# Polynomial-time Algorithmfor Maximum Weight Independent Set on P6-free Graphs

Polynomial-time Algorithmfor Maximum Weight Independent Set on P6-free Graphs In the classic Maximum Weight Independent Set problem, we are given a graph G with a nonnegative weight function on its vertices, and the goal is to find an independent set in G of maximum possible weight. While the problem is NP-hard in general, we give a polynomial-time algorithm working on any P6-free graph, that is, a graph that has no path on 6 vertices as an induced subgraph. This improves the polynomial-time algorithm on P5-free graphs of Lokshtanov et al. [15] and the quasipolynomial-time algorithm on P6-free graphs of Lokshtanov et al. [14]. The main technical contribution leading to our main result is enumeration of a polynomial-size family ℱ of vertex subsets with the following property: For every maximal independent set I in the graph, ℱ contains all maximal cliques of some minimal chordal completion of G that does not add any edge incident to a vertex of I. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM Transactions on Algorithms (TALG) Association for Computing Machinery

# Polynomial-time Algorithmfor Maximum Weight Independent Set on P6-free Graphs

, Volume 18 (1): 57 – Jan 22, 2022
57 pages

/lp/association-for-computing-machinery/polynomial-time-algorithmfor-maximum-weight-independent-set-on-p6-free-l1N77DNmzq
Publisher
Association for Computing Machinery
ISSN
1549-6325
eISSN
1549-6333
DOI
10.1145/3414473
Publisher site
See Article on Publisher Site

### Abstract

In the classic Maximum Weight Independent Set problem, we are given a graph G with a nonnegative weight function on its vertices, and the goal is to find an independent set in G of maximum possible weight. While the problem is NP-hard in general, we give a polynomial-time algorithm working on any P6-free graph, that is, a graph that has no path on 6 vertices as an induced subgraph. This improves the polynomial-time algorithm on P5-free graphs of Lokshtanov et al. [15] and the quasipolynomial-time algorithm on P6-free graphs of Lokshtanov et al. [14]. The main technical contribution leading to our main result is enumeration of a polynomial-size family ℱ of vertex subsets with the following property: For every maximal independent set I in the graph, ℱ contains all maximal cliques of some minimal chordal completion of G that does not add any edge incident to a vertex of I.

### Journal

ACM Transactions on Algorithms (TALG)Association for Computing Machinery

Published: Jan 22, 2022

Keywords: P6-free graphs

### References

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