Tree decompositions meet induced matchings: beyond Max Weight Independent Set
CoRR(2024)
Abstract
For a tree decomposition 𝒯 of a graph G, by μ(𝒯)
we denote the size of a largest induced matching in G all of whose edges
intersect one bag of 𝒯. Induced matching treewidth of a graph G
is the minimum value of μ(𝒯) over all tree decompositions
𝒯 of G. Yolov [SODA 2018] proved that Max Weight Independent Set
can be solved in polynomial time for graphs of bounded induced matching
treewidth.
In this paper we explore what other problems are tractable in such classes of
graphs. As our main result, we give a polynomial-time algorithm for Min Weight
Feedback Vertex Set. We also provide some positive results concerning packing
induced subgraphs, which in particular imply a PTAS for the problem of finding
a largest induced subgraph of bounded treewidth.
These results suggest that in graphs of bounded induced matching treewidth,
one could find in polynomial time a maximum-weight induced subgraph of bounded
treewidth satisfying a given CMSO_2 formula. We conjecture that such a result
indeed holds and prove it for graphs of bounded tree-independence number, which
form a rich and important family of subclasses of graphs of bounded induced
matching treewidth.
We complement these algorithmic results with a number of complexity and
structural results concerning induced matching treewidth.
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