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
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