Improved lower bounds for the online bin packing problem with cardinality constraints

The bin packing problem has been extensively studied and numerous variants have been considered. The k-item bin packing problem is one of the variants introduced by Krause et al. (J ACM 22:522-550, 1975). In addition to the formulation of the classical bin packing problem, this problem imposes a cardinality constraint that the number of items packed into each bin must be at most k. For the online setting of this problem, in which the items are given one by one, Babel et al. (Discret Appl Math 143: 238-251, 2004) provided lower boundsv root 2 approximate to 1.41421 and 1.5 on the asymptotic competitive ratio for k = 2 and 3, respectively. For k >= 4, some lower bounds (e.g., by van Vliet (Inf Process Lett 43:277-284, 1992) for the online bin packing problem, i.e., a problem without cardinality constraints, can be applied to this problem. In this paper we consider the online k-item bin packing problem. First, we improve the previous lower bound 1.41421 to 1.42764 for k = 2. Moreover, we propose a new method to derive lower bounds for general k and present improved bounds for various cases of k >= 4. For example, we improve 1.33333 to 1.5 for k = 4, and 1.33333 to 1.47058 for k = 5.ArticleJOURNAL OF COMBINATORIAL OPTIMIZATION. 29(1): 67-87 (2015)journal articl

Lower bounds for several online variants of bin packing

We consider several previously studied online variants of bin packing and prove new and improved lower bounds on the asymptotic competitive ratios for them. For that, we use a method of fully adaptive constructions. In particular, we improve the lower bound for the asymptotic competitive ratio of online square packing significantly, raising it from roughly 1.68 to above 1.75.Comment: WAOA 201

Packing Returning Secretaries

We study online secretary problems with returns in combinatorial packing domains with $n$ candidates that arrive sequentially over time in random order. The goal is to accept a feasible packing of candidates of maximum total value. In the first variant, each candidate arrives exactly twice. All $2n$ arrivals occur in random order. We propose a simple 0.5-competitive algorithm that can be combined with arbitrary approximation algorithms for the packing domain, even when the total value of candidates is a subadditive function. For bipartite matching, we obtain an algorithm with competitive ratio at least $0.5721 - o(1)$ for growing $n$, and an algorithm with ratio at least $0.5459$ for all $n \ge 1$. We extend all algorithms and ratios to $k \ge 2$ arrivals per candidate. In the second variant, there is a pool of undecided candidates. In each round, a random candidate from the pool arrives. Upon arrival a candidate can be either decided (accept/reject) or postponed (returned into the pool). We mainly focus on minimizing the expected number of postponements when computing an optimal solution. An expected number of $\Theta(n \log n)$ is always sufficient. For matroids, we show that the expected number can be reduced to $O(r \log (n/r))$, where $r \le n/2$ is the minimum of the ranks of matroid and dual matroid. For bipartite matching, we show a bound of $O(r \log n)$, where $r$ is the size of the optimum matching. For general packing, we show a lower bound of $\Omega(n \log \log n)$, even when the size of the optimum is $r = \Theta(\log n)$.Comment: 23 pages, 5 figure

Online Bin Packing with Cardinality Constraints Resolved

Cardinality constrained bin packing or bin packing with cardinality constraints is a basic bin packing problem. In the online version with the parameter k >= 2, items having sizes in (0,1] associated with them are presented one by one to be packed into unit capacity bins, such that the capacities of bins are not exceeded, and no bin receives more than k items. We resolve the online problem in the sense that we prove a lower bound of 2 on the overall asymptotic competitive ratio. This closes the long standing open problem of finding the value of the best possible overall asymptotic competitive ratio, since an algorithm of an absolute competitive ratio 2 for any fixed value of k is known. Additionally, we significantly improve the known lower bounds on the asymptotic competitive ratio for every specific value of k. The novelty of our constructions is based on full adaptivity that creates large gaps between item sizes. Thus, our lower bound inputs do not follow the common practice for online bin packing problems of having a known in advance input consisting of batches for which the algorithm needs to be competitive on every prefix of the input. Last, we show a lower bound strictly larger than 2 on the asymptotic competitive ratio of the online 2-dimensional vector packing problem, and thus provide for the first time a lower bound larger than 2 on the asymptotic competitive ratio for the vector packing problem in any fixed dimension

Bounds for online bin packing with cardinality constraints

Abstract We study a bin packing problem in which a bin can contain at most k items of total size at most 1, where k ≥ 2 is a given parameter. Items are presented one by one in an online fashion. We analyze the best absolute competitive ratio of the problem and prove tight bounds of 2 for any k ≥ 4 . Additionally, we present bounds for relatively small values of k with respect to the asymptotic competitive ratio and the absolute competitive ratio. In particular, we provide tight bounds on the absolute competitive ratio of First Fit for k = 2 , 3 , 4 , and improve the known lower bounds on asymptotic competitive ratios for multiple values of k. Our method for obtaining a lower bound on the asymptotic competitive ratio using a certain type of an input is general, and we also use it to obtain an alternative proof of the known lower bound on the asymptotic competitive ratio of standard online bin packing