Abstract
In the Edge Bipartization problem one is given an undirected graph G and an integer k, and the question is whether k edges can be deleted from G so that it becomes bipartite. Guo et al. (J Comput Syst Sci 72(8):1386–1396, 2006) proposed an algorithm solving this problem in time mathcal {O}(2^kcdot {m}^2); today, this algorithm is a textbook example of an application of the iterative compression technique. Despite extensive progress in the understanding of the parameterized complexity of graph separation problems in the recent years, no significant improvement upon this result has been yet reported. We present an algorithm for Edge Bipartization that works in time mathcal {O}(1.977^kcdot {nm}), which is the first algorithm with the running time dependence on the parameter better than 2^k. To this end, we combine the general iterative compression strategy of Guo et al. (2006), the technique proposed by Wahlström (in: Proceedings of SODA’14, SIAM, 2014) of using a polynomial-time solvable relaxation in the form of a Valued Constraint Satisfaction Problem to guide a bounded-depth branching algorithm, and an involved Measure&Conquer analysis of the recursion tree.
Highlights
The Edge Bipartization problem asks, for a given graph G and integer k, whether one can turn G into a bipartite graph using at most k edge deletions
To prove Theorem 1.1, we pursue the approach proposed by Guo et al [14] and use iterative compression to reduce solving Edge Bipartization to solving Terminal
In this work we have developed an algorithm for Edge Bipartization that has running time O(1.977k · nm), which is the first one to achieve a dependence on the parameter better than 2k
Summary
The Edge Bipartization problem asks, for a given graph G and integer k, whether one can turn G into a bipartite graph using at most k edge deletions. Together with its Algorithmica (2019) 81:917–966 close relative Odd Cycle Transversal (OCT), where one deletes vertices instead of edges, Edge Bipartization was one of the first problems shown to admit a fixed-parameter (FPT) algorithm using the technique of iterative compression. In a breakthrough paper [28] that introduces this methodology, Reed et al showed how to solve OCT in time O(3k · kmn).. In a breakthrough paper [28] that introduces this methodology, Reed et al showed how to solve OCT in time O(3k · kmn).1 This was the first FPT algorithm for OCT. Guo et al [14] applied iterative compression to show fixed-parameter tractability of several closely related problems Among other results, they designed an algorithm for Edge Bipartization with running time O(2k · m2). Both the algorithms of Reed et al and of Guo et al are textbook examples of the iterative compression technique [5,11]
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