51 research outputs found
Best Fit Bin Packing with Random Order Revisited
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.Comment: Full version of MFCS 2020 pape
Optimal Online Edge Coloring of Planar Graphs with Advice
Using the framework of advice complexity, we study the amount of knowledge
about the future that an online algorithm needs to color the edges of a graph
optimally, i.e., using as few colors as possible. For graphs of maximum degree
, it follows from Vizing's Theorem that bits of
advice suffice to achieve optimality, where is the number of edges. We show
that for graphs of bounded degeneracy (a class of graphs including e.g. trees
and planar graphs), only bits of advice are needed to compute an optimal
solution online, independently of how large is. On the other hand, we
show that bits of advice are necessary just to achieve a
competitive ratio better than that of the best deterministic online algorithm
without advice. Furthermore, we consider algorithms which use a fixed number of
advice bits per edge (our algorithm for graphs of bounded degeneracy belongs to
this class of algorithms). We show that for bipartite graphs, any such
algorithm must use at least bits of advice to achieve
optimality.Comment: CIAC 201
Online Bin Covering: Expectations vs. Guarantees
Bin covering is a dual version of classic bin packing. Thus, the goal is to
cover as many bins as possible, where covering a bin means packing items of
total size at least one in the bin.
For online bin covering, competitive analysis fails to distinguish between
most algorithms of interest; all "reasonable" algorithms have a competitive
ratio of 1/2. Thus, in order to get a better understanding of the combinatorial
difficulties in solving this problem, we turn to other performance measures,
namely relative worst order, random order, and max/max analysis, as well as
analyzing input with restricted or uniformly distributed item sizes. In this
way, our study also supplements the ongoing systematic studies of the relative
strengths of various performance measures.
Two classic algorithms for online bin packing that have natural dual versions
are Harmonic and Next-Fit. Even though the algorithms are quite different in
nature, the dual versions are not separated by competitive analysis. We make
the case that when guarantees are needed, even under restricted input
sequences, dual Harmonic is preferable. In addition, we establish quite robust
theoretical results showing that if items come from a uniform distribution or
even if just the ordering of items is uniformly random, then dual Next-Fit is
the right choice.Comment: IMADA-preprint-c
Improved Online Algorithms for Knapsack and GAP in the Random Order Model
The knapsack problem is one of the classical problems in combinatorial optimization: Given a set of items, each specified by its size and profit, the goal is to find a maximum profit packing into a knapsack of bounded capacity. In the online setting, items are revealed one by one and the decision, if the current item is packed or discarded forever, must be done immediately and irrevocably upon arrival. We study the online variant in the random order model where the input sequence is a uniform random permutation of the item set.
We develop a randomized (1/6.65)-competitive algorithm for this problem, outperforming the current best algorithm of competitive ratio 1/8.06 [Kesselheim et al. SIAM J. Comp. 47(5)]. Our algorithm is based on two new insights: We introduce a novel algorithmic approach that employs two given algorithms, optimized for restricted item classes, sequentially on the input sequence. In addition, we study and exploit the relationship of the knapsack problem to the 2-secretary problem.
The generalized assignment problem (GAP) includes, besides the knapsack problem, several important problems related to scheduling and matching. We show that in the same online setting, applying the proposed sequential approach yields a (1/6.99)-competitive randomized algorithm for GAP. Again, our proposed algorithm outperforms the current best result of competitive ratio 1/8.06 [Kesselheim et al. SIAM J. Comp. 47(5)]
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