Structural parameterizations for boxicity

The boxicity of a graph $G$ is the least integer $d$ such that $G$ has an intersection model of axis-aligned $d$-dimensional boxes. Boxicity, the problem of deciding whether a given graph $G$ has boxicity at most $d$, is NP-complete for every fixed $d \ge 2$. We show that boxicity is fixed-parameter tractable when parameterized by the cluster vertex deletion number of the input graph. This generalizes the result of Adiga et al., that boxicity is fixed-parameter tractable in the vertex cover number. Moreover, we show that boxicity admits an additive $1$-approximation when parameterized by the pathwidth of the input graph. Finally, we provide evidence in favor of a conjecture of Adiga et al. that boxicity remains NP-complete when parameterized by the treewidth.Comment: 19 page

Parameterized Algorithmics for Computational Social Choice: Nine Research Challenges

Computational Social Choice is an interdisciplinary research area involving Economics, Political Science, and Social Science on the one side, and Mathematics and Computer Science (including Artificial Intelligence and Multiagent Systems) on the other side. Typical computational problems studied in this field include the vulnerability of voting procedures against attacks, or preference aggregation in multi-agent systems. Parameterized Algorithmics is a subfield of Theoretical Computer Science seeking to exploit meaningful problem-specific parameters in order to identify tractable special cases of in general computationally hard problems. In this paper, we propose nine of our favorite research challenges concerning the parameterized complexity of problems appearing in this context

Parameterized Complexity of Fair Vertex Evaluation Problems

A prototypical graph problem is centered around a graph-theoretic property for a set of vertices and a solution to it is a set of vertices for which the desired property holds. The task is to decide whether, in the given graph, there exists a solution of a certain quality, where we use size as a quality measure. In this work, we are changing the measure to the fair measure (cf. Lin and Sahni [Li-Shin Lin and Sartaj Sahni, 1989]). The fair measure of a set of vertices S is (at most) k if the number of neighbors in the set S of any vertex (in the input graph) does not exceed k. One possible way to study graph problems is by defining the property in a certain logic. For a given objective, an evaluation problem is to find a set (of vertices) that simultaneously minimizes the assumed measure and satisfies an appropriate formula. More formally, we study the {MSO} Fair Vertex Evaluation, where the graph-theoretic property is described by an {MSO} formula. In the presented paper we show that there is an FPT algorithm for the {MSO} Fair Vertex Evaluation problem for formulas with one free variable parameterized by the twin cover number of the input graph and the size of the formula. One may define an extended variant of {MSO} Fair Vertex Evaluation for formulas with l free variables; here we measure a maximum number of neighbors in each of the l sets. However, such variant is {W[1]}-hard for parameter l even on graphs with twin cover one. Furthermore, we study the Fair Vertex Cover (Fair VC) problem. Fair VC is among the simplest problems with respect to the demanded property (i.e., the rest forms an edgeless graph). On the negative side, Fair VC is {W[1]}-hard when parameterized by both treedepth and feedback vertex set of the input graph. On the positive side, we provide an FPT algorithm for the parameter modular width

Streaming Kernelization

Kernelization is a formalization of preprocessing for combinatorially hard problems. We modify the standard definition for kernelization, which allows any polynomial-time algorithm for the preprocessing, by requiring instead that the preprocessing runs in a streaming setting and uses $\mathcal{O}(poly(k)\log|x|)$ bits of memory on instances $(x,k)$. We obtain several results in this new setting, depending on the number of passes over the input that such a streaming kernelization is allowed to make. Edge Dominating Set turns out as an interesting example because it has no single-pass kernelization but two passes over the input suffice to match the bounds of the best standard kernelization

On Sparse Hitting Sets: From Fair Vertex Cover to Highway Dimension

We consider the Sparse Hitting Set (Sparse-HS) problem, where we are given a set system (V,?,?) with two families ?,? of subsets of the universe V. The task is to find a hitting set for ? that minimizes the maximum number of elements in any of the sets of ?. This generalizes several problems that have been studied in the literature. Our focus is on determining the complexity of some of these special cases of Sparse-HS with respect to the sparseness k, which is the optimum number of hitting set elements in any set of ? (i.e., the value of the objective function). For the Sparse Vertex Cover (Sparse-VC) problem, the universe is given by the vertex set V of a graph, and ? is its edge set. We prove NP-hardness for sparseness k ? 2 and polynomial time solvability for k = 1. We also provide a polynomial-time 2-approximation algorithm for any k. A special case of Sparse-VC is Fair Vertex Cover (Fair-VC), where the family ? is given by vertex neighbourhoods. For this problem it was open whether it is FPT (or even XP) parameterized by the sparseness k. We answer this question in the negative, by proving NP-hardness for constant k. We also provide a polynomial-time (2-1/k)-approximation algorithm for Fair-VC, which is better than any approximation algorithm possible for Sparse-VC or the Vertex Cover problem (under the Unique Games Conjecture). We then switch to a different set of problems derived from Sparse-HS related to the highway dimension, which is a graph parameter modelling transportation networks. In recent years a growing literature has shown interesting algorithms for graphs of low highway dimension. To exploit the structure of such graphs, most of them compute solutions to the r-Shortest Path Cover (r-SPC) problem, where r > 0, ? contains all shortest paths of length between r and 2r, and ? contains all balls of radius 2r. It is known that there is an XP algorithm that computes solutions to r-SPC of sparseness at most h if the input graph has highway dimension h. However it was not known whether a corresponding FPT algorithm exists as well. We prove that r-SPC and also the related r-Highway Dimension (r-HD) problem, which can be used to formally define the highway dimension of a graph, are both W[1]-hard. Furthermore, by the result of Abraham et al. [ICALP 2011] there is a polynomial-time O(log k)-approximation algorithm for r-HD, but for r-SPC such an algorithm is not known. We prove that r-SPC admits a polynomial-time O(log n)-approximation algorithm

Parameterized complexity of the MINCCA problem on graphs of bounded decomposability

In an edge-colored graph, the cost incurred at a vertex on a path when two incident edges with different colors are traversed is called reload or changeover cost. The "Minimum Changeover Cost Arborescence" (MINCCA) problem consists in finding an arborescence with a given root vertex such that the total changeover cost of the internal vertices is minimized. It has been recently proved by G\"oz\"upek et al. [TCS 2016] that the problem is FPT when parameterized by the treewidth and the maximum degree of the input graph. In this article we present the following results for the MINCCA problem: - the problem is W[1]-hard parameterized by the treedepth of the input graph, even on graphs of average degree at most 8. In particular, it is W[1]-hard parameterized by the treewidth of the input graph, which answers the main open problem of G\"oz\"upek et al. [TCS 2016]; - it is W[1]-hard on multigraphs parameterized by the tree-cutwidth of the input multigraph; - it is FPT parameterized by the star tree-cutwidth of the input graph, which is a slightly restricted version of tree-cutwidth. This result strictly generalizes the FPT result given in G\"oz\"upek et al. [TCS 2016]; - it remains NP-hard on planar graphs even when restricted to instances with at most 6 colors and 0/1 symmetric costs, or when restricted to instances with at most 8 colors, maximum degree bounded by 4, and 0/1 symmetric costs.Comment: 25 pages, 11 figure