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

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

A tight analysis of Kierstead-Trotter algorithm for online unit interval coloring

Kierstead and Trotter (Congressus Numerantium 33, 1981) proved that their algorithm is an optimal online algorithm for the online interval coloring problem. In this paper, for online unit interval coloring, we show that the number of colors used by the Kierstead-Trotter algorithm is at most $3 \omega(G) - 3$, where $\omega(G)$ is the size of the maximum clique in a given graph $G$, and it is the best possible.Comment: 4 page

An easy subexponential bound for online chain partitioning

Bosek and Krawczyk exhibited an online algorithm for partitioning an online poset of width $w$ into $w^{14\lg w}$ chains. We improve this to $w^{6.5 \lg w + 7}$ 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

On-line partitioning of width w posets into w^O(log log w) chains

An on-line chain partitioning algorithm receives the elements of a poset one at a time, and when an element is received, irrevocably assigns it to one of the chains. In this paper, we present an on-line algorithm that partitions posets of width $w$ into $w^{O(\log{\log{w}})}$ chains. This improves over previously best known algorithms using $w^{O(\log{w})}$ chains by Bosek and Krawczyk and by Bosek, Kierstead, Krawczyk, Matecki, and Smith. Our algorithm runs in $w^{O(\sqrt{w})}n$ time, where $w$ is the width and $n$ is the size of a presented poset.Comment: 16 pages, 10 figure

Complexity of Grundy coloring and its variants

The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of GRUNDY COLORING, the problem of determining whether a given graph has Grundy number at least $k$. We also study the variants WEAK GRUNDY COLORING (where the coloring is not necessarily proper) and CONNECTED GRUNDY COLORING (where at each step of the greedy coloring algorithm, the subgraph induced by the colored vertices must be connected). We show that GRUNDY COLORING can be solved in time $O^*(2.443^n)$ and WEAK GRUNDY COLORING in time $O^*(2.716^n)$ on graphs of order $n$. While GRUNDY COLORING and WEAK GRUNDY COLORING are known to be solvable in time $O^*(2^{O(wk)})$ for graphs of treewidth $w$ (where $k$ is the number of colors), we prove that under the Exponential Time Hypothesis (ETH), they cannot be solved in time $O^*(2^{o(w\log w)})$. We also describe an $O^*(2^{2^{O(k)}})$ algorithm for WEAK GRUNDY COLORING, which is therefore \fpt for the parameter $k$. Moreover, under the ETH, we prove that such a running time is essentially optimal (this lower bound also holds for GRUNDY COLORING). Although we do not know whether GRUNDY COLORING is in \fpt, we show that this is the case for graphs belonging to a number of standard graph classes including chordal graphs, claw-free graphs, and graphs excluding a fixed minor. We also describe a quasi-polynomial time algorithm for GRUNDY COLORING and WEAK GRUNDY COLORING on apex-minor graphs. In stark contrast with the two other problems, we show that CONNECTED GRUNDY COLORING is \np-complete already for $k=7$ colors.Comment: 24 pages, 7 figures. This version contains some new results and improvements. A short paper based on version v2 appeared in COCOON'1

The on-line width of various classes of posets.

An on-line chain partitioning algorithm receives a poset, one element at a time, and irrevocably assigns the element to one of the chains. Over 30 years ago, Szemer\\u27edi proved that any on-line algorithm could be forced to use $\binom{w+1}{2}$ chains to partition a poset of width $w$. The maximum number of chains that can be forced on any on-line algorithm remains unknown. In the survey paper by Bosek et al., variants of the problem were studied where the class is restricted to posets of bounded dimension or where the poset is presented via a realizer of size $d$. We prove two results for this problem. First, we prove that any on-line algorithm can be forced to use $(2-o(1))\binom{w+1}{2}$ chains to partition a $2$-dimensional poset of width $w$. Second, we prove that any on-line algorithm can be forced to use $(2-\frac{1}{d-1}-o(1))\binom{w+1}{2}$ chains to partition a poset of width $w$ presented via a realizer of size $d$. Chrobak and \\u27Slusarek considered variants of the on-line chain partitioning problem in which the elements are presented as intervals and intersecting intervals are incomparable. They constructed an on-line algorithm which uses at most $3w-2$ chains, where $w$ is the width of the interval order, and showed that this algorithm is optimal. They also considered the problem restricted to intervals of unit-length and while they showed that first-fit needs at most $2w-1$ chains, over $30$ years later, it remains unknown whether a more optimal algorithm exists. We improve upon previously known bounds and show that any on-line algorithm can be forced to use $\lceil\frac{3}{2}w\rceil$ chains to partition a semi-order presented in the form of its unit-interval representation. As a consequence, we completely solve the problem for $w=3$. We also consider entirely new variants and present the results for those

Optimales Sortieren von Objekten

This thesis is concerned with the problem of optimally rearranging objects, in particular, railcars in a rail yard. The work is motivated by a research project of the Institute of Mathematical Optimization at Technische Universität Braunschweig, together with our project partner BASF, The Chemical Company, in Ludwigshafen. For many variants of such rearrangement problems - including the real-world application at BASF - we state the computational complexity by exploiting their equivalence to particular graph coloring, scheduling, and bin packing problems. We present mathematical optimization methods for determining schedules that are either optimal or close to optimal, and computational results are discussed from both a theoretical and practical point of view. In addition to the railway industry, there are other fields of application in which efficiently rearranging, sorting, or stacking is an important issue. For instance, the results obtained in this thesis could also be applied to solving certain piling problems in warehouses or container terminals.Die Dissertation beschäftigt sich mit dem optimalen Sortieren von Objekten, insbesondere von Güterwagen in Rangierbahnhöfen. Motiviert wurde diese Arbeit durch ein BMBF-gefördertes Projekt mit der BASF, The Chemical Company, in Ludwigshafen. Zahlreiche Varianten derartiger Sortierprobleme werden mathematisch formuliert und komplexitätstheoretisch eingeordnet. Für viele Varianten wird deren Äquivalenz zu bestimmten Graphenfärbungs-, Scheduling- sowie Bin-Packing-Problemen gezeigt. Für mehrere als theoretisch schwer bewiesene Fälle werden schnelle approximative Algorithmen vorgeschlagen, die Lösungen mit einer beweisbaren Güte liefern. Neben heuristischen Methoden werden auch exakte Verfahren zur Bestimmung optimaler Lösungen vorgestellt. Unter anderem handelt es sich bei den eingesetzten exakten Ansätzen um LP- sowie Lagrange-basierte Branch-and-Bound-Verfahren, die auf verschiedenen binären Modellen beruhen. Die Lösungsmethoden werden durch die Auswertung von Rechenergebnissen für reale Daten evaluiert. Den Abschluss der Dissertation bildet eine Kompetitivitätsanalyse diverser Online-Varianten, die dadurch gekennzeichnet sind, dass nicht alle relevanten Informationen zu Beginn der Planung vorliegen. Es sei auf das Verwertungspotenzial der in dieser Arbeit vorgestellten Optimierungsverfahren innerhalb anderer Anwendungsbereiche, in denen Sortieren, Stapeln, Lagern oder Verstauen eine Rolle spielen, hingewiesen