Half-integrality, LP-branching and FPT Algorithms

A recent trend in parameterized algorithms is the application of polytope tools (specifically, LP-branching) to FPT algorithms (e.g., Cygan et al., 2011; Narayanaswamy et al., 2012). However, although interesting results have been achieved, the methods require the underlying polytope to have very restrictive properties (half-integrality and persistence), which are known only for few problems (essentially Vertex Cover (Nemhauser and Trotter, 1975) and Node Multiway Cut (Garg et al., 1994)). Taking a slightly different approach, we view half-integrality as a \emph{discrete} relaxation of a problem, e.g., a relaxation of the search space from $\{0,1\}^V$ to $\{0,1/2,1\}^V$ such that the new problem admits a polynomial-time exact solution. Using tools from CSP (in particular Thapper and \v{Z}ivn\'y, 2012) to study the existence of such relaxations, we provide a much broader class of half-integral polytopes with the required properties, unifying and extending previously known cases. In addition to the insight into problems with half-integral relaxations, our results yield a range of new and improved FPT algorithms, including an $O^*(|\Sigma|^{2k})$-time algorithm for node-deletion Unique Label Cover with label set $\Sigma$ and an $O^*(4^k)$-time algorithm for Group Feedback Vertex Set, including the setting where the group is only given by oracle access. All these significantly improve on previous results. The latter result also implies the first single-exponential time FPT algorithm for Subset Feedback Vertex Set, answering an open question of Cygan et al. (2012). Additionally, we propose a network flow-based approach to solve some cases of the relaxation problem. This gives the first linear-time FPT algorithm to edge-deletion Unique Label Cover.Comment: Added results on linear-time FPT algorithms (not present in SODA paper

Linear-Time Kernelization for Feedback Vertex Set

In this paper, we give an algorithm that, given an undirected graph G of m edges and an integer k, computes a graph G\u27 and an integer k\u27 in O(k^4 m) time such that (1) the size of the graph G\u27 is O(k^2), (2) k\u27 leq k, and (3) G has a feedback vertex set of size at most k if and only if G\u27 has a feedback vertex set of size at most k\u27. This is the first linear-time polynomial-size kernel for Feedback Vertex Set. The size of our kernel is 2k^2+k vertices and 4k^2 edges, which is smaller than the previous best of 4k^2 vertices and 8k^2 edges. Thus, we improve the size and the running time simultaneously. We note that under the assumption of NP notsubseteq coNP/poly, Feedback Vertex Set does not admit an O(k^{2-epsilon})-size kernel for any epsilon>0. Our kernel exploits k-submodular relaxation, which is a recently developed technique for obtaining efficient FPT algorithms for various problems. The dual of k-submodular relaxation of Feedback Vertex Set can be seen as a half-integral variant of A-path packing, and to obtain the linear-time complexity, we give an efficient augmenting-path algorithm for this problem. We believe that this combinatorial algorithm is of independent interest. A solver based on the proposed method won first place in the 1st Parameterized Algorithms and Computational Experiments (PACE) challenge

Parameterized Algorithms for Zero Extension and Metric Labelling Problems

We consider the problems Zero Extension and Metric Labelling under the paradigm of parameterized complexity. These are natural, well-studied problems with important applications, but have previously not received much attention from this area. Depending on the chosen cost function mu, we find that different algorithmic approaches can be applied to design FPT-algorithms: for arbitrary mu we parameterize by the number of edges that cross the cut (not the cost) and show how to solve Zero Extension in time O(|D|^{O(k^2)} n^4 log n) using randomized contractions. We improve this running time with respect to both parameter and input size to O(|D|^{O(k)} m) in the case where mu is a metric. We further show that the problem admits a polynomial sparsifier, that is, a kernel of size O(k^{|D|+1}) that is independent of the metric mu. With the stronger condition that mu is described by the distances of leaves in a tree, we parameterize by a gap parameter (q - p) between the cost of a true solution q and a `discrete relaxation\u27 p and achieve a running time of O(|D|^{q-p} |T|m + |T|phi(n,m)) where T is the size of the tree over which mu is defined and phi(n,m) is the running time of a max-flow computation. We achieve a similar result for the more general Metric Labelling, while also allowing mu to be the distance metric between an arbitrary subset of nodes in a tree using tools from the theory of VCSPs. We expect the methods used in the latter result to have further applications

Experimental Evaluation of Parameterized Algorithms for Feedback Vertex Set

Feedback Vertex Set is a classic combinatorial optimization problem that asks for a minimum set of vertices in a given graph whose deletion makes the graph acyclic. From the point of view of parameterized algorithms and fixed-parameter tractability, Feedback Vertex Set is one of the landmark problems: a long line of study resulted in multiple algorithmic approaches and deep understanding of the combinatorics of the problem. Because of its central role in parameterized complexity, the first edition of the Parameterized Algorithms and Computational Experiments Challenge (PACE) in 2016 featured Feedback Vertex Set as the problem of choice in one of its tracks. The results of PACE 2016 on one hand showed large discrepancy between performance of different classic approaches to the problem, and on the other hand indicated a new approach based on half-integral relaxations of the problem as probably the most efficient approach to the problem. In this paper we provide an exhaustive experimental evaluation of fixed-parameter and branching algorithms for Feedback Vertex Set

Representative set statements for delta-matroids and the Mader delta-matroid

We present representative sets-style statements for linear delta-matroids, which are set systems that generalize matroids, with important connections to matching theory and graph embeddings. Furthermore, our proof uses a new approach of sieving polynomial families, which generalizes the linear algebra approach of the representative sets lemma to a setting of bounded-degree polynomials. The representative sets statements for linear delta-matroids then follow by analyzing the Pfaffian of the skew-symmetric matrix representing the delta-matroid. Applying the same framework to the determinant instead of the Pfaffian recovers the representative sets lemma for linear matroids. Altogether, this significantly extends the toolbox available for kernelization. As an application, we show an exact sparsification result for Mader networks: Let $G=(V,E)$ be a graph and $\mathcal{T}$ a partition of a set of terminals $T \subseteq V(G)$, $|T|=k$. A $\mathcal{T}$-path in $G$ is a path with endpoints in distinct parts of $\mathcal{T}$ and internal vertices disjoint from $T$. In polynomial time, we can derive a graph $G'=(V',E')$ with $T \subseteq V(G')$, such that for every subset $S \subseteq T$ there is a packing of $\mathcal{T}$-paths with endpoints $S$ in $G$ if and only if there is one in $G'$, and $|V(G')|=O(k^3)$. This generalizes the (undirected version of the) cut-covering lemma, which corresponds to the case that $\mathcal{T}$ contains only two blocks. To prove the Mader network sparsification result, we furthermore define the class of Mader delta-matroids, and show that they have linear representations. This should be of independent interest