. We consider variants of the classic bin packing and multiple knapsack problems, in which sets of items of different classes (colors) need to be placed in bins; the items may have different sizes and values. Each bin has a limited capacity, and a bound on the number of distinct classes of items it can hold. In the class-constrained multiple knapsack (CCMK) problem, our goal is to maximize the total value of packed items, whereas in the class-constrained bin-packing (CCBP), we seek to minimize the number of (identical) bins, needed for packing all the items. We give a polynomial time approximation scheme (PTAS) for CCMK and a dual PTAS for CCBP. We also show that the 0-1 class-constrained knapsack admits a fully polynomial time approximation scheme, even when the number of distinct colors of items depends on the input size. Finally, we introduce the generalized class-constrained packing problem (GCCP), where each item may have more than one color. We show that GCCP is APX...

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The bin packing problem is one of the classical NP-hard optimization problems. Even though there are many excellent theoretical results, including polynomial approximation schemes, there is still a lack of methods that are able to solve practical instances optimally. In this paper, we present a fast and simple generic approach for obtaining new lower bounds, based on dual feasible functions. Worst case analysis as well as computational results show that one of our classes clearly outperforms the currently best known "economical" lower bound for the bin packing problem by Martello and Toth, which can be understood as a special case. This indicates the usefulness of our results in a branch and bound framework.

The Multiple Subset Sum Problem (MSSP) is the selection of items from a given ground set and their packing into a given number of identical bins such that the sum of the item weights in every bin does not exceed the bin capacity and the total sum of the weights of the items packed is as large as possible. This problem is a relevant special case of the multiple knapsack problem, for which the existence of a Polynomial-Time Approximation Scheme (PTAS) is an important open question in the eld of knapsack problems. One main result of the present paper is the construction of a PTAS for MSSP. For the bottleneck case of the problem, where the minimum total weight contained in any bin is to be maximized, we describe a 2=3-approximation algorithm and show that this is the best possible approximation ratio. Moreover, PTASs are derived for the special cases in which either the number of bins or the number of dierent item weights is constant. We nally show that, even for the case of only two bins, no fully polynomial-time approximation scheme exists for both versions of the problem. Key words. multiple subset sum problem, approximation scheme, knapsack problem AMS Subject Classication. 90C27, 90C10, 90C59 1

this paper was presented at the twentieth Allerton onference on ommunication, ontrol and omputing held in onticello, llinois, in ctober, . epartment of omputer cience, e as A niversit , ollege tation, . epartment of omputer cience, niversit of ennessee, no ville, -- and epartment of omputer cience, ashington tate niversit , Pullman, A -- . his author's research has been supported in part b the ational cience oundation under grants P-- and P-- , and b the ce of aval esearch under contract -- -- -- . 2 1. Introduction

This paper deals with vector covering problems in d-dimensional space. The input to a vector covering problem consists of a set X of d-dimensional vectors in [0, 1] d. The goal is to partition X into a maximum number of parts, subject to the constraint that in every part the sum of all vectors is at least one in every coordinate. This problem is known to be NP-complete, and we are mainly interested in its on-line and off-line approximability. For the on-line version, we construct approximation algorithms with worst case guarantee arbitrarily close to 1/(2d) in d ≥ 2 dimensions. This result contradicts a statement of Csirik and Frenk (1990) in [5] where it is claimed that for d ≥ 2, no on-line algorithm can have a worst case ratio better than zero. Moreover, we prove that for d ≥ 2, no on-line algorithm can have worst case ratio better than 2/(2d + 1). For the off-line version, we derive polynomial time approximation algorithms with worst case guarantee Θ(1 / log d). For d = 2, we present a very fast and very simple off-line approximation algorithm that has worst case ratio 1/2. Moreover, we show that a method from the area of compact vector summation can be used to construct off-line approximation algorithms with worst case ratio 1/d for every d ≥ 2.

This paper is a chapter of the forthcoming Handbook of Combinatorics, to be published by North-Holland. It surveys the basic techniques and methods in combinatorial optimization. We organize our material according to the fundamental algorithmic techniques and illustrate them on problems to which these methods have been applied successfully. Special attention is given to approximation algorithms and fast (primal and dual) heuristics.

this paper, not to mention the bin packing literature, is confined to the one dimensional problem with equal bin capacities, which we normalize to 1 for convenience; in this case, any set of items whose total size is at most 1 fits into a bin. Correspondingly, items are always assumed to be numbers drawn from an interval [0; ff] for some ff 1. We keep with these classical assumptions throughout the next section; the third and last section will cover more general versions of bin packing. We note once and for all that the versions of bin packing of interest to us here are NP-hard, except where stated otherwise.

this paper. The general form is not needed, therefore only a special case is given here. In general the A--cycle problem is to decide whether there exists a cycle containing all elements of A, where A is a collection of vertices and edges from the graph. But let us restrict ourselves to the case when A ` V (G). For sake of simplicity we will suppose that the elements of A are pairwise non-adjacent. Otherwise a 2-degree vertex is put into the middle of the edge connecting the two vertices. (This new vertex is not in A in this case.) We say that a pair (X; Y ), where X ` V (G \Gamma A) and Y ` E(G \Gamma A \Gamma X), disconnects A,

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