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)

Simple PTAS's for families of graphs excluding a minor

We show that very simple algorithms based on local search are polynomial-time approximation schemes for Maximum Independent Set, Minimum Vertex Cover and Minimum Dominating Set, when the input graphs have a fixed forbidden minor.Comment: To appear in Discrete Applied Mathematic

Fully Polynomial Time Approximation Schemes for Stochastic Dynamic Programs

We present a framework for obtaining fully polynomial time approximation schemes (FPTASs) for stochastic univariate dynamic programs with either convex or monotone single-period cost functions. This framework is developed through the establishment of two sets of computational rules, namely, the calculus of K-approximation functions and the calculus of K-approximation sets. Using our framework, we provide the first FPTASs for several NP-hard problems in various fields of research such as knapsack models, logistics, operations management, economics, and mathematical finance. Extensions of our framework via the use of the newly established computational rules are also discussed

Polynomial-Time Approximation Schemes for Independent Packing Problems on Fractionally Tree-Independence-Number-Fragile Graphs

We investigate a relaxation of the notion of treewidth-fragility, namely tree-independence-number-fragility. In particular, we obtain polynomial-time approximation schemes for independent packing problems on fractionally tree-independence-number-fragile graph classes. Our approach unifies and extends several known polynomial-time approximation schemes on seemingly unrelated graph classes, such as classes of intersection graphs of fat objects in a fixed dimension or proper minor-closed classes. We also study the related notion of layered tree-independence number, a relaxation of layered treewidth