When recursion is better than iteration: A linear-time algorithm for directed acyclicity with few error vertices
Planarity, bipartiteness and (directed) acyclicity are basic graph properties with classic linear-time recognition algorithms. However, the problems of testing whether a given graph has k vertices whose deletion makes it planar, bipartite or a directed acyclic graph (DAG) are all fundamental NP-comp...
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| Veröffentlicht in: | Journal of combinatorial theory. Series B Jg. 177; S. 143 - 185 |
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| Abstract | Planarity, bipartiteness and (directed) acyclicity are basic graph properties with classic linear-time recognition algorithms. However, the problems of testing whether a given 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 every fixed k, we can test in linear time whether a graph is k vertices away from being planar or bipartite, the best known algorithms in the case of directed acyclicity are the algorithm of Garey and Tarjan [IPL 1978], which runs in time O(nk−1⋅m) and the algorithm of Chen, Liu, Lu, O'Sullivan and Razgon [JACM 2008], which runs in time O(k!⋅4k⋅k4⋅n⋅m), where n and m are the number of vertices and arcs in the input digraph, respectively. In other words, it has remained open whether it is possible to recognize in linear time, a graph that is two 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!⋅4k⋅k5⋅(n+m)). That is, for every fixed k, our algorithm runs in time O(m+n), thus mirroring the case for planarity and bipartiteness.
We obtain our algorithm by introducing a general methodology that shaves off a factor of n from certain 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 FPT algorithms for directed cut problems and (ii) it can be used in combination with future improvements in algorithms for the so-called compression version of other well-studied cut problems such as Multicut and Directed Subset Feedback Vertex Set. |
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| AbstractList | Planarity, bipartiteness and (directed) acyclicity are basic graph properties with classic linear-time recognition algorithms. However, the problems of testing whether a given 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 every fixed k, we can test in linear time whether a graph is k vertices away from being planar or bipartite, the best known algorithms in the case of directed acyclicity are the algorithm of Garey and Tarjan [IPL 1978], which runs in time O(nk−1⋅m) and the algorithm of Chen, Liu, Lu, O'Sullivan and Razgon [JACM 2008], which runs in time O(k!⋅4k⋅k4⋅n⋅m), where n and m are the number of vertices and arcs in the input digraph, respectively. In other words, it has remained open whether it is possible to recognize in linear time, a graph that is two 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!⋅4k⋅k5⋅(n+m)). That is, for every fixed k, our algorithm runs in time O(m+n), thus mirroring the case for planarity and bipartiteness.
We obtain our algorithm by introducing a general methodology that shaves off a factor of n from certain 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 FPT algorithms for directed cut problems and (ii) it can be used in combination with future improvements in algorithms for the so-called compression version of other well-studied cut problems such as Multicut and Directed Subset Feedback Vertex Set. |
| Author | Lokshtanov, Daniel Ramanujan, M.S. Saurabh, Saket |
| Author_xml | – sequence: 1 givenname: Daniel surname: Lokshtanov fullname: Lokshtanov, Daniel email: daniello@ucsb.edu organization: Department of Computer Science, University of California Santa Barbara, USA – sequence: 2 givenname: M.S. surname: Ramanujan fullname: Ramanujan, M.S. email: r.maadapuzhi-sridharan@warwick.ac.uk organization: Department of Computer Science, University of Warwick, UK – sequence: 3 givenname: Saket surname: Saurabh fullname: Saurabh, Saket email: saket@imsc.res.in organization: The Institute of Mathematical Sciences, HBNI, Chennai, India |
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| Cites_doi | 10.1137/S0097539793251219 10.1016/j.jctb.2004.08.001 10.1016/j.jcss.2003.07.008 10.1137/S0097539792228228 10.1016/j.orl.2003.10.009 10.1145/44483.44491 10.1007/s00224-007-1345-z 10.1016/S0095-8956(03)00067-4 10.1137/130947374 10.1145/2500119 10.1145/2566616 10.1145/210332.210337 10.1137/110855247 10.1006/jctb.1995.1006 10.1137/0201010 10.1137/140961808 10.1016/0020-0190(78)90015-7 10.4153/CJM-1956-045-5 10.1145/3155299 10.1137/140962838 10.1145/3128600 10.1145/1411509.1411511 |
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| SubjectTerms | Directed graphs Feedback vertex set Fixed-parameter algorithms |
| Title | When recursion is better than iteration: A linear-time algorithm for directed acyclicity with few error vertices |
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