Worst-Case Analysis of Heuristics for the Bin Packing Problem with General Cost Structures

Abstract

We consider the famous bin packing problem where a set of items must be stored in bins of equal capacity. In the classical version, the objective is to minimize the number of bins used. Motivated by several optimization problems that occur in the context of the storage of items, we study a more general cost structure where the cost of a bin is a concave function of the number of items in the bin. The objective is to store the items in such a way that total cost is minimized. Such cost functions can greatly alter the way the items should be assigned to the bins. We show that some of the best heuristics developed for the classical bin packing problem can perform poorly under the general cost structure. On the other hand, the so-called next-fit increasing heuristic has an absolute worst-case bound of no more than 1.75 and an asymptotic worst-case bound of 1.691 for any concave and monotone cost function. Our analysis also provides a new worst-case bound for the well studied next-tit decreasing heuristic when the objective is to minimize the number of bins used.

Scheduling with tool changes to minimize total completion time: A study of heuristics and their performance

“…Minimizing Z 2 entails the minimization of the total weighted cardinality over all the blocks; this can be viewed as an extension of the bin-packing problem. The worst case performance ratio of the SPT rule for the bin-packing problem is 1.7, compared to 1.22 of the First Fit Decreasing (FFD) rule [4] and [8]. We conjecture that as T C increases, the SPT schedule becomes progressively worse in the case of our problem.…”

Scheduling with tool changes to minimize total completion time: Basic results and SPT performance

“…If mðBIN jþ1 Þ is the number of tools used for jobs 1 through j in an optimal bin-packing solution, it is known from Anily et al [6] that mðSPTÞ=mðBINÞ 6 1:75.…”

Parallel Execution Optimization of GPU-aware Components in Embedded Systems

“…The optimization challenge is an NP-hard combinatorial problem [3]. Therefore, heuristic techniques need to be employed in order to find solutions.…”

“…This type of problem, where bin sizes are changed based on their items and their connection, is NP-hard [3], i.e., an exact solution cannot be calculated in feasible time (unless P = N P ).…”

“…Furthermore, the method provides optimized solutions considering component extra-functional properties (i.e., execution time). Resembling with the bin-packing problem and the multiprocessor scheduling problem which are combinatorial NP-hard problems [3], heuristics need to be utilized in finding solutions. For the implementation part, we use a mixed-integer nonlinear programming (MINLP) heuristic approach to compute execution schemes (Section VII).…”