When Recursion is Better than Iteration: A Linear-Time Algorithm for Acyclicity with Few Error Vertices

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 k k 4 nm). 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!4 k k 5 (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

Parameterized Algorithms for Generalizations of Directed Feedback Vertex Set

“…The question was finally resolved by Chen et al who gave a 4 k k!k 4 • O(nm)-time algorithm for graphs with n vertices and m arcs. Recently, an algorithm for DFVS with run time 4 k k!k 5 •O(n+m) was given by Lokshtanov et al [15]. It is well-known that the arc deletion variant is parameter-equivalent to the vertex deletion variant and hence Directed Feedback Arc Set (DFAS) can also be solved in time 4 k k!k 5 • O(n + m).…”

Parameterized algorithms for generalizations of Directed Feedback Vertex Set

“…The question was finally resolved by Chen et al who gave a 4 k k!k 4 • O(nm)-time algorithm for graphs with n vertices and m arcs. Recently, an algorithm for DFVS with run time 4 k k!k 5 •O(n+m) was given by Lokshtanov et al [14]. It is well-known that the arc deletion variant is parameter-equivalent to the vertex deletion variant and hence Directed Feedback Arc Set (DFAS) can also be solved in time 4 k k!k 5 • O(n + m).…”

Hitting Long Directed Cycles is Fixed-Parameter Tractable

“…It was a long-standing open problem whether DFVS admits such an algorithm; the question was finally resolved by Chen et al who gave a 4 k k!k 4 ¨Opnmq-time algorithm for graphs with n vertices and m edges. Recently, an algorithm for DFVS with run time 4 k k!k 5 ¨Opn `mq was given by Lokshtanov et al [23]. A fruitful research direction is trying to extend the algorithm to more general problems than DFVS.…”