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Approximate Minimum Sum Colorings and Maximum $k$-Colorable Subgraphs of Chordal Graphs

Ian DeHaan, Zachary Friggstad

TL;DR

The first polynomial-time approximation scheme for the maximum $k$-colorable subgraph problem in chordal graphs and the first polynomial-time approximation scheme for the maximum $k$-colorable subgraph problem in chordal graphs are designed.

Abstract

We give a $(1.796+ε)$-approximation for the minimum sum coloring problem on chordal graphs, improving over the previous 3.591-approximation by Gandhi et al. [2005]. To do so, we also design the first polynomial-time approximation scheme for the maximum $k$-colorable subgraph problem in chordal graphs.

Approximate Minimum Sum Colorings and Maximum $k$-Colorable Subgraphs of Chordal Graphs

TL;DR

The first polynomial-time approximation scheme for the maximum -colorable subgraph problem in chordal graphs and the first polynomial-time approximation scheme for the maximum -colorable subgraph problem in chordal graphs are designed.

Abstract

We give a -approximation for the minimum sum coloring problem on chordal graphs, improving over the previous 3.591-approximation by Gandhi et al. [2005]. To do so, we also design the first polynomial-time approximation scheme for the maximum -colorable subgraph problem in chordal graphs.

Paper Structure

This paper contains 9 sections, 10 theorems, 20 equations, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Techniques
  3. An LP-Based Approximation Algorithm for MSC
  4. The Configuration LP
  5. The Rounding Algorithm and Analysis
  6. A PTAS for M$k$CS in Chordal Graphs
  7. Analysis
  8. APX-Hardness for M$k$CS in Perfect Graphs
  9. Conclusion

Key Result

theorem thmcountertheorem

For any constant $\epsilon > 0$, there is a polynomial-time $\frac{\mu^{\star}}{2} + \epsilon \approx 1.796 + \epsilon$ approximation for weighted MSC on chordal graphs.

Theorems & Definitions (19)

  • definition thmcounterdefinition
  • theorem thmcountertheorem
  • definition thmcounterdefinition
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • lemma thmcounterlemma
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • proof
  • ...and 9 more