Faster Graph bipartization☆
Introduction
Graph bipartization is a fundamental NP-complete problem with several applications [4], [29], [23]. This problem, which is also known as Odd Cycle Transversal, is formally defined as follows.
The parameterized complexity of OCT was a well known open problem for a long time. In other words, it was open whether OCT could be solved in time for a computable function f. Such algorithms, i.e., algorithms whose running time can be bounded by for some computable function f, are called fixed-parameter (FPT) algorithms. In a breakthrough paper, Reed, Smith and Vetta [28] resolved this question by presenting an algorithm for OCT running in time . We use n and m to denote the number of vertices and edges of the input graph, respectively. In fact, this work introduced the powerful iterative compression technique that has been instrumental in resolving the parameterized complexity of numerous other problems in the area, such as Directed Feedback Vertex Set, Almost 2-SAT, and Multicut [3], [27], [20]. The two main questions that arose from this work of Reed, Smith and Vetta, were the following.
- 1.
Can the dependence on the parameter be improved?
- 2.
Can the dependence on the input size be improved to , even at the cost of a worse dependence on k?
The first question remained unresolved until the work of Narayanaswamy et al. [22], who obtained an algorithm with an improved dependence on the parameter k. This algorithm is based on a branching guided by linear programming and runs in time . This was later improved by Lokshtanov et al. [17] to . In related work, Pilipczuk et al. [24] developed an algorithm for the edge version of this problem running in time , improving upon the previous best bound of [11].
In a parallel line of research attempting to address the second question, Fiorini et al. [5] showed that when the input is restricted to planar graphs there is a time algorithm, thus settling this question on planar graphs. This result was then improved by Lokshtanov et al. [18], who obtained a time algorithm. In the case of general graphs, Kawarabayashi and Reed [12] obtained an algorithm for OCT running in time , using tools from graph minors and odd variants of graph minors. Here, is the inverse of the Ackermann function (see Tarjan [30]) and is at least a triple-exponential function. Finally, Iwata et al. [13] and Ramanujan and Saurabh [26] answered the second question in the affirmative by giving and algorithms respectively. One of the main contributions of this paper is a significant improvement over these algorithms for OCT. To be specific, we prove the following result.
Theorem 1 OCT can be solved in time .
The central component of our result is the following new parameterized approximation algorithm for OCT. We refer to a set whose deletion makes the input graph bipartite, as an odd cycle transversal of the graph.
Lemma 1 There is an algorithm that, given an instance of OCT, runs in time and either returns an odd cycle transversal of G of size at most or correctly concludes that G has no odd cycle transversal of size at most k. Here , .
In order to prove Theorem 1, we combine Lemma 1 and the compression routine from [28], formally described as follows.
Proposition 1 [28] There is an algorithm that, given an instance of OCT and a set , which is an odd cycle transversal of G of size at most , runs in time and either correctly concludes that G has no odd cycle transversal of size at most k (in which case it returns No) or returns an odd cycle transversal of G of size at most k.
Proof (Theorem 1) Let be the given instance of OCT and let . For each , let denote the graph , i.e., the subgraph of G induced by the first i vertices. We first execute the algorithm of Lemma 1 on the instance to either conclude that G has no odd cycle transversal of size at most k or compute a set of size at most that is an odd cycle transversal of G. In the former case, we return the same. Otherwise, assume without loss of generality that , where . That is, the vertices of S appear contiguously and towards the end of the ordering of . Let . Clearly, is a bipartite graph. We now define as follows. Each is either a vertex set or No. Initially, . We now compute in this order by iterating over as follows. If No, then we may conclude that G has no odd cycle transversal of size at most k. Therefore, we set No. Otherwise, is defined as the output of the algorithm of Proposition 1 on input and . The correctness of Proposition 1 guarantees that if G has an odd cycle transversal of size at most k then for every , is a set of size at most k and is an odd cycle transversal of . Conversely, if G has no odd cycle transversal of size at most k then for some , would be set to No. Since , we conclude that if is No then G has no odd cycle transversal of size at most k and conversely if is a set then it is an odd cycle transversal of G of size at most k and we return Yes. Note that the algorithm of Lemma 1 is executed exactly once and takes time , while the algorithm of Proposition 1 is executed exactly times and takes time for each execution. Consequently, the total time taken by our algorithm for OCT is , which is dominated by the second term. This completes the proof of the theorem. □
We note that it is fairly straightforward to use existing results to obtain a polynomial-time approximation for Odd Cycle Transversal with a better approximation factor than the one in Lemma 1. Indeed, if then one could simply use the -approximation algorithm of Agarwal et al. [1] and otherwise, the FPT-algorithms in [28], [13], [26] already run in polynomial time. Lemma 1 therefore, trades off a worse approximation factor for a better dependence on the input-size.
We now focus on describing our approximation algorithm for OCT. In fact, we obtain Lemma 1 as a corollary of a more general result for the d-Skew-Symmetric Multicut problem, which was first introduced by Ramanujan and Saurabh [26]. A skew-symmetric graph is a directed graph D with a specific kind of involution σ (σ is an involution if for every domain element x) on the set of vertices and arcs. Here, for all , if and only if . That is, vertices are mapped to vertices and arcs are mapped to arcs. The function σ can be extended in a natural way to subsets of as well. Flows on skew-symmetric graphs have been used to generalize maximum flow and maximum matching problems on graphs, initially by Tutte [31], and later by Goldberg and Karzanov [9]. Aspvall et al. [2] characterized satisfiable 2-cnf formulas in terms of forbidden closed walks in an associated skew-symmetric graph, implying a linear-time algorithm for 2-SAT.
The d-Skew-Symmetric Multicut problem is a variant of the classic Multicut problem on skew-symmetric graphs and is formally defined as follows. Note that a d-set of vertices is simply a vertex subset with exactly d vertices.
A set S satisfying the properties above (except for the size constraint ) is called a skew-symmetric multicut of and if additionally, , then we call it a solution for this instance. Alternatively, we also use the term approximate solution to refer to a skew-symmetric multicut of , whose size is not necessarily at most 2k. We design an approximation algorithm for the d-Skew-Symmetric Multicut problem that runs in polynomial time such that any approximate solution returned by the algorithm has size bounded quadratically in the parameter. Specifically, we prove the following result.
Theorem 2 There is an algorithm that, given an instance of d-Skew-Symmetric Multicut, runs in time and either returns a skew-symmetric multicut of size at most or correctly concludes that no such set of size at most 2k exists. Here , , and ℓ, the length of the family , is defined as .
This algorithm, combined with known linear-time parameter-preserving reductions (reductions where the parameter does not undergo a change) from OCT, Almost 2-SAT and Deletion q-Horn Backdoor Set Detection ([26]) gives approximation algorithms for all these problems in the spirit of Lemma 1. In particular, OCT and Almost 2-SAT are special cases of 1-Skew-Symmetric Multicut and so we get -approximation algorithms for these two problems. Note that OCT itself is known to be a special case of Almost 2-SAT [14], [20]. Similarly, Deletion q-Horn Backdoor Set Detection is a special case of 3-Skew-Symmetric Multicut and so we get a -approximation algorithm for this problem. Gaspers et al. [8] showed that Deletion q-Horn Backdoor Set Detection has a -approximation algorithm running in time . As a result, our corollary for Deletion q-Horn Backdoor Set Detection improves on the exponential dependence on the parameter in the running time of the algorithm of Gaspers et al. [8] significantly with only a constant factor multiplicative increase in the approximation factor.
We mainly build upon the work of Gaspers et al. [8] and Ramanujan and Saurabh [26]. However, we need to introduce and work with a combinatorial object that is specifically tailored to obtaining approximation algorithms on skew-symmetric graphs. This object, called an -set (L is a subset of vertices such that ), is effectively a subgraph of the input graph and has two crucial structural properties that point us to an approximate solution.
- •
Every -set has a “boundary” of size and intersects every solution for the given instance.
- •
Removing the “boundary” of any -set leads to a residual instance with a strictly smaller solution.
We show that given these two properties, if one could always find an -set (if it exists) in polynomial time, then it leads to an algorithm that, either correctly concludes that there is no solution of the required size or outputs an approximate solution whose size is bounded by . As a result, once we prove that -sets satisfy both structural properties, we present a subroutine designed to compute one such -set at every step of our algorithm. It must be noted that the running time of this subroutine is allowed to have only a polynomial dependence on the parameter and linear dependence on the input size.
To achieve this, we build upon a lemma from [26] that can only check for the existence of a special kind of -sets (called -components) and enhance this algorithm to check for the existence of a general -set and moreover, compute one if it exists. The notion of -sets itself builds upon that of -components. However, -components are too restrictive to guarantee either of the two desirable properties listed above. The reader familiar with the notion of important separators [19] and associated terminology may interpret the boundary of an -component as a skew-symmetric analogue of the unique smallest important L- separator (minus the uniqueness). In the same spirit, the boundary of an -set may be interpreted as a skew-symmetric analogue of an arbitrary (but small) important L- separator such that there is no other such separator ‘further from L’.
Our idea of using arbitrary furthest important separators to obtain approximation algorithms is also applicable to other cut problems. We refer the reader to Section 8.3 in [25] for an expository application of this idea to the Node Multiway Cut problem to obtain a -approximation. We note that Node Multiway Cut already has a 2-approximation [7] and so the result in [25] does not give any improvements.
Section snippets
Preliminaries
In this section we give some basic definitions and set up the notations used in the paper.
Digraphs. Let be a directed graph. For an arc , we refer to u as the tail of this arc and we refer to v as the head of this arc. For a set of vertices , we let denote the set of arcs with both endpoints in the set . For a set of vertices , we let denote the set of arcs that have their tail in and their head in . Similarly, we let denote the set of arcs that
Skew-symmetric graphs, separators and components
Below, we only consider arc-separators of disjoint vertex sets.
Definition 1 Let be a directed graph and let be disjoint subsets of V. A set is an X-Y separator if there is no X-Y path in the graph . We say that S is a minimal X-Y separator if no proper subset of S is an X-Y separator.
We require the following observations and definitions from [26] regarding skew-symmetric graphs.
Observation 1 Let be a skew-symmetric graph and let . There is a v-u path in D if and only if there is a -
Structural properties and computation of -sets
In this section, we first give a formal proof of the utility of -sets. Following this, we give an algorithm for the computation of an -set.
Lemma 3 Let be a yes-instance of d-Skew-Symmetric Multicut and let L be a regular set of vertices such that there is an L- path in D. If there is a solution S for the given instance that is an L- self-conjugate separator in D, then the following hold. For every -set Z, and the instance is
The approximation algorithm for d-Skew-Symmetric Multicut
In this section we design our approximation algorithm for d-Skew-Symmetric Multicut.
From this point onwards, we assume that an instance of d-Skew-Symmetric Multicut is of the form where L is a regular set of vertices and the question is to determine whether there is a solution for the given instance that is an L- self-conjugate separator. To solve the problem on the input instance (as required by the formal definition of the problem), we simply solve it on
Conclusion
In this work, we presented an algorithm for the Odd Cycle Transversal problem via a new approximation algorithm. We achieve this by introducing an approach applicable to the more general d-Skew-Symmetric Multicut problem.
Our general result for d-Skew-Symmetric Multicut could find further applications in the realm of kernelization. A kernel of size for a parameterized problem Π is simply a polynomial-time algorithm that takes as input an instance and outputs another instance
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
We would like to thank the anonymous referees for their valuable comments and helpful suggestions.
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The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 306992.