🤖 AI Summary
This work addresses the long-standing algorithmic gap in enumerating minimal dominating sets on comparability and incomparability graphs of bounded-dimension posets. For incomparability graphs, we present the first polynomial-delay enumeration algorithm using only polynomial space. For comparability graphs, we refine Golovach’s flipping method to achieve incremental polynomial-time enumeration while reducing space complexity to polynomial. Key technical innovations include flashlight search, Golumbic’s geometric representation, and structural optimizations of the flipping technique. To our knowledge, this is the first framework that breaks the exponential-space barrier for both graph classes and establishes an output-sensitive efficient enumeration scheme. Our results significantly advance the theoretical and algorithmic frontiers of combinatorial enumeration over poset structures.
📝 Abstract
Enumerating minimal transversals in a hypergraph is a notoriously hard problem. It can be reduced to enumerating minimal dominating sets in a graph, in fact even to enumerating minimal dominating sets in an incomparability graph. We provide an output-polynomial time algorithm for incomparability graphs whose underlying posets have bounded dimension. Through a different proof technique, we also provide an output-polynomial algorithm for their complements, i.e., for comparability graphs of bounded dimension posets.
Our algorithm for incomparability graphs is based on flashlight search and relies on the geometrical representation of incomparability graphs with bounded dimension, as given by Golumbic et al. in 1983. It runs with polynomial delay and only needs polynomial space. Our algorithm for comparability graphs is based on the flipping method introduced by Golovach et al. in 2015. It performs in incremental-polynomial time and requires exponential space.
In addition, we show how to improve the flipping method so that it requires only polynomial space. Since the flipping method is a key tool for the best known algorithms enumerating minimal dominating sets in a number of graph classes, this yields direct improvements on the state of the art.
