14,363 research outputs found
First-Fit is Linear on Posets Excluding Two Long Incomparable Chains
A poset is (r + s)-free if it does not contain two incomparable chains of
size r and s, respectively. We prove that when r and s are at least 2, the
First-Fit algorithm partitions every (r + s)-free poset P into at most
8(r-1)(s-1)w chains, where w is the width of P. This solves an open problem of
Bosek, Krawczyk, and Szczypka (SIAM J. Discrete Math., 23(4):1992--1999, 2010).Comment: v3: fixed some typo
First-Fit coloring of Cartesian product graphs and its defining sets
Let the vertices of a Cartesian product graph be ordered by an
ordering . By the First-Fit coloring of we mean the
vertex coloring procedure which scans the vertices according to the ordering
and for each vertex assigns the smallest available color. Let
be the number of colors used in this coloring. By
introducing the concept of descent we obtain a sufficient condition to
determine whether , where and
are arbitrary orders. We study and obtain some bounds for , where is any quasi-lexicographic ordering. The First-Fit
coloring of does not always yield an optimum coloring. A
greedy defining set of is a subset of vertices in the
graph together with a suitable pre-coloring of such that by fixing the
colors of the First-Fit coloring of yields an optimum
coloring. We show that the First-Fit coloring and greedy defining sets of
with respect to any quasi-lexicographic ordering (including the known
lexicographic order) are all the same. We obtain upper and lower bounds for the
smallest cardinality of a greedy defining set in , including some
extremal results for Latin squares.Comment: Accepted for publication in Contributions to Discrete Mathematic
An Improved Bound for First-Fit on Posets Without Two Long Incomparable Chains
It is known that the First-Fit algorithm for partitioning a poset P into
chains uses relatively few chains when P does not have two incomparable chains
each of size k. In particular, if P has width w then Bosek, Krawczyk, and
Szczypka (SIAM J. Discrete Math., 23(4):1992--1999, 2010) proved an upper bound
of ckw^{2} on the number of chains used by First-Fit for some constant c, while
Joret and Milans (Order, 28(3):455--464, 2011) gave one of ck^{2}w. In this
paper we prove an upper bound of the form ckw. This is best possible up to the
value of c.Comment: v3: referees' comments incorporate
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On the monotonicity of certain bin packing algorithms
This paper examines the monotonicity of the approximation bin packing algorithms Worst-Fit (WF), Worst-Fit Decreasing (WFD), Best-Fit (BF), Best-Fit Decreasing (BFD), and Next-Fit-k (NF-k). Let X and Y be two sets of items such that the set X can be derived from the set Y by possibly deleting some members of Y or by reducing the size of some members of Y. If an algorithm never uses more bins to pack X than it uses to pack Y we say that algorithm is monotonic. It is shown that NF and NF-2 are monotonic. It was already known that First-Fit and First-Fit Decreasing were non-monotonic and we give examples which show BF, BFD, WF, and WFD also suffer from this anomaly. One may consider First-Fit as the limiting case of NF-k. We notice that NF-1 is monotonic while First-Fit is not, suggesting there exists some critical k for which NF-k' is monotonic, for k' k. We establish that this is indeed the case and determine that critical k. An upper bound on the non-monotonicity of selected algorithms is also provided
First Fit bin packing: A tight analysis
In the bin packing problem we are given an instance consisting of a sequence of items with sizes between 0 and 1. The objective is to pack these items into the smallest possible number of bins of unit size. FirstFit algorithm packs each item into the first bin where it fits, possibly opening a new bin if the item cannot fit into any currently open bin. In early seventies it was shown that the asymptotic approximation ratio of FirstFit bin packing is equal to 1.7.
We prove that also the absolute approximation ratio for FirstFit bin packing is exactly 1.7. This means that if the optimum needs OPT bins, FirstFit always uses at most lfloor 1.7 OPT rfloor bins.
Furthermore we show matching lower bounds for a majority of values of OPT, i.e., we give instances on which FirstFit uses exactly lfloor 1.7 OPT rfloor bins.
Such matching upper and lower bounds were previously known only for finitely many small values of OPT. The previous published bound on the absolute approximation ratio of FirstFit was 12/7 approx 1.7143. Recently a bound of 101/59 approx 1.7119 was claimed
Two-dimensional rectangle packing: on-line methods and results
The first algorithms for the on-line two-dimensional rectangle packing problem were introduced by Coppersmith and Raghavan. They showed that for a family of heuristics 13/4 is an upper bound for the asymptotic worst-case ratios. We have investigated the Next Fit and the First Fit variants of their method. We proved that the asymptotic worst-case ratio equals 13/4 for the Next Fit variant and that 49/16 is an upper bound of the asymptotic worst-case ratio for the First Fit variant.
An easy subexponential bound for online chain partitioning
Bosek and Krawczyk exhibited an online algorithm for partitioning an online
poset of width into chains. We improve this to with a simpler and shorter proof by combining the work of Bosek &
Krawczyk with work of Kierstead & Smith on First-Fit chain partitioning of
ladder-free posets. We also provide examples illustrating the limits of our
approach.Comment: 23 pages, 11 figure
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