Online variable-sized bin packing

AbstractThe classical bin packing problem is one of the best-known and most widely studied problems of combinatorial optimization. Efficient offline approximation algorithms have recently been designed and analyzed for the more general and realistic model in which bins of differing capacities are allowed (Friesen and Langston (1986)). In this paper, we consider fast online algorithms for this challenging model. Selecting either the smallest or the largest available bin size to begin a new bin as pieces arrive turns out to yield a tight worst-case ratio of 2. We devise a slightly more complicated scheme that uses the largest available bin size for small pieces, and selects bin sizes for large pieces based on a user-specified fill factor ƒ≥12, and prove that this strategy guarantees a worst-case bound not exceeding 1.5+ƒ2

On variable sized vector packing

One of the open problems in on-line packing is the gap between the lower bound Ω(l) and the upper bound O(d) for vector packing of d-dimensional items into d-dimensional bins. We address a more general packing problem with variable sized bins. In this problem, the set of allowed bins contains the traditional "all-1" vector, but also a finite number of other d-dimensional vectors. The study of this problem can be seen as a first step towards solving the classical problem. It is not hard to see that a simple greedy algorithm achieves competitive ratio O(d) for every set of bins. We show that for all small ε > 0 there exists a set of bins for which the competitive ratio is 1 + ε. On the other hand we show that there exists a set of bins for which every deterministic or randomized algorithm has competitive ratio Ω(d). We also study one special case for d = 2

Use of genetic algorithm as a solution of unidimensional cutting rationalization problem

The following research is dedicated to studying an example of a solution of unidimension cutting rationalization problem. The possibility to use the elements of GA (genetic algorithm) theory at its determination is shown. The specific of the following problem is that in the process of solution search the number of frontages and cutting details is steady, but their order is variable. The modifications of the classical genetic algorithm allowed us to develop a new algorithm of solution search. The results of the solution were used to create a DLL-library, which is easily integrated in different kinds of application software

Bounded space on-line variable-sized bin packing

In this paper we consider the fc-bounded space on-line bin packing problem. Some efficient approximation algorithms are described and analyzed. Selecting either the smallest or the largest available bin size to start a new bin as items arrive turns out to yield a worst-case performance bound of 2. By packing large items into appropriate bins, an efficient approximation algorithm is derived from fc-bounded space on-line bin packing algorithms and its worst-case performance bounds is 1.7 for k > 3

Parameterized Complexity of Conflict-Free Matchings and Paths

An input to a conflict-free variant of a classical problem Gamma, called Conflict-Free Gamma, consists of an instance I of Gamma coupled with a graph H, called the conflict graph. A solution to Conflict-Free Gamma in (I,H) is a solution to I in Gamma, which is also an independent set in H. In this paper, we study conflict-free variants of Maximum Matching and Shortest Path, which we call Conflict-Free Matching (CF-Matching) and Conflict-Free Shortest Path (CF-SP), respectively. We show that both CF-Matching and CF-SP are W[1]-hard, when parameterized by the solution size. Moreover, W[1]-hardness for CF-Matching holds even when the input graph where we want to find a matching is itself a matching, and W[1]-hardness for CF-SP holds for conflict graph being a unit-interval graph. Next, we study these problems with restriction on the conflict graphs. We give FPT algorithms for CF-Matching when the conflict graph is chordal. Also, we give FPT algorithms for both CF-Matching and CF-SP, when the conflict graph is d-degenerate. Finally, we design FPT algorithms for variants of CF-Matching and CF-SP, where the conflicting conditions are given by a (representable) matroid

Shortest Path with Positive Disjunctive Constraints -- a Parameterized Perspective

We study the SHORTEST PATH problem with positive disjunctive constraints from the perspective of parameterized complexity. For positive disjunctive constraints, there are certain pair of edges such that any feasible solution must contain at least one edge from every such pair. In this paper, we initiate the study of SHORTEST PATH problem subject to some positive disjunctive constraints the classical version is known to be NP-Complete. Formally, given an undirected graph G = (V, E) with a forcing graph H = (E, F) such that the vertex set of H is same as the edge set of G. The goal is to find a set S of at most k edges from G such that S forms a vertex cover in H and there is a path from s to t in the subgraph of G induced by the edge set S. In this paper, we consider two natural parameterizations for this problem. One natural parameter is the solution size, i.e. k for which we provide a kernel with O(k^5) vertices when both G and H are general graphs. Additionally, when either G or H (but not both) belongs to some special graph classes, we provied kernelization results with O(k^3) vertices . The other natural parameter we consider is structural properties of H, i.e. the size of a vertex deletion set of H to some special graph classes. We provide some fixed-parameter tractability results for those structural parameterizations.Comment: 14 page

Cloud storage and online bin packing

Cloud storage is the service provided by some corporations (such as Mozy and Carbonite) to store and backup computer files. We study the problem of allocating memory of servers in a data center based on online requests for storage. Over-the-net data backup has become increasingly easy and cheap due to cloud storage. Given an online sequence of storage requests and a cost associated with serving the request by allocating space on a certain server one seeks to select the minimum number of servers as to minimize total cost. We use two different algorithms and propose a third algorithm; we show that all algorithms perform well when the requests are random. The work here is related to bin packing , a well studied problem in theoretical computer science. As an aside the thesis will survey some of the literature related to bin packing

The Generalized Bin Packing Problem with bin-dependent item profits

In this paper, we introduce the Generalized Bin Packing Problem with bin-dependent item profits (GBPPI), a variant of the Generalized Bin Packing Problem. In GBPPI, various bin types are available with their own capacities and costs. A set of compulsory and non-compulsory items are also given, with volume and bin-dependent profits. The aim of GBPPI is to determine an assignment of items to bins such that the overall net cost is minimized. The importance of GBPPI is confirmed by a number of applications. The introduction of bin-dependent item profits enables the application of GBPPI to cross-country and multi-modal transportation problems at strategic and tactical levels as well as in last-mile logistic environments. Having provided a Mixed Integer Programming formulation of the problem, we introduce efficient heuristics that can effectively address GBPPI for instances involving up to 1000 items and problems with a mixed objective function. Extensive computational tests demonstrate the accuracy of the proposed heuristics. Finally, we present a case study of a well-known international courier operating in northern Italy. The problem approached by the international courier is GBPPI. In this case study, our methodology outperforms the policies of the company