New Width Parameters for Independent Set: Onesidedmimwidth and Neighbordepth
Abstract
We study the tractability of the maximum independent set problem from the viewpoint of graph width parameters, with the goal of defining a width parameter that is as general as possible and allows to solve independent set in polynomialtime on graphs where the parameter is bounded. We introduce two new graph width parameters: onesided maximum induced matchingwidth (omimwidth) and neighbordepth. Omimwidth is a graph parameter that is more general than the known parameters mimwidth and treeindependence number, and we show that independent set and feedback vertex set can be solved in polynomialtime given a decomposition with bounded omimwidth. Omimwidth is the first width parameter that gives a common generalization of chordal graphs and graphs of bounded cliquewidth in terms of tractability of these problems. The parameter omimwidth, as well as the related parameters mimwidth and simwidth, have the limitation that no algorithms are known to compute boundedwidth decompositions in polynomialtime. To partially resolve this limitation, we introduce the parameter neighbordepth. We show that given a graph of neighbordepth $k$, independent set can be solved in time $n^{O(k)}$ even without knowing a corresponding decomposition. We also show that neighbordepth is bounded by a polylogarithmic function on the number of vertices on large classes of graphs, including graphs of bounded omimwidth, and more generally graphs of bounded simwidth, giving a quasipolynomialtime algorithm for independent set on these graph classes. This resolves an open problem asked by Kang, Kwon, Strømme, and Telle [TCS 2017].
 Publication:

arXiv eprints
 Pub Date:
 February 2023
 DOI:
 10.48550/arXiv.2302.10643
 arXiv:
 arXiv:2302.10643
 Bibcode:
 2023arXiv230210643B
 Keywords:

 Computer Science  Data Structures and Algorithms
 EPrint:
 26 pages, 1 figure. This is the full version of an extended abstract that will appear in WG2023