A Note on First-Fit Coloring of Interval Graphs

Narayanaswamy, N. S.; Babu, R. Subhash

doi:10.1007/s11083-008-9076-6

Cited by 36 publications

(26 citation statements)

References 3 publications

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“…al. modified the analysis slightly to achieve the bound of 8 · χ(G). Also, Narayanaswamy and Subhash Babu [Narayanaswamy and Babu 2004] presented a different modification which also leads to an 8 · χ(G) upper bound. We present our analysis from [Pemmaraju et al 2004] as improved by [Narayanaswamy and Babu 2004], which leads to a 8-approximation algorithm for max-coloring on interval graphs.…”

Section: Analysis Of First-fitmentioning

confidence: 99%

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Max-coloring and online coloring with bandwidths on interval graphs

Pemmaraju

Raman

Varadarajan

2011

ACM Trans. Algorithms

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Given a graph G = (V, E) and positive integral vertex weights w : V → N, the max-coloring problem seeks to find a proper vertex coloring of G whose color classes C 1 , C 2 , . . . , C k , minimizemax v∈C i w(v). This problem, restricted to interval graphs, arises whenever there is a need to design dedicated memory managers that provide better performance than the general purpose memory management of the operating system.Though this problem seems similar to the dynamic storage allocation problem, there are fundamental differences. We make a connection between max-coloring and on-line graph coloring and use this to devise a simple 2-approximation algorithm for max-coloring on interval graphs. We also show that a simple first-fit strategy, that is a natural choice for this problem, yields an 8-approximation algorithm. We show this result by proving that the first-fit algorithm for on-line coloring an interval graph G uses no more than 8 · χ(G) colors, significantly improving the bound of 26·χ(G) by Kierstead and Qin (Discrete Math., 144, 1995). We also show that the max-coloring problem is NP-hard.The problem of online coloring of intervals with bandwidths is a simultaneous generalization of online interval coloring and online bin packing. The input is a set I of intervals, each interval i ∈ I having an associated bandwidth b(i) ∈ (0, 1]. We seek an online algorithm that produces a coloring of the intervals such that for any color c and any real r, the sum of the bandwidths of intervals containing r and colored c is at most 1. Motivated by resource allocation problems, Adamy and Erlebach (Proceedings of the First International Workshop on Online and Approximation Algorithms, 2003, LNCS 2909, pp 1-12 ) consider this problem and present an algorithm that uses at most 195 times the number of colors used by an optimal off-line algorithm. Using the new analysis of first-fit coloring of interval graphs, we show that the Adamy-Erlebach algorithm is 35-competitive. Finally, we generalize the Adamy-Erlebach algorithm to a class of algorithms and show that a different instance from this class is 30-competitive.

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Section: Analysis Of First-fitmentioning

confidence: 99%

“…We thank Narayanaswamy and Subhash Babu for letting us present their clever improvement [Narayanaswamy and Babu 2004] of our analysis of First-Fit in the preliminary version [Pemmaraju et al 2004]. We also thank Brightwell, Kierstead, and Trotter for sharing their improved analysis [Brightwell et al 2003] with us.…”

Section: Acknowledgementsmentioning

confidence: 99%

Max-coloring and online coloring with bandwidths on interval graphs

Pemmaraju

Raman

Varadarajan

2011

ACM Trans. Algorithms

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“…We note that the elegant argument given by Joret and Milans is an extension of the column labeling method introduced by Pemmaraju, Raman and Varadarajan [17]. Sharpening this labeling tool was a central component in the approaches taken by Brightwell, Kierstead and Trotter in [2] and by Narayanaswary and Babu [16].…”

Section: Background Materialsmentioning

confidence: 98%

“…In [1], this argument is presented, together with an improved lower bound of (1 − o (1) Moreover, it is known that First Fit performs reasonably well in coloring interval graphs. In fact, in a series of papers [22], [11], [13], [17], [2] and [16] and [8], incremental improvements have been made in analyzing the performance of First Fit in the coloring of interval graphs with maximum clique size at most k, with the current upper bound being 8k − 4.…”

Section: Background Materialsmentioning

confidence: 99%

On-Line Dimension for Posets Excluding Two Long Incomparable Chains

Felsner

Krawczyk

Trotter

2011

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Abstract. For a positive integer k, let k + k denote the poset consisting of two disjoint k-element chains, with all points of one chain incomparable with all points of the other. Bosek, Krawczyk and Szczypka showed that for each k ≥ 1, there exists a constant c k so that First Fit will use at most c k w 2 chains in partitioning a poset P of width at most w, provided the poset excludes k+k as a subposet. This result played a key role in the recent proof by Bosek and Krawczyk that O(w 16 log w ) chains are sufficient to partition on-line a poset of width w into chains. This result was the first improvement in Kierstead's exponential bound: (5 w − 1)/4 in nearly 30 years. Subsequently, Joret and Milans improved the Bosek-Krawczyk-Szczypka bound for the performance of First Fit to 8(k − 1) 2 w, which in turn yields the modest improvement to O(w 14 log w ) for the general on-line chain partitioning result. In this paper, we show that this class of posets admits a notion of on-line dimension. Specifically, we show that when k and w are positive integers, there exists an integer t = t(k, w) and an on-line algorithm that will construct an on-line realizer of size t for any poset P having width at most w, provided that the poset excludes k + k as a subposet.

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“…Later, Pemmaraju, Raman, and Varadarajan [16] (see also [17]) proved that FF(w) ≤ 10w with an elegant argument known as the Column Construction Method. Their proof was later refined by Brightwell, Kierstead, and Trotter [5] and independently by Narayanaswamy and Babu [15] to show that FF(w) ≤ 8w.…”

Section: Introductionmentioning

confidence: 99%