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We consider two coloring problems: interval coloring and max-coloring for chordal graphs. Given a graph G = (V , E) and positive-integral vertex weights w : V → N, the interval-coloring problem seeks to find an assignment of a real interval I (u) to each vertex u ∈ V , such that two constraints are satisfied: (i) for every vertex u ∈ V , |I (u)| = w(u) and (ii) for every pair of adjacent vertices u and v, I (u) ∩ I (v) = ∅. The goal is to minimize the span | ∪ v∈V I (v)|. The max-coloring problem seeks to find a proper vertex coloring of G whose color classes C 1 , C 2 , . . . , C k , minimize the sum of the weights of the heaviest vertices in the color classes, that is, k i=1 max v∈C i w(v). Both problems arise in efficient memory allocation for programs. The interval-coloring problem models the compiletime memory allocation problem and has a rich history dating back at least to the 1970s. The max-coloring problem arises in minimizing the total buffer size needed by a dedicated memory manager for programs. In another application, this problem models scheduling of conflicting jobs in batches to minimize the makespan. Both problems are NP-complete even for interval graphs, although there are constant-factor approximation algorithms for both problems on interval graphs. In this paper, we consider these problems for chordal graphs, a subclass of perfect graphs. These graphs naturally generalize interval graphs and can be defined as the class of graphs that have no induced cycle of length > 3. Recently, a 4-approximation algorithm (which we call GeomFit) has been presented for the max-coloring problem on perfect graphs (Pemmaraju and Raman 2005). This algorithm can be used to obtain an interval coloring as well, but without the constant-factor approximation guarantee. In fact, there is no known constant-factor approximation algorithm for the interval-coloring problem on perfect graphs. We study the performance of GeomFit and several simple O(log(n))-factor approximation algorithms for both problems. We experimentally evaluate and compare four simple heuristics: first-fit, best-fit, GeomFit, and a heuristic based on partitioning the graph into vertex sets of similar weight. Both for max-coloring and for interval coloring, GeomFit deviates from OPT by about 1.5%, on average. The performance of first-fit comes close second, deviating from OPT by less than 6%, on average, for both problems. Best-fit comes third and graphpartitioning heuristic comes a distant last. Our basic data comes from about 10,000 runs of each of the heuristics for each of the two problems on randomly generated chordal graphs of various sizes, sparsity, and structure.

In the \emph{tollbooth problem}, we are given a tree $\bT=(V,E)$ with $n$ edges, and a set of $m$ customers, each of whom is interested in purchasing a path on the tree. Each customer has a fixed budget, and the objective is to price the edges of $\bT$ such that the total revenue made by selling the paths to the customers that can afford them is maximized. An important special case of this problem, known as the \emph{highway problem}, is when $\bT$ is restricted to be a line. For the tollbooth problem, we present a randomized $O(\log n)$-approximation, improving on the current best $O(\log m)$-approximation, since $n\leq 3m$ can be assumed. We also study a special case of the tollbooth problem, when all the paths that customers are interested in purchasing go towards a fixed root of $\bT$. In this case, we present an algorithm that returns a $(1-\epsilon)$-approximation, for any $\epsilon > 0$, and runs in quasi-polynomial time. On the other hand, we rule out the existence of an FPTAS by showing that even for the line case, the problem is strongly NP-hard

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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Given a system of constraints i ≤ a T i x ≤ u i , where a i ∈ {0, 1} n , and i , u i ∈ R + , for i = 1, . . . , m, we consider the problem Mrfs of finding the largest subsystem for which there exists a feasible solution x ≥ 0. We present approximation algorithms and inapproximability results for this problem, and study some important special cases. Our main contributions are :1. In the general case, where a i ∈ {0, 1} n , a sharp separation in the approximability between the case when L = max{ 1 , · · · , m } is bounded above by a polynomial in n and m, and the case when it is not.2. In the case where A is an interval matrix, a sharp separation in approximability between the case where we allow a violation of the upper bounds by at most a (1 + ) factor, for any fixed > 0 and the case where no violations are allowed.Along the way, we prove that the induced matching problem on bipartite graphs is inapproximable beyond a factor of Ω(n 1 3 − ), for any > 0 unless NP=ZPP. Finally, we also show applications of Mrfs to some recently studied pricing problems.

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