We consider the two-dimensional bin packing and strip packing problem, where a list of rectangles has to be packed into a minimal number of rectangular bins or a strip of minimal height, respectively. All packings have to be non-overlapping and orthogonal, i.e., axisparallel. Our algorithm for strip packing has an absolute approximation ratio of 1.9396 and is the first algorithm to break the ap...

In textile industry, marker planning is one of the main operations in the cutting fabric stage. Marker packing is usually used to maximize cloth exploitation and minimize its waste. In this research, a method is used based on new found meta-heuristic imperialist competitive algorithm (ICA) and Bottom-Left-Fill Algorithm (BLF) to achieve optimal marker packing. Function of the proposed method wa...

The Earliest Deadline First (EDF) scheduling algorithm has the least runtime complexity among joblevel fixed-priority algorithms for scheduling tasks on multiprocessor architectures. However, EDF suffers from suboptimality in multiprocessor systems. This paper proposes a new restricted migration-based scheduling algorithm for multiprocessor real-time systems, called Two-level Hierarchical Sched...

We present a new lemma stating that, given an arbitrary packing of a set of rectangles into a larger rectangle, a “structured” packing of nearly the same set of rectangles exists. This lemma has several implications on the approximability of 2-dimensional packing problems. In this paper, we use it to show the existence of a polynomial-time approximation scheme for 2-dimensional geometric knapsa...

In previous chapters we have seen the definition of a constant factor approximation algorithm. In this chapter, we will introduce the notion of a polynomial time approximation scheme (PTAS), which allows approximability to any required degree. To illustrate how PTAS works, we will study two examples, including the knapsack problem and the bin packing problem. The dynamic programming technique w...

Following the work of Anily et al., we consider a variant of bin packing, called BIN PACKING WITH GENERAL COST STRUCTURES (GCBP) and design an asymptotic fully polynomial time approximation scheme (AFPTAS) for this problem. In the classic bin packing problem, a set of one-dimensional items is to be assigned to subsets of total size at most 1, that is, to be packed into unit sized bins. However,...

Given a simple graph G = (V,E), a subset of E is called a triangle cover if it intersects each triangle of G. Let νt(G) and τt(G) denote the maximum number of pairwise edge-disjoint triangles in G and the minimum cardinality of a triangle cover of G, respectively. Tuza conjectured in 1981 that τt(G)/νt(G) ≤ 2 holds for every graph G. In this paper, using a hypergraph approach, we design polynom...

We investigate the problem of packing and covering odd (u, v)-trails in a graph. A (u, v)-trail is a (u, v)-walk that is allowed to have repeated vertices but no repeated edges. We call a trail odd if the number of edges in the trail is odd. Let ν(u, v) denote the maximum number of edge-disjoint odd (u, v)-trails, and τ(u, v) denote the minimum size of an edge-set that intersects every odd (u, ...

A new lower bound for the average probability or error for a two-user discrete memoryless (DM) multiple-access channel (MAC) is derived. This bound has a structure very similar to the well-known sphere packing packing bound derived by Haroutunian. However, since explicitly imposes independence of the users’ input distributions (conditioned on the time-sharing auxiliary variable) results in a ti...

The bin packing problem is one of the classical NP-hard optimization problems. In this paper, we present a simple generic approach for obtaining new fast lower bounds, based on dual feasible functions. Worst-case analysis as well as computational results show that one of our classes clearly outperforms the previous best “economical” lower bound for the bin packing problem by Martello and Toth, ...