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When recursion is better than iteration : a linear-time algorithm for acyclicity with few error vertices
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Lokshtanov, Daniel, Ramanujan, Maadapuzhi Sridharan and Saket, Saurabh (2018) When recursion is better than iteration : a linear-time algorithm for acyclicity with few error vertices. In: 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, Louisiana, USA. , New Orleans, Louisiana, USA, 7-10 Jan 2018. Published in: Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms pp. 1-18. ISBN 9781611975031. doi:10.1137/1.9781611975031.125
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Official URL: https://doi.org/10.1137/1.9781611975031.125
Abstract
Planarity, bipartiteness and (directed) acyclicity are basic graph properties with classic linear time recognition algorithms. However, the problems of testing whether a given (di)graph has k vertices whose deletion makes it planar, bipartite or a directed acyclic graph (DAG) are all fundamental NP-complete problems when k is part of the input. As a result, a significant amount of research has been devoted to understanding whether, for every fixed k, these problems admit a polynomial time algorithm (where the exponent in the polynomial is independent of k) and in particular, whether they admit linear time algorithms.
While we now know that for any fixed k, we can test in linear time whether a graph is k vertices away from being planar [FOCS 2009, SODA 2014] or bipartite [SODA 2014, SICOMP 2016], the best known algorithms in the case of directed acyclicity are the algorithm of Garey and Tarjan [IPL 78] which runs in time O(nk–1m) and the algorithm of Chen, Liu, Lu, O'Sullivan and Razgon [JACM 2008] which runs in time O(k!4kk4nm). In other words, it has remained open whether it is possible to recognize in linear time, a graph which is 2 vertices away from being acyclic!
In this paper, we settle this question by giving an algorithm that decides whether a given graph is k vertices away from being acyclic, in time O(k!4kk5(n + m)). That is, for every fixed k, our algorithm runs in time O(m + n), thus mirroring the case for planarity and bipartiteness.
Our algorithm is designed via a general methodology that shaves off a factor of n from some algorithms that use the powerful technique of iterative compression. The two main features of our methodology are: (i) This is the first generic technique for designing linear time algorithms for directed cut-problems and (ii) it can be used in combination with future improvements in algorithms for the compression version of other well-studied cut-problems such as Multicut and Directed Subset Feedback Vertex Set.
| Item Type: | Conference Item (Paper) | ||||||||||||
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| Subjects: | Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software | ||||||||||||
| Divisions: | Faculty of Science > Computer Science | ||||||||||||
| Library of Congress Subject Headings (LCSH): | Computer algorithms., Graph theory., Computer science -- Mathematics, Computer programming., Computational complexity. | ||||||||||||
| Journal or Publication Title: | Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms | ||||||||||||
| Publisher: | SIAM | ||||||||||||
| ISBN: | 9781611975031 | ||||||||||||
| Official Date: | 10 January 2018 | ||||||||||||
| Dates: |
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| Page Range: | pp. 1-18 | ||||||||||||
| DOI: | 10.1137/1.9781611975031.125 | ||||||||||||
| Status: | Peer Reviewed | ||||||||||||
| Publication Status: | Published | ||||||||||||
| Access rights to Published version: | Open Access | ||||||||||||
| RIOXX Funder/Project Grant: |
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| Conference Paper Type: | Paper | ||||||||||||
| Title of Event: | 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, Louisiana, USA. | ||||||||||||
| Type of Event: | Conference | ||||||||||||
| Location of Event: | New Orleans, Louisiana, USA | ||||||||||||
| Date(s) of Event: | 7-10 Jan 2018 | ||||||||||||
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