Max-coloring and online coloring with bandwidths on interval graphs

Abstract

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.

An Improved Bound for First-Fit on Posets Without Two Long Incomparable Chains

“…First we prove that the incomparability graph of every poset of width w without k + k has small pathwidth, namely pathwidth at most (2k − 3)w − 1. Then we show that the fact that First-Fit uses at most 8w chains on interval orders of width w, as proved in [20], implies that First-Fit uses at most 8(p + 1) chains on posets whose incomparability graphs have pathwidth p. Combining these two results, we obtain an upper bound of 8(2k − 3)w on the number of chains used by First-Fit on posets of width w without k + k.…”

“…If no such chain is found, then a new chain containing only v is added at the end of the current chain partition. The performance of First-Fit on various classes of posets has been studied extensively [1,3,4,5,6,10,14,16,22,18,19,20]. In particular, Kierstead [14] showed that First-Fit can be forced to use an unbounded number of chains even on posets of width 2.…”

“…Nevertheless, First-Fit behaves well on some restricted classes of posets. A prominent example are interval orders, for which Kierstead [15] obtained a linear bound of 40w, which was subsequently improved by Kierstead and Qin [16] to 25.8w, and then by Pemmaraju, Raman and Varadarajan [19,20] to 10w. It was later shown by Brightwell, Kierstead and Trotter [5] and by Narayanaswamy and Babu [18] that the proof method of Pemmaraju et al actually gives a bound of 8w.…”

First-Fit is Linear on Posets Excluding Two Long Incomparable Chains

“…Kierstead and Qin [11] subsequently improved the bound, showing that FF(w) ≤ 25.8w. 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.…”

“…As far as we know, this also provides the first proof that some on-line algorithm uses o(w 2 ) chains on (r + s)-free posets. Our proof is strongly influenced by the Column Construction Method of Pemmaraju et al [17] and can be viewed as a generalization of that technique from interval orders to (r + s)-free posets.…”