13,476 research outputs found
Approximability of Sparse Integer Programs
The main focus of this paper is a pair of new approximation algorithms for certain integer programs. First, for covering integer programs {min cx:Ax≥b,0≤x≤d} where A has at most k nonzeroes per row, we give a k-approximation algorithm. (We assume A,b,c,d are nonnegative.) For any k≥2 and ε>0, if P≠NP this ratio cannot be improved to k−1−ε, and under the unique games conjecture this ratio cannot be improved to k−ε. One key idea is to replace individual constraints by others that have better rounding properties but the same nonnegative integral solutions; another critical ingredient is knapsack-cover inequalities. Second, for packing integer programs {max cx:Ax≤b,0≤x≤d} where A has at most k nonzeroes per column, we give a (2k 2+2)-approximation algorithm. Our approach builds on the iterated LP relaxation framework. In addition, we obtain improved approximations for the second problem when k=2, and for both problems when every A ij is small compared to b i. Finally, we demonstrate a 17/16-inapproximability for covering integer programs with at most two nonzeroes per colum
A new binary floating-point division algorithm and its software implementation on the ST231 processor
This paper deals with the design and implementation of low latency software for binary floating-point division with correct rounding to nearest. The approach we present here targets a VLIW integer processor of the ST200 family, and is based on fast and accurate programs for evaluating some particular bivariate polynomials. We start by giving approximation and evaluation error conditions that are sufficient to ensure correct rounding. Then we describe the heuristics used to generate such evaluation programs, as well as those used to automatically validate their accuracy. Finally, we propose, for the binary32 format, a complete C implementation of the resulting division algorithm. With the ST200 compiler and compared to previous implementations, the speed-up observed with our approach is by a factor of almost 1.8
Approximability of Sparse Integer Programs
The main focus of this paper is a pair of new approximation algorithms for
certain integer programs. First, for covering integer programs {min cx: Ax >=
b, 0 <= x <= d} where A has at most k nonzeroes per row, we give a
k-approximation algorithm. (We assume A, b, c, d are nonnegative.) For any k >=
2 and eps>0, if P != NP this ratio cannot be improved to k-1-eps, and under the
unique games conjecture this ratio cannot be improved to k-eps. One key idea is
to replace individual constraints by others that have better rounding
properties but the same nonnegative integral solutions; another critical
ingredient is knapsack-cover inequalities. Second, for packing integer programs
{max cx: Ax <= b, 0 <= x <= d} where A has at most k nonzeroes per column, we
give a (2k^2+2)-approximation algorithm. Our approach builds on the iterated LP
relaxation framework. In addition, we obtain improved approximations for the
second problem when k=2, and for both problems when every A_{ij} is small
compared to b_i. Finally, we demonstrate a 17/16-inapproximability for covering
integer programs with at most two nonzeroes per column.Comment: Version submitted to Algorithmica special issue on ESA 2009. Previous
conference version: http://dx.doi.org/10.1007/978-3-642-04128-0_
Approximation Algorithms for Covering/Packing Integer Programs
Given matrices A and B and vectors a, b, c and d, all with non-negative
entries, we consider the problem of computing min {c.x: x in Z^n_+, Ax > a, Bx
< b, x < d}. We give a bicriteria-approximation algorithm that, given epsilon
in (0, 1], finds a solution of cost O(ln(m)/epsilon^2) times optimal, meeting
the covering constraints (Ax > a) and multiplicity constraints (x < d), and
satisfying Bx < (1 + epsilon)b + beta, where beta is the vector of row sums
beta_i = sum_j B_ij. Here m denotes the number of rows of A.
This gives an O(ln m)-approximation algorithm for CIP -- minimum-cost
covering integer programs with multiplicity constraints, i.e., the special case
when there are no packing constraints Bx < b. The previous best approximation
ratio has been O(ln(max_j sum_i A_ij)) since 1982. CIP contains the set cover
problem as a special case, so O(ln m)-approximation is the best possible unless
P=NP.Comment: Preliminary version appeared in IEEE Symposium on Foundations of
Computer Science (2001). To appear in Journal of Computer and System Science
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
The parallel approximability of a subclass of quadratic programming
In this paper we deal with the parallel approximability of a special class of Quadratic Programming (QP), called Smooth Positive Quadratic Programming. This subclass of QP is obtained by imposing restrictions on the coefficients of the QP instance. The Smoothness condition restricts the magnitudes of the coefficients while the positiveness requires that all the coefficients be non-negative. Interestingly, even with these restrictions several combinatorial problems can be modeled by Smooth QP. We show NC Approximation Schemes for the instances of Smooth Positive QP. This is done by reducing the instance of QP to an instance of Positive Linear Programming, finding in NC an approximate fractional solution to the obtained program, and then rounding the fractional solution to an integer approximate solution for the original problem. Then we show how to extend the result for positive instances of bounded degree to Smooth Integer Programming problems. Finally, we formulate several important combinatorial problems as Positive Quadratic Programs (or Positive Integer Programs) in packing/covering form and show that the techniques presented can be used to obtain NC Approximation Schemes for "dense" instances of such problems.Peer ReviewedPostprint (published version
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