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

Fast Biclustering by Dual Parameterization

We study two clustering problems, Starforest Editing, the problem of adding and deleting edges to obtain a disjoint union of stars, and the generalization Bicluster Editing. We show that, in addition to being NP-hard, none of the problems can be solved in subexponential time unless the exponential time hypothesis fails. Misra, Panolan, and Saurabh (MFCS 2013) argue that introducing a bound on the number of connected components in the solution should not make the problem easier: In particular, they argue that the subexponential time algorithm for editing to a fixed number of clusters (p-Cluster Editing) by Fomin et al. (J. Comput. Syst. Sci., 80(7) 2014) is an exception rather than the rule. Here, p is a secondary parameter, bounding the number of components in the solution. However, upon bounding the number of stars or bicliques in the solution, we obtain algorithms which run in time O(2^{3*sqrt(pk)} + n + m) for p-Starforest Editing and O(2^{O(p * sqrt(k) * log(pk))} + n + m) for p-Bicluster Editing. We obtain a similar result for the more general case of t-Partite p-Cluster Editing. This is subexponential in k for a fixed number of clusters, since p is then considered a constant. Our results even out the number of multivariate subexponential time algorithms and give reasons to believe that this area warrants further study

A Faster Parameterized Algorithm for Treedepth

The width measure \emph{treedepth}, also known as vertex ranking, centered coloring and elimination tree height, is a well-established notion which has recently seen a resurgence of interest. We present an algorithm which---given as input an $n$-vertex graph, a tree decomposition of the graph of width $w$, and an integer $t$---decides Treedepth, i.e. whether the treedepth of the graph is at most $t$, in time $2^{O(wt)} \cdot n$. If necessary, a witness structure for the treedepth can be constructed in the same running time. In conjunction with previous results we provide a simple algorithm and a fast algorithm which decide treedepth in time $2^{2^{O(t)}} \cdot n$ and $2^{O(t^2)} \cdot n$, respectively, which do not require a tree decomposition as part of their input. The former answers an open question posed by Ossona de Mendez and Nesetril as to whether deciding Treedepth admits an algorithm with a linear running time (for every fixed $t$) that does not rely on Courcelle's Theorem or other heavy machinery. For chordal graphs we can prove a running time of $2^{O(t \log t)}\cdot n$ for the same algorithm.Comment: An extended abstract was published in ICALP 2014, Track

A General Kernelization Technique for Domination and Independence Problems in Sparse Classes

We unify and extend previous kernelization techniques in sparse classes [Jochen Alber et al., 2004; Pilipczuk and Siebertz, 2018] by defining water lilies and show how they can be used in bounded expansion classes to construct linear bikernels for (r,c)-Dominating Set, (r,c)-Scattered Set, Total r-Domination, r-Roman Domination, and a problem we call (r,[?,?])-Domination (implying a bikernel for r-Perfect Code). At the cost of slightly changing the output graph class our bikernels can be turned into kernels. We also demonstrate how these constructions can be combined to create "multikernels", meaning graphs that represent kernels for multiple problems at once

Domination Above r-Independence: Does Sparseness Help?

Inspired by the potential of improving tractability via gap- or above-guarantee parametrisations, we investigate the complexity of Dominating Set when given a suitable lower-bound witness. Concretely, we consider being provided with a maximal r-independent set X (a set in which all vertices have pairwise distance at least r+1) along the input graph G which, for r >= 2, lower-bounds the minimum size of any dominating set of G. In the spirit of gap-parameters, we consider a parametrisation by the size of the "residual" set R := V(G) N[X]. Our work aims to answer two questions: How does the constant r affect the tractability of the problem and does the restriction to sparse graph classes help here? For the base case r = 2, we find that the problem is paraNP-complete even in apex- and bounded-degree graphs. For r = 3, the problem is W[2]-hard for general graphs but in FPT for nowhere dense classes and it admits a linear kernel for bounded expansion classes. For r >= 4, the parametrisation becomes essentially equivalent to the natural parameter, the size of the dominating set

Fast Biclustering by Dual Parameterization

We study two clustering problems, Starforest Editing, the problem of adding and deleting edges to obtain a disjoint union of stars, and the generalization Bicluster Editing. We show that, in addition to being NP-hard, none of the problems can be solved in subexponential time unless the exponential time hypothesis fails. Misra, Panolan, and Saurabh (MFCS 2013) argue that introducing a bound on the number of connected components in the solution should not make the problem easier: In particular, they argue that the subexponential time algorithm for editing to a fixed number of clusters (p-Cluster Editing) by Fomin et al. (J. Comput. Syst. Sci., 80(7) 2014) is an exception rather than the rule. Here, p is a secondary parameter, bounding the number of components in the solution. However, upon bounding the number of stars or bicliques in the solution, we obtain algorithms which run in time $2^{5 \sqrt{pk}} + O(n+m)$ for p-Starforest Editing and $2^{O(p \sqrt{k} \log(pk))} + O(n+m)$ for p-Bicluster Editing. We obtain a similar result for the more general case of t-Partite p-Cluster Editing. This is subexponential in k for fixed number of clusters, since p is then considered a constant. Our results even out the number of multivariate subexponential time algorithms and give reasons to believe that this area warrants further study.Comment: Accepted for presentation at IPEC 201

When You Come at the King You Best Not Miss

A tournament is an orientation of a complete graph. We say that a vertex x in a tournament T controls another vertex y if there exists a directed path of length at most two from x to y. A vertex is called a king if it controls every vertex of the tournament. It is well known that every tournament has a king. We follow Shen, Sheng, and Wu [Jian Shen et al., 2003] in investigating the query complexity of finding a king, that is, the number of arcs in T one has to know in order to surely identify at least one vertex as a king. The aforementioned authors showed that one always has to query at least ?(n^{4/3}) arcs and provided a strategy that queries at most O(n^{3/2}). While this upper bound has not yet been improved for the original problem, [Biswas et al., 2017] proved that with O(n^{4/3}) queries one can identify a semi-king, meaning a vertex which controls at least half of all vertices. Our contribution is a novel strategy which improves upon the number of controlled vertices: using O(n^{4/3} polylog n) queries, we can identify a (1/2+2/17)-king. To achieve this goal we use a novel structural result for tournaments