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Book ChapterDOI

Approximation Algorithms for the Weight-Reducible Knapsack Problem

TL;DR: This work considers the weight-reducible knapsack problem, where one is given a limited budget that can be used to decrease item weights, and would like to optimize theknapsack objective value using such weight improvements.
Abstract: We consider the weight-reducible knapsack problem, where we are given a limited budget that can be used to decrease item weights, and we would like to optimize the knapsack objective value using such weight improvements.
Citations
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Journal ArticleDOI
TL;DR: The notion of query competitiveness for strict robustness is introduced to evaluate the quality of an algorithm for this problem, and lower and upper bounds on this competitiveness for interval-based uncertainty are obtained.

38 citations

Book
01 Jan 2014
TL;DR: This paper considers two types of decomposition approaches: separate recovery and combined recovery, and investigates their application to recoverable robust optimization problems by the technique of column generation.
Abstract: Real-life planning problems are often complicated by the occurrence of disturbances, which imply that the original plan cannot be followed anymore and some recovery action must be taken to cope with the disturbance. In such a situation it is worthwhile to arm yourself against possible disturbances by including recourse actions in your planning strategy. Well-known approaches to create plans that take possible, common disturbances into account are robust optimization and stochastic programming. More recently, another approach has been developed that combines the best of these two: recoverable robustness. In this paper, we solve recoverable robust optimization problems by the technique of column generation. We consider two types of decomposition approaches: separate recovery and combined recovery. We investigate our approach for two example problems: the size robust knapsack problem, in which the knapsack size may get reduced, and the demand robust shortest path problem, in which the sink is uncertain and the cost of edges may increase. For each problem, we present elaborate computational experiments. We think that our approach is very promising and can be generalized to many other problems.

10 citations


Cites background from "Approximation Algorithms for the We..."

  • ...Disser et al [14] consider policies for packing a knapsack with unknown capacity and Goerigk et al [15] consider the knapsack problem in which there is a limited budget to decrease item weights....

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Journal ArticleDOI
TL;DR: In this article, a branch-and-price-based approach is proposed to solve recoverable robust optimization problems, where the LP relaxation combined recovery dominates separate recovery with respect to the value of LP-relaxation.

9 citations

Posted Content
TL;DR: The result in this paper improves the 0/1 cost upgrade model of budget constrained network upgradeable problems and presents a randomized algorithm for the problem of weight upgradeable budget constrained maximum spanning tree on a general graph.
Abstract: We study budget constrained network upgradeable problems. We are given an undirected edge weighted graph $G=(V,E)$ where the weight an edge $e \in E$ can be upgraded for a cost $c(e)$. Given a budget $B$ for improvement, the goal is to find a subset of edges to be upgraded so that the resulting network is optimum for $B$. The results obtained in this paper include the following. Maximum Weight Constrained Spanning Tree We present a randomized algorithm for the problem of weight upgradeable budget constrained maximum spanning tree on a general graph. This returns a spanning tree $\mathcal{T}^{'}$ which is feasible within the budget $B$, such that $\Pr [ l(\mathcal{T}^{'}) \geq (1-\epsilon)\text{OPT}\text{ , } c(\mathcal{T}^{'} ) \leq B] \ge 1-\frac{1}{n}$ (where $l$ and $c$ denote the length and cost of the tree respectively), for any fixed $\epsilon >0$, in time polynomial in $|V|=n$, $|E|=m$. Our results extend to the minimization version also. Previously Krumke et. al. \cite{krumke} presented a$(1+\frac{1}{\gamma}, 1+ \gamma)$ bicriteria approximation algorithm for any fixed $\gamma >0$ for this problem in general graphs for a more general cost upgrade function. The result in this paper improves their 0/1 cost upgrade model. Longest Path in a DAG We consider the problem of weight improvable longest path in a $n$ vertex DAG and give a $O(n^3)$ algorithm for the problem when there is a bound on the number of improvements allowed. We also give a $(1-\epsilon)$-approximation which runs in $O(\frac{n^4}{\epsilon})$ time for the budget constrained version. Similar results can be achieved also for the problem of shortest paths in a DAG.

1 citations

Posted Content
TL;DR: A detailed analysis for several cases of improvable knapsack problems is presented, presenting constant factor approximation algorithms and two PTAS.
Abstract: We consider a variant of the knapsack problem, where items are available with different possible weights. Using a separate budget for these item improvements, the question is: Which items should be improved to which degree such that the resulting classic knapsack problem yields maximum profit? We present a detailed analysis for several cases of improvable knapsack problems, presenting constant factor approximation algorithms and two PTAS.

Cites methods from "Approximation Algorithms for the We..."

  • ...alysis for several cases of improvable knapsack problems, presenting constant factor approximation algorithms and two PTAS. Parts of this paper have been published in the extended conference abstract [GSSS14]. 1 Introduction We consider an extension of the knapsack problem which allows to use different versions of the same item, where the weight of an item can be reduced. Each such improvement has associat...

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References
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Book
02 Jul 2001
TL;DR: Covering the basic techniques used in the latest research work, the author consolidates progress made so far, including some very recent and promising results, and conveys the beauty and excitement of work in the field.
Abstract: Covering the basic techniques used in the latest research work, the author consolidates progress made so far, including some very recent and promising results, and conveys the beauty and excitement of work in the field. He gives clear, lucid explanations of key results and ideas, with intuitive proofs, and provides critical examples and numerous illustrations to help elucidate the algorithms. Many of the results presented have been simplified and new insights provided. Of interest to theoretical computer scientists, operations researchers, and discrete mathematicians.

4,290 citations


"Approximation Algorithms for the We..." refers result in this paper

  • ...which is similar to the classic FPTAS for Knapsack [6]....

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Journal ArticleDOI
TL;DR: An algorithm is presented which finds for any 0 < e < 1 an approximate solution P satisfying (P* P)/P* < ~, where P* is the desired optimal sum.
Abstract: Given a positive integer M and n pairs of positive integers (p~, cD, , (p. , c.), maximize the s u m ~ ~p~ subject to the cons t ramts~ ~c, < M and ~, = 0 or 1 This is the well-known 0/1 knapsack problem An algorithm is presented which finds for any 0 < e < 1 an approximate solution P satisfying (P* P)/P* < ~, where P* is the desired optimal sum Moreover, for any fixed e, the algorithm has time complexity 0(n log n) and space complexity O(n) Modification of the algorithm for the unbounded knapsack problem where the ~,'s can be any nonnegative integer results in a O(n) computing time A hnear-time algorithm is also obtained for a special class of 0/1 knapsack problems having the property that p,/c, is the same for all 1 < z < n

999 citations


Additional excerpts

  • ..., [5] for the knapsack problem....

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Proceedings ArticleDOI
23 Oct 2010
TL;DR: A new {\em swap rounding} technique which can be applied in a variety of settings including matroids and matroid intersection, while providing Chernoff-type concentration bounds for linear and sub modular functions of the rounded solution is described.
Abstract: We consider the problem of randomly rounding a fractional solution $x$ in an integer polytope $P \subseteq [0,1]^n$ to a vertex $X$ of $P$, so that $\E[X] = x$. Our goal is to achieve {\em concentration properties} for linear and sub modular functions of the rounded solution. Such dependent rounding techniques, with concentration bounds for linear functions, have been developed in the past for two polytopes: the assignment polytope (that is, bipartite matchings and $b$-matchings)~\cite{S01, GKPS06, KMPS09}, and more recently for the spanning tree polytope~\cite{AGMGS10}. These schemes have led to a number of new algorithmic results. In this paper we describe a new {\em swap rounding} technique which can be applied in a variety of settings including {\em matroids} and {\em matroid intersection}, while providing Chernoff-type concentration bounds for linear and sub modular functions of the rounded solution. In addition to existing techniques based on negative correlation, we use a martingale argument to obtain an exponential tail estimate for monotone sub modular functions. The rounding scheme explicitly exploits {\em exchange properties} of the underlying combinatorial structures, and highlights these properties as the basis for concentration bounds. Matroids and matroid intersection provide a unifying framework for several known applications~\cite{GKPS06, KMPS09, CCPV09, KST09, AGMGS10} as well as new ones, and their generality allows a richer set of constraints to be incorporated easily. We give some illustrative examples, with a more comprehensive discussion deferred to a later version of the paper.

272 citations


"Approximation Algorithms for the We..." refers background or methods in this paper

  • ...For a constant number of knapsack constraints, as is the case here, one can apply the randomized rounding methods proposed in [4] to obtain a randomized e/(e− 1)-approximation algorithm....

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  • ...It may be noted that, no explicit analysis is presented in [4] except for a claim that the method takes polynomial time....

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Journal ArticleDOI
TL;DR: The main ideas contained in the PTAS are used to derivePTAS's for the knapsack problem and its multi-dimensional generalization which improve on the previously proposed PTAS's.

193 citations


"Approximation Algorithms for the We..." refers background or methods in this paper

  • ...Considering the special case of only one improvement per item and constant improvement costs, we further presented a linear-time 3-approximation based on the algorithm of [7] for the related k-CKP, and strengthened the analysis of the previous 3-approximation for the general problem, to be a 2-approximation for the special case....

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  • ...[7] and [8] present algorithms for this kind of problem....

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  • ...We first note that a linear-time 4-approximation algorithm can be easiliy achieved: Following [7], there is a 2-approximation algorithm for CKP, with linear runtime due to [9]....

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Journal ArticleDOI
TL;DR: Pseudopolynomial algorithms for two special graph classes, namely graphs of bounded treewidth (including trees and series-parallel graphs) and chordal graphs are presented and from these algorithms the authors can easily derive fully polynomial time approximation schemes (FPTAS).
Abstract: We extend the classical 0-1 knapsack problem by introducing disjunctive constraints for pairs of items which are not allowed to be packed together into the knapsack. These constraints are represented by edges of a conict graph whose vertices correspond to the items of the knapsack problem. Similar conditions were treated in the literature for bin packing and scheduling problems. For the knapsack problem with conict graphs, exact and heuristic algorithms were proposed in the past. While the problem is strongly NP-hard in general, we present pseudopolynomial algorithms for two special graph classes, namely graphs of bounded treewidth (including trees and series-parallel graphs) and chordal graphs. From these algorithms we can easily derive fully polynomial time approximation schemes (FPTAS).

122 citations


"Approximation Algorithms for the We..." refers background in this paper

  • ...Note that we may consider this problem as a special case of a multi-dimensional knapsack problem with conflict constraints (for the one-dimensional case, we refer to [2,3])....

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