Fixed-parameter Tractability Research Articles

New algorithms for fair k-center problem with outliers and capacity constraints

The fair k-center problem has been paid lots of attention recently. In the fair k-center problem, we are given a set X of points in a metric space and a parameter k∈Z+, where the points in X are divided into several groups, and each point is assigned a color to denote which group it is in. The goal is to partition X into k clusters such that the number of cluster centers with each color is equal to a given value, and the k-center problem objective is minimized. In this paper, we consider the fair k-center problem with outliers and capacity constraints, denoted as the fair k-center with outliers (FkCO) problem and the capacitated fair k-center (CFkC) problem, respectively. The outliers constraints allow up to z outliers to be discarded when computing the objective function, while the capacity constraints require that each cluster has size no more than L. In this paper, we design an Fixed-Parameter Tractability (FPT) approximation algorithm and a polynomial approximation algorithm for the above two problems. In particular, our algorithms give (1+ϵ)-approximations with FPT time for the FkCO and CFkC problems in doubling metric space. Moreover, we also propose a 3-approximation algorithm in polynomial time for the FkCO problem with some reasonable assumptions.

Exact Algorithms and Lowerbounds for Multiagent Path Finding: Power of Treelike Topology

In the Multiagent Path Finding (MAPF for short) problem, we focus on efficiently finding non-colliding paths for a set of k agents on a given graph G, where each agent seeks a path from its source vertex to a target. An important measure of the quality of the solution is the length of the proposed schedule l, that is, the length of a longest path (including the waiting time). In this work, we propose a systematic study under the parameterized complexity framework. The hardness results we provide align with many heuristics used for this problem, whose running time could potentially be improved based on our Fixed-Parameter Tractability (FPT) results. We show that MAPF is W[1]-hard with respect to k (even if k is combined with the maximum degree of the input graph). The problem remains NP-hard in planar graphs even if the maximum degree and the makespan l are fixed constants. On the positive side, we show an FPT algorithm for k+l. As we continue, the structure of G comes into play. We give an FPT algorithm for parameter k plus the diameter of the graph G. The MAPF problem is W[1]-hard for cliquewidth of G plus l while it is FPT for treewidth of G plus l.

A General Theoretical Framework for Learning Smallest Interpretable Models

We develop a general algorithmic framework that allows us to obtain fixed-parameter tractability for computing smallest symbolic models that represent given data. Our framework applies to all ML model types that admit a certain extension property. By showing this extension property for decision trees, decision sets, decision lists, and binary decision diagrams, we obtain that minimizing these fundamental model types is fixed-parameter tractable. Our framework even applies to ensembles, which combine individual models by majority decision.

Grouped domination parameterized by vertex cover, twin cover, and beyond

A dominating set S of graph G is called an r-grouped dominating set if S can be partitioned into S1,S2,…,Sk such that the size of each unit Si is r and the subgraph of G induced by Si is connected. The concept of r-grouped dominating sets generalizes several well-studied variants of dominating sets with requirements for connected component sizes, such as the ordinary dominating sets (r=1), paired dominating sets (r=2), and connected dominating sets (r is arbitrary and k=1). In this paper, we investigate the computational complexity of r-Grouped Dominating Set, which is the problem of deciding whether a given graph has an r-grouped dominating set with at most k units. For general r, r-Grouped Dominating Set is hard to solve in various senses because the hardness of the connected dominating set is inherited. We thus focus on the case in which r is a constant or a parameter, but we see that r-Grouped Dominating Set for every fixed r>0 is still hard to solve. From the observations about the hardness, we consider the parameterized complexity concerning well-studied graph structural parameters. We first see that r-Grouped Dominating Set is fixed-parameter tractable for r and treewidth, which is derived from the fact that the condition of r-grouped domination for a constant r can be represented as monadic second-order logic (MSO2). This fixed-parameter tractability is good news, but the running time is not practical. We then design an O⁎(min⁡{(2τ(r+1))τ,(2τ)2τ})-time algorithm for general r≥2, where τ is the twin cover number, which is a parameter between vertex cover number and clique-width. For paired dominating set and trio dominating set, i.e., r∈{2,3}, we can speed up the algorithm, whose running time becomes O⁎((r+1)τ). We further argue the relationship between FPT results and graph parameters, which draws the parameterized complexity landscape of r-Grouped Dominating Set.

Long directed detours: Reduction to 2-Disjoint Paths

In the Longest(s,t)-Detour problem, we look for an (s,t)-path that is at least k vertices longer than a shortest one. We study the parameterized complexity of Longest(s,t)-Detour when parameterized by k: this falls into the research paradigm of ‘parameterization above guarantee’. Whereas the problem is known to be fixed-parameter tractable (FPT) on undirected graphs, the status of Longest(s,t)-Detour on directed graphs remains highly unclear: it is not even known to be solvable in polynomial time for k=1. Recently, Fomin et al. made progress in this direction by showing that the problem is FPT on every class of directed graphs where the 3-Disjoint Paths problem is solvable in polynomial time. We improve upon their result by weakening this assumption: we show that only a polynomial-time algorithm for 2-Disjoint Paths is required.

An impossible combinatorial counting method in distance geometry

The Distance Geometry Problem asks for a geometric representation of a given weighted graph in RK so that vertices are points and edges are segments with lengths equal to the corresponding weights. Two problem variants are defined by a vertex order given as part of the input, which allows the use of a branching algorithm based on K-lateration: find two possible positions for the next vertex j using the positions of K predecessors and their distances to j, then explore each position recursively, pruning positions at need. Whereas the first variant only requires the K predecessors to exist, the second variant also requires them to be contiguous and immediately preceding j. For this variant, fixed-parameter tractability of the algorithm can be established by means of a solution counting method that only depends on the graph edges (rather than their weights). Only in the first variant, however, it is possible to efficiently construct a suitable vertex order directly from the graph. Since both fixed-parameter tractability and efficient vertex order construction are desirable properties, one would need an analogous solution counting method for the first variant. In this paper we prove that such a counting method cannot be devised for the first variant.

On the Parameterized Intractability of Determinant Maximization

AbstractIn the Determinant Maximization problem, given an $$n \times n$$ n × n positive semi-definite matrix $${\textbf {A}} $$ A in $$\mathbb {Q}^{n \times n}$$ Q n × n and an integer k, we are required to find a $$k \times k$$ k × k principal submatrix of $${\textbf {A}} $$ A having the maximum determinant. This problem is known to be -hard and further proven to be [1]-hard with respect to k by Koutis (Inf Process Lett 100:8–13, 2006); i.e., a $$f(k)n^{{{\,\mathrm{\mathcal {O}}\,}}(1)}$$ f ( k ) n O ( 1 ) -time algorithm is unlikely to exist for any computable function f. However, there is still room to explore its parameterized complexity in the restricted case, in the hope of overcoming the general-case parameterized intractability. In this study, we rule out the fixed-parameter tractability of Determinant Maximization even if an input matrix is extremely sparse or low rank, or an approximate solution is acceptable. We first prove that Determinant Maximization is -hard and [1]-hard even if an input matrix is an arrowhead matrix; i.e., the underlying graph formed by nonzero entries is a star, implying that the structural sparsity is not helpful. By contrast, Determinant Maximization is known to be solvable in polynomial time on tridiagonal matrices (Al-Thani and Lee, in: LAGOS, 2021). Thereafter, we demonstrate the [1]-hardness with respect to the rankr of an input matrix. Our result is stronger than Koutis’ result in the sense that any $$k \times k$$ k × k principal submatrix is singular whenever $$k > r$$ k > r . We finally give evidence that it is [1]-hard to approximate Determinant Maximization parameterized by k within a factor of $$2^{-c\sqrt{k}}$$ 2 - c k for some universal constant $$c > 0$$ c > 0 . Our hardness result is conditional on the Parameterized Inapproximability Hypothesis posed by Lokshtanov et al. (in: SODA, 2020), which asserts that a gap version of Binary Constraint Satisfaction Problem is [1]-hard. To complement this result, we develop an $$\varepsilon $$ ε -additive approximation algorithm that runs in $$\varepsilon ^{-r^2} \cdot r^{{{\,\mathrm{\mathcal {O}}\,}}(r^3)} \cdot n^{{{\,\mathrm{\mathcal {O}}\,}}(1)}$$ ε - r 2 · r O ( r 3 ) · n O ( 1 ) time for the rank r of an input matrix, provided that the diagonal entries are bounded.

Characterization of matrices with bounded Graver bases and depth parameters and applications to integer programming

AbstractAn intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix A and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of A, and when parameterized by the dual tree-depth and the entry complexity of A; both these parameterization imply that A is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively. We study preconditioners transforming a given matrix to a row-equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse row-equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the $$\ell _1$$ ℓ 1 -norm of the Graver basis is bounded by a function of the maximum $$\ell _1$$ ℓ 1 -norm of a circuit of A. We use our results to design a parameterized algorithm that constructs a matrix row-equivalent to an input matrix A that has small primal/dual tree-depth and entry complexity if such a row-equivalent matrix exists. Our results yield parameterized algorithms for integer programming when parameterized by the $$\ell _1$$ ℓ 1 -norm of the Graver basis of the constraint matrix, when parameterized by the $$\ell _1$$ ℓ 1 -norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix row-equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix row-equivalent to the constraint matrix.

Orthologic with Axioms

We study the proof theory and algorithms for orthologic, a logical system based on ortholattices, which have shown practical relevance in simplification and normalization of verification conditions. Ortholattices weaken Boolean algebras while having polynomial-time equivalence checking that is sound with respect to Boolean algebra semantics. We generalize ortholattice reasoning and obtain an algorithm for proving a larger class of classically valid formulas. As the key result, we analyze a proof system for orthologic augmented with axioms. An important feature of the system is that it limits the number of formulas in a sequent to at most two, which makes the extension with axioms non-trivial. We show a generalized form of cut elimination for this system, which implies a sub-formula property. From there we derive a cubic-time algorithm for provability from axioms, or equivalently, for validity in finitely presented ortholattices. We further show that propositional resolution of width 5 proves all formulas provable in orthologic with axioms. We show that orthologic system subsumes resolution of width 2 and arbitrarily wide unit resolution and is complete for reasoning about generalizations of propositional Horn clauses. Moving beyond ground axioms, we introduce effectively propositional orthologic (by analogy with EPR for classical logic), presenting its semantics as well as a sound and complete proof system. Our proof system implies the decidability of effectively propositional orthologic, as well as its fixed-parameter tractability for a bounded maximal number of variables in each axiom. As a special case, we obtain a generalization of Datalog with negation and disjunction.

Disentangling the Computational Complexity of Network Untangling

We study the network untangling problem introduced by Rozenshtein et al. (Data Min. Knowl. Disc. 35(1), 213–247, 2021), which is a variant of Vertex Cover on temporal graphs–graphs whose edge set changes over discrete time steps. They introduce two problem variants. The goal is to select at most k time intervals for each vertex such that all time-edges are covered and (depending on the problem variant) either the maximum interval length or the total sum of interval lengths is minimized. This problem has data mining applications in finding activity timelines that explain the interactions of entities in complex networks. Both variants of the problem are NP-hard. In this paper, we initiate a multivariate complexity analysis involving the following parameters: number of vertices, lifetime of the temporal graph, number of intervals per vertex, and the interval length bound. For both problem versions, we (almost) completely settle the parameterized complexity for all combinations of those four parameters, thereby delineating the border of fixed-parameter tractability.

Partitioning subclasses of chordal graphs with few deletions

In the (Vertex)k-Way Cut problem, input is an undirected graph G, an integer s, and the goal is to find a subset S of edges (vertices) of size at most s, such that G−S has at least k connected components. Downey et al. [Electr. Notes Theor. Comput. Sci. 2003] showed that k-Way Cut is W[1]-hard parameterized by k. However, Kawarabayashi and Thorup [FOCS 2011] showed that the problem is fixed-parameter tractable (FPT) in general graphs with respect to the parameter s and provided a O(ssO(s)n2) time algorithm, where n denotes the number of vertices in G. The best-known algorithm for this problem runs in time sO(s)nO(1) given by Lokshtanov et al. [ACM Tran. of Algo. 2021]. On the other hand, Vertexk-Way Cut is W[1]-hard with respect to either of the parameters, k or s or k+s. These algorithmic results motivate us to look at the problems on special classes of graphs. In this paper, we consider the (Vertex)k-Way Cut problem on subclasses of chordal graphs and obtain the following results.•We first give a sub-exponential FPT algorithm for k-Way Cut running in time 2O(slog⁡s)nO(1) on chordal graphs.•It is “known” that Vertexk-Way Cut is W[1]-hard on chordal graphs, in fact on split graphs, parameterized by k+s. We complement this hardness result by designing polynomial-time algorithms for Vertexk-Way Cut on interval graphs, circular-arc graphs and permutation graphs.

Parameterised and Fine-Grained Subgraph Counting, Modulo 2

Given a class of graphs H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {H}}$$\\end{document}, the problem ⊕Sub(H)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\oplus \ ext {{Sub}}({\\mathcal {H}})$$\\end{document} is defined as follows. The input is a graph H∈H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H\\in {\\mathcal {H}}$$\\end{document} together with an arbitrary graph G. The problem is to compute, modulo 2, the number of subgraphs of G that are isomorphic to H. The goal of this research is to determine for which classes H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {H}}$$\\end{document} the problem ⊕Sub(H)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\oplus \ ext {{Sub}}({\\mathcal {H}})$$\\end{document} is fixed-parameter tractable (FPT), i.e., solvable in time f(|H|)·|G|O(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(|H|)\\cdot |G|^{O(1)}$$\\end{document}. Curticapean, Dell, and Husfeldt (ESA 2021) conjectured that ⊕Sub(H)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\oplus \ ext {{Sub}}({\\mathcal {H}})$$\\end{document} is FPT if and only if the class of allowed patterns H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {H}}$$\\end{document} is matching splittable, which means that for some fixed B, every H∈H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H \\in {\\mathcal {H}}$$\\end{document} can be turned into a matching (a graph in which every vertex has degree at most 1) by removing at most B vertices. Assuming the randomised Exponential Time Hypothesis, we prove their conjecture for (I) all hereditary pattern classes H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {H}}$$\\end{document}, and (II) all tree pattern classes, i.e., all classes H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {H}}$$\\end{document} such that every H∈H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H\\in {\\mathcal {H}}$$\\end{document} is a tree. We also establish almost tight fine-grained upper and lower bounds for the case of hereditary patterns (I).

Solving the Workflow Satisfiability Problem Using General Purpose Solvers

The workflow satisfiability problem (WSP) is a well-studied problem in access control seeking allocation of authorised users to every step of the workflow, subject to workflow specification constraints. It was noticed that the number <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> of steps is typically small compared to the number of users in the real-world instances of WSP; therefore <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> is considered as the parameter in WSP parametrised complexity research. While WSP in general was shown to be W[1]-hard, WSP restricted to a special case of user-independent (UI) constraints is fixed-parameter tractable (FPT). However, restriction to the UI constraints might be impractical. To efficiently handle non-UI constraints, we introduce the notion of branching factor of a constraint. As long as the branching factors of the constraints are relatively small and the number of non-UI constraints is reasonable, WSP can be solved in FPT time. Extending the results from Karapetyan et al. (2019), we demonstrate that general-purpose solvers are capable of achieving FPT-like performance on WSP with arbitrary constraints when used with appropriate formulations. This enables one to tackle most of practical WSP instances. While important on its own, we hope that this result will also motivate researchers to look for FPT-aware formulations of other FPT problems.

Filling crosswords is very hard

We revisit a classical crossword filling puzzle which already appeared in Garey& Jonhson's book. We are given a grid with n vertical and horizontal slots and a dictionary with m words. We are asked to place words from the dictionary in the slots so that shared cells are consistent. We attempt to pinpoint the source of intractability of this problem by carefully taking into account the structure of the grid graph, which contains a vertex for each slot and an edge if two slots intersect. Our main approach is to consider the case where this graph has a tree-like structure. Unfortunately, if we impose the common rule that words cannot be reused, we discover that the problem remains NP-hard under very severe structural restrictions, namely, if the grid graph is a union of stars and the alphabet has size 2, or the grid graph is a matching (so the crossword is a collection of disjoint crosses) and the alphabet has size 3. The problem does become slightly more tractable if word reuse is allowed, as we obtain an mtw algorithm in this case, where tw is the treewidth of the grid graph. However, even in this case, we show that our algorithm cannot be improved to obtain fixed-parameter tractability. More strongly, we show that under the ETH the problem cannot be solved in time mo(k), where k is the number of horizontal slots of the instance (which trivially bounds tw).Motivated by these mostly negative results, we also consider the much more restricted case where the problem is parameterized by the number of slots n. Here, we show that the problem does become FPT (if the alphabet has constant size), but the parameter dependence is exponential in n2. We show that this dependence is also justified: the existence of an algorithm with running time 2o(n2), even for binary alphabet, would contradict the randomized ETH. After that, we consider an optimization version of the problem, where we seek to place as many words on the grid as possible. Here it is easy to obtain a 12-approximation, even on weighted instances, simply by considering only horizontal or only vertical slots. We show that this trivial algorithm is also likely to be optimal, as obtaining a better approximation ratio in polynomial time would contradict the Unique Games Conjecture. The latter two results apply whether word reuse is allowed or not.Finally, we present some special cases where the problem is decidable in polynomial time. In particular, we present three reductions, one to 2-SAT, the second to Maximum Matching and the third to Exact Matching.

Orienting undirected phylogenetic networks

This paper studies the relationship between undirected (unrooted) and directed (rooted) phylogenetic networks. We describe a polynomial-time algorithm for deciding whether an undirected nonbinary phylogenetic network, given the locations of the root and reticulation vertices, can be oriented as a directed nonbinary phylogenetic network. Moreover, we characterize when this is possible and show that, in such instances, the resulting directed nonbinary phylogenetic network is unique. In addition, without being given the location of the root and the reticulation vertices, we describe an algorithm for deciding whether an undirected binary phylogenetic network N can be oriented as a directed binary phylogenetic network of a certain class. The algorithm is fixed-parameter tractable (FPT) when the parameter is the level of N and is applicable to classes of directed phylogenetic networks that satisfy certain conditions. As an example, we show that the well-studied class of binary tree-child networks satisfies these conditions.

On structural parameterizations of load coloring

Given a graph G and a positive integer k, the 2-Load Coloring problem is to check whether there is a 2-coloring f:V(G)→{r,b} of G such that for every i∈{r,b}, there are at least k edges with both end vertices colored i. It is known that the problem is NP-complete even on special classes of graphs like regular graphs. Gutin and Jones (2014) showed that the problem is fixed-parameter tractable by giving a kernel with at most 7k vertices. Barbero et al. (2017) obtained a kernel with less than 4k vertices and O(k) edges, improving the earlier result.In this paper, we study the parameterized complexity of the problem with respect to structural graph parameters. We show that 2-Load Coloring cannot be solved in time f(w)no(w), unless ETH fails and it can be solved in time nO(w), where n is the size of the input graph, w is the clique-width of the graph and f is an arbitrary function of w. Next, we consider the parameters distance to cluster graphs, distance to co-cluster graphs and distance to threshold graphs, which are weaker than the parameter clique-width and show that the problem is fixed-parameter tractable (FPT) with respect to these parameters. Finally, we show that 2-Load Coloring is NP-complete even on bipartite graphs and split graphs.

Small Vertex Cover Helps in Fixed-Parameter Tractability of Graph Deletion Problems over Data Streams

In the study of parameterized streaming complexity on graph problems, the main goal is to design streaming algorithms for parameterized problems such that O(f(k)logO(1)n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(f(k) \\log ^{\\mathcal {O}(1)} n)$$\\end{document} space is enough, where f is an arbitrary computable function depending only on the parameter k. However, in the past few years very few positive results have been established. Most of the graph problems that do have streaming algorithms of the above nature are ones where localized checking is required, like Vertex Cover or Maximum Matching parameterized by the size k of the solution we are seeking. Chitnis et al. (SODA’16) have shown that many important parameterized problems that form the backbone of traditional parameterized complexity are known to require Ω(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega (n)$$\\end{document} bits of storage for any streaming algorithm; e.g. Feedback Vertex Set, Even Cycle Transversal, Odd Cycle Transversal, Triangle Deletion or the more general F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{F}$$\\end{document}-Subgraph Deletion when parameterized by solution size k. Our contribution lies in overcoming the obstacles to efficient parameterized streaming algorithms in graph deletion problems by utilizing the power of parameterization. We focus on the vertex cover size K as the parameter for the parameterized graph deletion problems we consider. In this work, we consider the four most well-studied streaming models: the Ea, Dea, Va (vertex arrival) and Al (adjacency list) models. Surprisingly, the consideration of vertex cover size K in the different models leads to a classification of positive and negative results for problems like F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{F}$$\\end{document}-Subgraph Deletion and F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{F}$$\\end{document}-Minor Deletion.

Clustering what Matters: Optimal Approximation for Clustering with Outliers

Clustering with outliers is one of the most fundamental problems in Computer Science. Given a set X of n points and two numbers k, m, the clustering with outliers aims to exclude m points from X and partition the remaining points into k clusters that minimizes a certain cost function. In this paper, we give a general approach for solving clustering with outliers, which results in a fixed-parameter tractable (FPT) algorithm in k and m—i.e., an algorithm with running time of the form f(k, m) · nO(1) for some function f—that almost matches the approximation ratio for its outlier-free counterpart. As a corollary, we obtain FPT approximation algorithms with optimal approximation ratios for k-Median and k-Means with outliers in general and Euclidean metrics. We also exhibit more applications of our approach to other variants of the problem that impose additional constraints on the clustering, such as fairness or matroid constraints.

Answer Counting under Guarded TGDs

We study the complexity of answer counting for ontology-mediated queries and for querying under constraints, considering conjunctive queries and unions thereof (UCQs) as the query language and guarded TGDs as the ontology and constraint language, respectively. Our main result is a classification according to whether answer counting is fixed-parameter tractable (FPT), W[1]-equivalent, #W[1]-equivalent, #W[2]-hard, or #A[2]-equivalent, lifting a recent classification for UCQs without ontologies and constraints due to Dell et al. The classification pertains to various structural measures, namely treewidth, contract treewidth, starsize, and linked matching number. Our results rest on the assumption that the arity of relation symbols is bounded by a constant and, in the case of ontology-mediated querying, that all symbols from the ontology and query can occur in the data (so-called full data schema). We also study the meta-problems for the mentioned structural measures, that is, to decide whether a given ontology-mediated query or constraint-query specification is equivalent to one for which the structural measure is bounded.

Even More Effort Towards Improved Bounds and Fixed-Parameter Tractability for Multiwinner Rules

Even More Effort Towards Improved Bounds and Fixed-Parameter Tractability for Multiwinner Rules