# Clique Clustering Yields a PTAS for Max-Coloring Interval Graphs

Clique Clustering Yields a PTAS for Max-Coloring Interval Graphs We are given an interval graph $$G = (V,E)$$ G = ( V , E ) where each interval $$I \in V$$ I ∈ V has a weight $$w_I \in \mathbb {Q}^+$$ w I ∈ Q + . The goal is to color the intervals $$V$$ V with an arbitrary number of color classes $$C_1, C_2, \ldots , C_k$$ C 1 , C 2 , … , C k such that $$\sum _{i=1}^k \max _{I \in C_i} w_I$$ ∑ i = 1 k max I ∈ C i w I is minimized. This problem, called max-coloring interval graphs or weighted coloring interval graphs, contains the classical problem of coloring interval graphs as a special case for uniform weights, and it arises in many practical scenarios such as memory management. Pemmaraju, Raman, and Varadarajan showed that max-coloring interval graphs is NP-hard [21] and presented a 2-approximation algorithm. We settle the approximation complexity of this problem by giving a polynomial-time approximation scheme (PTAS), that is, we show that there is an $$(1+\epsilon )$$ ( 1 + ϵ ) -approximation algorithm for any $$\epsilon > 0$$ ϵ > 0 . The PTAS also works for the bounded case where the sizes of the color classes are bounded by some arbitrary $$k \le n$$ k ≤ n . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Algorithmica Springer Journals

# Clique Clustering Yields a PTAS for Max-Coloring Interval Graphs

Algorithmica, Volume 80 (10) – Aug 17, 2017
16 pages

/lp/springer_journal/clique-clustering-yields-a-ptas-for-max-coloring-interval-graphs-0vIQNNd3Rn
Publisher
Springer Journals
Subject
Computer Science; Algorithm Analysis and Problem Complexity; Theory of Computation; Mathematics of Computing; Algorithms; Computer Systems Organization and Communication Networks; Data Structures, Cryptology and Information Theory
ISSN
0178-4617
eISSN
1432-0541
D.O.I.
10.1007/s00453-017-0362-9
Publisher site
See Article on Publisher Site

### Abstract

We are given an interval graph $$G = (V,E)$$ G = ( V , E ) where each interval $$I \in V$$ I ∈ V has a weight $$w_I \in \mathbb {Q}^+$$ w I ∈ Q + . The goal is to color the intervals $$V$$ V with an arbitrary number of color classes $$C_1, C_2, \ldots , C_k$$ C 1 , C 2 , … , C k such that $$\sum _{i=1}^k \max _{I \in C_i} w_I$$ ∑ i = 1 k max I ∈ C i w I is minimized. This problem, called max-coloring interval graphs or weighted coloring interval graphs, contains the classical problem of coloring interval graphs as a special case for uniform weights, and it arises in many practical scenarios such as memory management. Pemmaraju, Raman, and Varadarajan showed that max-coloring interval graphs is NP-hard [21] and presented a 2-approximation algorithm. We settle the approximation complexity of this problem by giving a polynomial-time approximation scheme (PTAS), that is, we show that there is an $$(1+\epsilon )$$ ( 1 + ϵ ) -approximation algorithm for any $$\epsilon > 0$$ ϵ > 0 . The PTAS also works for the bounded case where the sizes of the color classes are bounded by some arbitrary $$k \le n$$ k ≤ n .

### Journal

AlgorithmicaSpringer Journals

Published: Aug 17, 2017

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