Dotted interval graphs

Yonatan Aumann; Moshe Lewenstein; Oren Melamud; Ron Pinter; Zohar Yakhini

doi:10.1145/2151171.2151172

Dotted interval graphs

Yonatan Aumann, Moshe Lewenstein, Oren Melamud, Ron Pinter, Zohar Yakhini

Bar-Ilan University, Technion - Israel Institute of Technology

Research output: Contribution to journal › Article › peer-review

Abstract

We introduce a generalization of interval graphs, which we call Dotted Interval Graphs (DIG). A dotted interval graph is an intersection graph of arithmetic progressions (dotted intervals). Coloring of dotted interval graphs naturally arises in the context of high throughput genotyping. We study the properties of dotted interval graphs, with a focus on coloring. We show that any graph is a DIG, but that DIGd graphs, that is, DIGs in which the arithmetic progressions have a jump of at most d, form a strict hierarchy. We show that coloring DIG _d graphs is NP-complete even for d = 2. For any fixed d, we provide a 5/6d + o(d) approximation for the coloring of DIG _d graphs. Finally, we show that finding the maximal clique in DIG _d graphs is fixed parameter tractable in d.

Original language	English
Article number	9
Journal	ACM Transactions on Algorithms
Volume	8
Issue number	2
DOIs	https://doi.org/10.1145/2151171.2151172
State	Published - Apr 2012

Keywords

Approximation algorithms
Circular arc graph
Graph coloring
Graph theory
Intersection graph
Interval graph

All Science Journal Classification (ASJC) codes

Mathematics (miscellaneous)

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10.1145/2151171.2151172

Cite this

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title = "Dotted interval graphs",

abstract = "We introduce a generalization of interval graphs, which we call Dotted Interval Graphs (DIG). A dotted interval graph is an intersection graph of arithmetic progressions (dotted intervals). Coloring of dotted interval graphs naturally arises in the context of high throughput genotyping. We study the properties of dotted interval graphs, with a focus on coloring. We show that any graph is a DIG, but that DIGd graphs, that is, DIGs in which the arithmetic progressions have a jump of at most d, form a strict hierarchy. We show that coloring DIG d graphs is NP-complete even for d = 2. For any fixed d, we provide a 5/6d + o(d) approximation for the coloring of DIG d graphs. Finally, we show that finding the maximal clique in DIG d graphs is fixed parameter tractable in d.",

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T1 - Dotted interval graphs

AU - Aumann, Yonatan

AU - Lewenstein, Moshe

AU - Melamud, Oren

AU - Pinter, Ron

AU - Yakhini, Zohar

PY - 2012/4

Y1 - 2012/4

N2 - We introduce a generalization of interval graphs, which we call Dotted Interval Graphs (DIG). A dotted interval graph is an intersection graph of arithmetic progressions (dotted intervals). Coloring of dotted interval graphs naturally arises in the context of high throughput genotyping. We study the properties of dotted interval graphs, with a focus on coloring. We show that any graph is a DIG, but that DIGd graphs, that is, DIGs in which the arithmetic progressions have a jump of at most d, form a strict hierarchy. We show that coloring DIG d graphs is NP-complete even for d = 2. For any fixed d, we provide a 5/6d + o(d) approximation for the coloring of DIG d graphs. Finally, we show that finding the maximal clique in DIG d graphs is fixed parameter tractable in d.

AB - We introduce a generalization of interval graphs, which we call Dotted Interval Graphs (DIG). A dotted interval graph is an intersection graph of arithmetic progressions (dotted intervals). Coloring of dotted interval graphs naturally arises in the context of high throughput genotyping. We study the properties of dotted interval graphs, with a focus on coloring. We show that any graph is a DIG, but that DIGd graphs, that is, DIGs in which the arithmetic progressions have a jump of at most d, form a strict hierarchy. We show that coloring DIG d graphs is NP-complete even for d = 2. For any fixed d, we provide a 5/6d + o(d) approximation for the coloring of DIG d graphs. Finally, we show that finding the maximal clique in DIG d graphs is fixed parameter tractable in d.

KW - Approximation algorithms

KW - Circular arc graph

KW - Graph coloring

KW - Graph theory

KW - Intersection graph

KW - Interval graph

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DO - 10.1145/2151171.2151172

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JO - ACM Transactions on Algorithms

JF - ACM Transactions on Algorithms

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Dotted interval graphs

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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