140 research outputs found
Randomized Strategies for Robust Combinatorial Optimization
In this paper, we study the following robust optimization problem. Given an
independence system and candidate objective functions, we choose an independent
set, and then an adversary chooses one objective function, knowing our choice.
Our goal is to find a randomized strategy (i.e., a probability distribution
over the independent sets) that maximizes the expected objective value. To
solve the problem, we propose two types of schemes for designing approximation
algorithms. One scheme is for the case when objective functions are linear. It
first finds an approximately optimal aggregated strategy and then retrieves a
desired solution with little loss of the objective value. The approximation
ratio depends on a relaxation of an independence system polytope. As
applications, we provide approximation algorithms for a knapsack constraint or
a matroid intersection by developing appropriate relaxations and retrievals.
The other scheme is based on the multiplicative weights update method. A key
technique is to introduce a new concept called -reductions for
objective functions with parameters . We show that our scheme
outputs a nearly -approximate solution if there exists an
-approximation algorithm for a subproblem defined by
-reductions. This improves approximation ratio in previous
results. Using our result, we provide approximation algorithms when the
objective functions are submodular or correspond to the cardinality robustness
for the knapsack problem
AFPTAS results for common variants of bin packing: A new method to handle the small items
We consider two well-known natural variants of bin packing, and show that
these packing problems admit asymptotic fully polynomial time approximation
schemes (AFPTAS). In bin packing problems, a set of one-dimensional items of
size at most 1 is to be assigned (packed) to subsets of sum at most 1 (bins).
It has been known for a while that the most basic problem admits an AFPTAS. In
this paper, we develop methods that allow to extend this result to other
variants of bin packing. Specifically, the problems which we study in this
paper, for which we design asymptotic fully polynomial time approximation
schemes, are the following. The first problem is "Bin packing with cardinality
constraints", where a parameter k is given, such that a bin may contain up to k
items. The goal is to minimize the number of bins used. The second problem is
"Bin packing with rejection", where every item has a rejection penalty
associated with it. An item needs to be either packed to a bin or rejected, and
the goal is to minimize the number of used bins plus the total rejection
penalty of unpacked items. This resolves the complexity of two important
variants of the bin packing problem. Our approximation schemes use a novel
method for packing the small items. This new method is the core of the improved
running times of our schemes over the running times of the previous results,
which are only asymptotic polynomial time approximation schemes (APTAS)
A Deterministic Fully Polynomial Time Approximation Scheme For Counting Integer Knapsack Solutions Made Easy
Given n elements with nonnegative integer weights w=(w_1,...,w_n), an integer capacity C and positive integer ranges u=(u_1,...,u_n), we consider the counting version of the classic integer knapsack problem: find the number of distinct multisets whose weights add up to at most C. We give a deterministic algorithm that estimates the number of solutions to within relative error epsilon in time polynomial in n, log U and 1/epsilon, where U=max_i u_i. More precisely, our algorithm runs in O((n^3 log^2 U)/epsilon) log (n log U)/epsilon) time. This is an improvement of n^2 and 1/epsilon (up to log terms) over the best known deterministic algorithm by Gopalan et al. [FOCS, (2011), pp. 817-826]. Our algorithm is relatively simple, and its analysis is rather elementary. Our results are achieved by means of a careful formulation of the problem as a dynamic program, using the notion of binding constraints
Hybrid Rounding Techniques for Knapsack Problems
We address the classical knapsack problem and a variant in which an upper
bound is imposed on the number of items that can be selected. We show that
appropriate combinations of rounding techniques yield novel and powerful ways
of rounding. As an application of these techniques, we present a linear-storage
Polynomial Time Approximation Scheme (PTAS) and a Fully Polynomial Time
Approximation Scheme (FPTAS) that compute an approximate solution, of any fixed
accuracy, in linear time. This linear complexity bound gives a substantial
improvement of the best previously known polynomial bounds.Comment: 19 LaTeX page
An FPTAS for the -modular multidimensional knapsack problem
It is known that there is no EPTAS for the -dimensional knapsack problem
unless . It is true already for the case, when . But, an
FPTAS still can exist for some other particular cases of the problem.
In this note, we show that the -dimensional knapsack problem with a
-modular constraints matrix admits an FPTAS, whose complexity bound
depends on linearly. More precisely, the proposed algorithm complexity
is
where is the linear programming complexity bound. In particular, for
fixed the arithmetical complexity bound becomes Our algorithm is actually a
generalisation of the classical FPTAS for the -dimensional case.
Strictly speaking, the considered problem can be solved by an exact
polynomial-time algorithm, when is fixed and grows as a polynomial
on . This fact can be observed combining previously known results. In this
paper, we give a slightly more accurate analysis to present an exact algorithm
with the complexity bound Note that
the last bound is non-linear by with respect to the given FPTAS
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