5,788 research outputs found
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
Applications of sparse approximation in communications
Sparse approximation problems abound in many scientific, mathematical, and engineering applications. These problems are defined by two competing notions: we approximate a signal vector as a linear combination of elementary atoms and we require that the approximation be both as accurate and as concise as possible. We introduce two natural and direct applications of these problems and algorithmic solutions in communications. We do so by constructing enhanced codebooks from base codebooks. We show that we can decode these enhanced codebooks in the presence of Gaussian noise. For MIMO wireless communication channels, we construct simultaneous sparse approximation problems and demonstrate that our algorithms can both decode the transmitted signals and estimate the channel parameters
On k-Column Sparse Packing Programs
We consider the class of packing integer programs (PIPs) that are column
sparse, i.e. there is a specified upper bound k on the number of constraints
that each variable appears in. We give an (ek+o(k))-approximation algorithm for
k-column sparse PIPs, improving on recent results of and
. We also show that the integrality gap of our linear programming
relaxation is at least 2k-1; it is known that k-column sparse PIPs are
-hard to approximate. We also extend our result (at the loss
of a small constant factor) to the more general case of maximizing a submodular
objective over k-column sparse packing constraints.Comment: 19 pages, v3: additional detail
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
Partial resampling to approximate covering integer programs
We consider column-sparse covering integer programs, a generalization of set
cover, which have a long line of research of (randomized) approximation
algorithms. We develop a new rounding scheme based on the Partial Resampling
variant of the Lov\'{a}sz Local Lemma developed by Harris & Srinivasan (2019).
This achieves an approximation ratio of , where is the minimum covering
constraint and is the maximum -norm of any column of the
covering matrix (whose entries are scaled to lie in ). When there are
additional constraints on the variable sizes, we show an approximation ratio of
(where is the maximum number
of non-zero entries in any column of the covering matrix). These results
improve asymptotically, in several different ways, over results of Srinivasan
(2006) and Kolliopoulos & Young (2005).
We show nearly-matching inapproximability and integrality-gap lower bounds.
We also show that the rounding process leads to negative correlation among the
variables, which allows us to handle multi-criteria programs
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