Computing Imbalance-Minimal Orderings for Bipartite Permutation Graphs and Threshold Graphs

Abstract

The Imbalance problem is an NP-complete graph ordering problem which is only known to be polynomially solvable on some very restricted graph classes such as proper interval graphs, trees, and bipartite graphs of low maximum degree. In this paper, we show that Imbalance can be solved in linear time for bipartite permutation graphs and threshold graphs, resolving two open questions of Gorzny and Buss [COCOON 2019]. The results rely on the fact that if a graph can be partitioned into a vertex cover and an independent set, there is an imbalance-minimal ordering for which each vertex in the independent set is as balanced as possible. Furthermore, like the previous results of Gorzny and Buss, the paper shows that optimal orderings for Imbalance are similar to optimal orderings for Cutwidth on these graph classes. We observe that approaches for Cutwidth  are applicable for Imbalance. In particular, we observe that there is fixed-parameter tractable (FPT) algorithm which runs in time $O(2^kn^{O(1)})$ where k is the size of a minimum vertex cover of the input graph G and n is the number of vertices of G. This FPT algorithm improves the best known algorithm for this problem and parameter. Finally, we observe that Imbalance has no polynomial kernel parameterized by vertex cover unless NP $\subseteq$ coNP/poly.