-adaptive) algorithm of Maehara and
Yamaguchi [SODA 2018] achieves a -approximation (resp.
-approximation) by querying up to edges per
vertex where denotes the maximum integer edge-weight. Our result is a
substantial improvement over this bound and has an appealing message: No matter
what the structure of the input graph is, one can get arbitrarily close to the
optimum solution by querying only a constant number of edges per vertex.
To obtain our results, we introduce novel properties of a generalization of
{\em augmenting paths} to weighted matchings that may be of independent
interest
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
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