Best Fit Bin Packing with Random Order Revisited

Abstract

Best Fit is a well known online algorithm for the bin packing problem, where a collection of one-dimensional items has to be packed into a minimum number of unit-sized bins. In a seminal work, Kenyon [SODA 1996] introduced the (asymptotic) random order ratio as an alternative performance measure for online algorithms. Here, an adversary specifies the items, but the order of arrival is drawn uniformly at random. Kenyon’s result establishes lower and upper bounds of 1.08 and 1.5, respectively, for the random order ratio of Best Fit. Although this type of analysis model became increasingly popular in the field of online algorithms, no progress has been made for the Best Fit algorithm after the result of Kenyon. We study the random order ratio of Best Fit and tighten the long-standing gap by establishing an improved lower bound of 1.10. For the case where all items are larger than 1/3, we show that the random order ratio converges quickly to 1.25. It is the existence of such large items that crucially determines the performance of Best Fit in the general case. Moreover, this case is closely related to the classical maximum-cardinality matching problem in the fully online model. As a side product, we show that Best Fit satisfies a monotonicity property on such instances, unlike in the general case. In addition, we initiate the study of the absolute random order ratio for this problem. In contrast to asymptotic ratios, absolute ratios must hold even for instances that can be packed into a small number of bins. We show that the absolute random order ratio of Best Fit is at least 1.3. For the case where all items are larger than 1/3, we derive upper and lower bounds of 21/16 and 1.2, respectively.

An Adaptive Hybrid Optimization Strategy for Resource Allocation in Network Function Virtualization

“…First, it prefers to select a server that has sufficient available resources corresponding to the application programs required by the SFC. Secondly, if there are multiple servers meeting the requirements under the same optimal conditions, the server with the lowest total available resources is selected based on the Best Fit algorithm [28] as the destination of the SFC. These principles are designed to minimize the number of servers used to handle SFCs for cost-saving and energy consumption reduction.…”

“…The Best Fit algorithm [28] is to find free resources from the entire free space that can satisfy the business requirements and have the smallest size. The best fit algorithm flow is as follows: first, all the free areas are sorted from small to large, and then the order is compared with the size of the required resources, the free area where the first available resource size meets the demand is the final selected free area.…”

Almost Tight Bounds for Online Facility Location in the Random-Order Model

“…For F ⊆ M and v ∈ M, d(F, v) is the minimal distance between a point in F and v, that is, d(F, v) = min u∈F d(u, v). 1 Given a set of open facilities F ⊆ M, the cost of assigning a demand v is d(F, v). The goal is to find a set of facilities F ⊆ M that minimizes the total cost |F| • f + v∈U d(F, v).…”

“…This is also the case for the classical FirstFit and BestFit algorithms for the online bin packing problem. Their performance in the worst-case was tightly analyzed in the 1970s [27] (see also [14,15]), while there are still gaps remaining in the random-order model [32,1]. In contrast, for DistProb, we overcome this challenge and give a tight analysis in the random order model.…”

Improved Approximation Algorithms for Weighted Edge Coloring of Graphs

“…When all the graph is a bipartite graph with only two vertices then the problem reduces to classical bin packing problem, which is well-studied in both offline [10,16,14] and online setting [3,1,2]. There are many other related generalizations of bin packing [5,4,12].…”