An Improved Branching Algorithm for Two-Layer Planarization Parameterized by the Feedback Edge Set Number

Abstract

Given an undirected graph G and an integer k ≥ 0, the NP-complete Two-Layer Planarization problem asks whether G can be transformed into a forest of caterpillar trees by removing at most k edges. Since transforming G into a forest of caterpillar trees requires breaking every cycle, the size f of a minimum feedback edge set is a natural parameter with f ≤ k. We refine and enhance ideas that led to previous algorithms running in O(3.562^k k + |G|) time and O(6^f f ² + f·|G|) time, respectively, to an improved branching algorithm running in O(3.8^f f ² + f·|G|) time. Since we expect f to be significantly smaller than k for a wide range of input instances, the presented algorithm can be considered superior to the previous algorithms. We present an empirical study of an implementation of our algorithm and compare it to implementations of previous algorithms. Our experiments show that even large instances can be solved as long as they are sparse.