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The notion of forbidden-transition graphs allows for a robust generalization of walks in graphs. In a forbidden-transition graph, every pair of edges incident to a common vertex is permitted or forbidden; a walk is compatible if all pairs of consecutive edges on the walk are permitted. Forbidden-transition graphs and related models have found applications in a variety of fields, such as routing in optical telecommunication networks, road networks, and bio-informatics. A widely-studied special case are edge-colored graphs, where a compatible walk is forbidden to take two edges of the same color in a row. We initiate the study of fundamental problems on finding paths, cycles and walks in forbidden-transition graphs from the point of view of parameterized complexity, including an in-depth study of tractability with regards to various graph-width parameters. Among several results, we prove that finding a simple compatible path between given endpoints in a forbidden-transition graph is W[1]-hard when parameterized by the vertex-deletion distance to a linear forest (so it is also hard when parameterized by pathwidth or treewidth). On the other hand, we show an algebraic trick that yields tractability when parameterized by treewidth for finding a compatible Hamiltonian cycle in the edge-colored graph setting.

Treewidth and hypertree width have proven to be highly successful structural parameters in the context of the Constraint Satisfaction Problem (CSP). When either of these parameters is bounded by a constant, then CSP becomes solvable in polynomial time. However, here the order of the polynomial in the running time depends on the width, and this is known to be unavoidable; therefore, the problem is not fixed-parameter tractable parameterized by either of these width measures. Here we introduce an enhancement of tree and hypertree width through a novel notion of thresholds, allowing the associated decompositions to take into account information about the computational costs associated with solving the given CSP instance. Aside from introducing these notions, we obtain efficient theoretical as well as empirical algorithms for computing threshold treewidth and hypertree width and show that these parameters give rise to fixed-parameter algorithms for CSP as well as other, more general problems. We complement our theoretical results with experimental evaluations in terms of heuristics as well as exact methods based on SAT/SMT encodings., Comment: 24 pages, 4 figures. An extended abstract appeared at IJCAI 2020. A full version appeared in the Journal of Artificial Intelligence Research

We study generalizations of stable matching in which agents may be matched fractionally; this models time-sharing assignments. We focus on the so-called ordinal stability and cardinal stability, and investigate the computational complexity of finding an ordinally stable or cardinally stable fractional matching which either maximizes the social welfare (i.e., the overall utilities of the agents) or the number of fully matched agents (i.e., agents whose matching values sum up to one). We complete the complexity classification of both optimization problems for both ordinal stability and cardinal stability, distinguishing between the marriage (bipartite) and roommates (non-bipartite) cases and the presence or absence of ties in the preferences. In particular, we prove a surprising result that finding a cardinally stable fractional matching with maximum social welfare is NP-hard even for the marriage case without ties. This answers an open question and exemplifies a rare variant of stable marriage that remains hard for preferences without ties. We also complete the picture of the relations of the stability notions and derive structural properties.

We study the NP-complete Minimum Shared Edges (MSE) problem. Given an undirected graph, a source and a sink vertex, and two integers p and k, the question is whether there are p paths in the graph connecting the source with the sink and sharing at most k edges. Herein, an edge is shared if it appears in at least two paths. We show that MSE is W[1]-hard when parameterized by the treewidth of the input graph and the number k of shared edges combined. We show that MSE is fixed-parameter tractable with respect to p, but does not admit a polynomial-size kernel (unless NP is contained in coNP/poly). In the proof of the fixed-parameter tractability of MSE parameterized by p, we employ the treewidth reduction technique due to Marx, O'Sullivan, and Razgon [ACM TALG 2013]., 35 pages, 16 figures

We study the computational complexity of routing multiple objects through a network while avoiding collision: Given a graph G with two distinct terminals and two positive integers p , k , the question is whether one can connect the terminals by at least p routes (walks, trails, paths; the latter two without repeated edges or vertices, respectively) such that in at most k edges it happens that we traverse them in more than one route at the same time. We prove that for paths and trails the problem is NP-hard on undirected and directed planar graphs even if the maximum vertex degree or k ≥ 0 is constant. For walks we prove polynomial-time solvability on undirected graphs for unbounded k and on directed graphs if k ≥ 0 is constant. We additionally study variants where the maximum length of a route is restricted. For walks this variant becomes NP-hard on undirected graphs.

A point visibility graph is a graph induced by a set of points in the plane where the vertices of the graph represent the points in the point set and two vertices are adjacent if and only if no other point from the point set lies on the line segment between the two corresponding points. The set of all point visibility graphs form a graph class which is examined from a computational complexity perspective in this paper. We show NP-hardness for several classic graph problems on point visibility graphs such as Feedback Vertex Set , Longest Induced Path , Bisection and F - Free Vertex Deletion (for certain sets F ). Furthermore, we consider the complexity of the Dominating Set problem on point visibility graphs of points on a grid.

We study Open image in new window , which Alcalde [Economic Design, 1995] introduced as an alternative solution concept for matching markets involving property rights, such as assigning persons to two-bed rooms. Here, a matching of a given Stable Marriage or Stable Roommates instance is called Open image in new window if it does not admit any Open image in new window , that is, a subset S of agents in which everyone prefers the partner of some other agent in S. The matching is Open image in new window if it does not admit any Open image in new window , that is, an exchange-blocking coalition of size two.

A fundamental graph problem is to recognize whether the vertex set of a graph $G$ can be bipartitioned into sets $A$ and $B$ such that $G[A]$ and $G[B]$ satisfy properties $\Pi_A$ and $\Pi_B$, respectively. This so-called $(\Pi_A,\Pi_B)$-Recognition problem generalizes amongst others the recognition of $3$-colorable, bipartite, split, and monopolar graphs. In this paper, we study whether certain fixed-parameter tractable $(\Pi_A,\Pi_B)$-Recognition problems admit polynomial kernels. In our study, we focus on the first level above triviality, where $\Pi_A$ is the set of $P_3$-free graphs (disjoint unions of cliques, or cluster graphs), the parameter is the number of clusters in the cluster graph $G[A]$, and $\Pi_B$ is characterized by a set $\mathcal{H}$ of connected forbidden induced subgraphs. We prove that, under the assumption that NP is not a subset of coNP/poly, \textsc{$(\Pi_A,\Pi_B)$-Recognition} admits a polynomial kernel if and only if $\mathcal{H}$ contains a graph with at most $2$ vertices. In both the kernelization and the lower bound results, we exploit the properties of a pushing process, which is an algorithmic technique used recently by Heggerness et al. and by Kanj et al. to obtain fixed-parameter algorithms for many cases of $(\Pi_A,\Pi_B)$-Recognition, as well as several other problems., Comment: Full version of the corresponding article in the Proceedings of the 26th Annual European Symposium on Algorithms (ESA '18), 35 pages, 7 figures

We introduce and study the Minimum Feasible Tileset problem: given a set of symbols and subsets of these symbols (scenarios), find a smallest possible number of pairs of symbols (tiles) such that each scenario can be formed by selecting at most one symbol from each tile. We show that this problem is $$\mathsf {APX}$$ -hard and that it is $$\mathsf {NP}$$ -hard even if each scenario contains at most three symbols. Our main result is a 4/3-approximation algorithm for the general case. In addition, we show that the Minimum Feasible Tileset problem is fixed-parameter tractable both when parameterized with the number of scenarios and with the number of symbols.

We propose two solution concepts for matchings under preferences: robustness and near stability. The former strengthens while the latter relaxes the classic definition of stability by Gale and Shapley (1962). Informally speaking, robustness requires that a matching must be stable in the classic sense, even if the agents slightly change their preferences. Near stability, on the other hand, imposes that a matching must become stable (again, in the classic sense) provided the agents are willing to adjust their preferences a bit. Both of our concepts are quantitative; together they provide means for a fine-grained analysis of stability for matchings. Moreover, our concepts allow to explore the trade-offs between stability and other criteria of societal optimality, such as the egalitarian cost and the number of unmatched agents. We investigate the computational complexity of finding matchings that implement certain predefined trade-offs. We provide a polynomial-time algorithm that given agents' preferences returns a socially optimal robust matching, and we prove that finding a socially optimal and nearly stable matching is computationally hard., Comment: 47 pages. Accepted for presentation at the 20th ACM Conference on Economics and Computation (EC '19)

An author's profile on Google Scholar consists of indexed articles and associated data, such as the number of citations and the H-index. The author is allowed to merge articles; this may affect the H-index. We analyze the (parameterized) computational complexity of maximizing the H-index using article merges. Herein, to model realistic manipulation scenarios, we define a compatibility graph whose edges correspond to plausible merges. Moreover, we consider several different measures for computing the citation count of a merged article. For the measure used by Google Scholar, we give an algorithm that maximizes the H-index in linear time if the compatibility graph has constant-size connected components. In contrast, if we allow to merge arbitrary articles (that is, for compatibility graphs that are cliques), then already increasing the H-index by one is NP-hard. Experiments on Google Scholar profiles of AI researchers show that the H-index can be manipulated substantially only if one merges articles with highly dissimilar titles., Manuscript accepted to Artificial Intelligence

Given an undirected graph $G=(V,E)$ the NP-hard Strong Triadic Closure (STC) problem asks for a labeling of the edges as \emph{weak} and \emph{strong} such that at most $k$ edges are weak and for each induced $P_3$ in $G$ at least one edge is weak. In this work, we study the following generalizations of STC with $c$ different strong edge colors. In Multi-STC an induced $P_3$ may receive two strong labels as long as they are different. In Edge-List Multi-STC and Vertex-List Multi-STC we may additionally restrict the set of permitted colors for each edge of $G$. We show that, under the Exponential Time Hypothesis (ETH), Edge-List Multi-STC and Vertex-List Multi-STC cannot be solved in time $2^{o(|V|^2)}$. We then proceed with a parameterized complexity analysis in which we extend previous fixed-parameter tractability results and kernelizations for STC [Golovach et al., Algorithmica '20, Gr\"uttemeier and Komusiewicz, Algorithmica '20] to the three variants with multiple edge colors or outline the limits of such an extension., Comment: 29 pages

Funnels are a new natural subclass of DAGs. Intuitively, a DAG is a funnel if every source-sink path can be uniquely identified by one of its arcs. Funnels are an analog to trees for directed graphs that is more restrictive than DAGs but more expressive than in-/out-trees. Computational problems such as finding vertex-disjoint paths or tracking the origin of memes remain NP-hard on DAGs while on funnels they become solvable in polynomial time. Our main focus is the algorithmic complexity of finding out how funnel-like a given DAG is. To this end, we study the NP-hard problem of computing the arc-deletion distance to a funnel of a given DAG. We develop efficient exact and approximation algorithms for the problem and test them on synthetic random graphs and real-world graphs.

We investigate the computational complexity of the Densestk-Subgraph problem, where the input is an undirected graph G=(V,E) and one wants to find a subgraph on exactly k vertices with the maximum number of edges. We extend previous work on Densestk-Subgraph by studying its parameterized complexity for parameters describing the sparseness of the input graph and for parameters related to the solution size k.On the positive side, we show that, when fixing some constant minimum density of the sought subgraph, Densestk-Subgraph becomes fixed-parameter tractable with respect to either of the parameters maximum degree of G and h-index of G. Furthermore, we obtain a fixed-parameter algorithm for Densestk-Subgraph with respect to the combined parameter degeneracy of G and |V|k.On the negative side, we find that Densestk-Subgraph is W[1]-hard with respect to the combined parameter solution size k and degeneracy of G. We furthermore strengthen a previous hardness result for Densestk-Subgraph (Cai, 2008) by showing that for every fixed , 0<

The centrality of a vertex v in a network intuitively captures how important v is for communication in the network. The task of improving the centrality of a vertex has many applications, as a higher centrality often implies a larger impact on the network or less transportation or administration cost. In this work we study the parameterized complexity of the NP-complete problems Closeness Improvement and Betweenness Improvement in which we ask to improve a given vertex’ closeness or betweenness centrality by a given amount through adding a given number of edges to the network. Herein, the closeness of a vertex v sums the multiplicative inverses of distances of other vertices to v and the betweenness sums for each pair of vertices the fraction of shortest paths going through v. Unfortunately, for the natural parameter “number of edges to add” we obtain hardness results, even in rather restricted cases. On the positive side, we also give an island of tractability for the parameter measuring the vertex deletion distance to cluster graphs.

Dynamics of interactions play an increasingly important role in the analysis of complex networks. A modeling framework to capture this is temporal graphs which consist of a set of vertices (entities in the network) and a set of time-stamped binary interactions between the vertices. We focus on enumerating Δ-cliques, an extension of the concept of cliques to temporal graphs: for a given time period Δ, a Δ-clique in a temporal graph is a set of vertices and a time interval such that all vertices interact with each other at least after every Δ time steps within the time interval. Viard, Latapy, and Magnien (ASONAM 2015, TCS 2016) proposed a greedy algorithm for enumerating all maximal Δ-cliques in temporal graphs. In contrast to this approach, we adapt the Bron–Kerbosch algorithm—an efficient, recursive backtracking algorithm which enumerates all maximal cliques in static graphs—to the temporal setting. We obtain encouraging results both in theory (concerning worst-case running time analysis based on the parameter “Δ-slice degeneracy” of the underlying graph) as well as in practice with experiments on real-world data. The latter culminates in an improvement for most interesting Δ-values concerning running time in comparison with the algorithm of Viard, Latapy, and Magnien.

Multi-layer graphs consist of several graphs (layers) over the same vertex set. They are motivated by real-world problems where entities (vertices) are associated via multiple types of relationships (edges in different layers). We chart the border of computational (in)tractability for the class of subgraph detection problems on multi-layer graphs, including fundamental problems such as maximum matching, finding certain clique relaxations (motivated by community detection), or path problems. Mostly encountering hardness results, sometimes even for two or three layers, we can also spot some islands of tractability.

We study the computational complexity of routing multiple objects through a network in such a way that only few collisions occur: Given a graph G with two distinct terminal vertices and two positive integers p and k, the question is whether one can connect the terminals by at least p routes (e.g. paths) such that at most k edges are time-wise shared among them. We study three types of routes: traverse each vertex at most once (paths), each edge at most once (trails), or no such restrictions (walks). We prove that for paths and trails the problem is \({\text {NP}}\)-complete on undirected and directed graphs even if k is constant or the maximum vertex degree in the input graph is constant. For walks, however, it is solvable in polynomial time on undirected graphs for arbitrary k and on directed graphs if k is constant. We additionally study for all route types a variant of the problem where the maximum length of a route is restricted by some given upper bound. We prove that this length-restricted variant has the same complexity classification with respect to paths and trails, but for walks it becomes \({\text {NP}}\)-complete on undirected graphs.

Golomb rulers are special rulers where for any two marks it holds that the distance between them is unique. They find applications in radio frequency selection, radio astronomy, data encryption, communication networks, and bioinformatics. An important subproblem in constructing "compact" Golomb rulers is Golomb Subruler (GSR), which asks whether it is possible to make a given ruler Golomb by removing at most $$k$$ k marks. We initiate a study of GSR from a parameterized complexity perspective. In particular, we consider a natural hypergraph characterization of rulers and investigate the construction and structure of the corresponding hypergraphs. We exploit their properties to derive polynomial-time data reduction rules that reduce a given instance of GSR to an equivalent one with $${{\mathrm{O}}}(k^3)$$ O ( k 3 ) marks. Finally, we complement a recent computational complexity study of GSR by providing a simplified reduction that shows NP-hardness even when all integers are bounded by a polynomial in the input length.

This work studies the parameterized complexity of finding secluded solutions to classical combinatorial optimization problems on graphs such as finding minimum s-t separators, feedback vertex sets, dominating sets, maximum independent sets, and vertex deletion problems for hereditary graph properties: Herein, one searches not only to minimize or maximize the size of the solution, but also to minimize the size of its neighborhood. This restriction has applications in secure routing and community detection., Compared to the previous version, this version additionally shows that Small Secluded s-t-Separator is fixed-parameter tractable parameterized by the combination of the solution size and the open neighborhood size (Theorem 3.5). To appear in Discrete Optimization

Visualizing hypergraphs, systems of subsets of some universe, has continuously attracted research interest in the last decades. We study a natural kind of hypergraph visualization called subdivision drawings. Dinkla et al. [Comput. Graph. Forum ’12] claimed that only few hypergraphs have a subdivision drawing. However, this statement seems to be based on the assumption (also used in previous work) that the input hypergraph does not contain twins, pairs of vertices which are in precisely the same hyperedges (subsets of the universe). We show that such vertices may be necessary for a hypergraph to admit a subdivision drawing. As a counterpart, we show that the number of such “necessary twins” is upper-bounded by a function of the number m of hyperedges and a further parameter r of the desired drawing related to its number of layers. This leads to a linear-time algorithm for determining such subdivision drawings if m and r are constant; in other words, the problem is linear-time fixed-parameter tractable with respect to the parameters m and r.

We provide a new characterization of the NP-hard arc routing problem Rural Postman in terms of a constrained variant of minimum-weight perfect matching on bipartite graphs. To this end, we employ a parameterized equivalence between Rural Postman and Eulerian Extension, a natural arc addition problem in directed multigraphs. We indicate the NP-hardness of the introduced matching problem. In particular, we use the matching problem to make partial progress towards answering the open question about the parameterized complexity of Rural Postman with respect to the parameter “number of weakly connected components in the graph induced by the required arcs”. This is a more than thirty years open and long-neglected question with significant practical relevance.

We propose new practical algorithms to find maximum-cardinality k-plexes in graphs. A k-plex denotes a vertex subset in a graph inducing a subgraph where every vertex has edges to all but at most k vertices in the k-plex. Cliques are 1-plexes. In analogy to the special case of finding maximum-cardinality cliques, finding maximum-cardinality k-plexes is NP-hard. Complementing previous work, we develop exact combinatorial algorithms, which are strongly based on methods from parameterized algorithmics. The experiments with our freely available implementation indicate the competitiveness of our approach, for many real-world graphs outperforming the previously used methods.

We prove that any polynomial-time $\alpha(n)$-approximation algorithm for the $n$-vertex metric asymmetric Traveling Salesperson Problem yields a polynomial-time $O(\alpha(C))$-approximation algorithm for the mixed and windy Capacitated Arc Routing Problem, where $C$ is the number of weakly connected components in the subgraph induced by the positive-demand arcs---a small number in many applications. In conjunction with known results, we obtain constant-factor approximations for $C\in O(\log n)$ and $O(\log C/\log\log C)$-approximations in general. Experiments show that our algorithm, together with several heuristic enhancements, outperforms many previous polynomial-time heuristics. Finally, since the solution quality achievable in polynomial time appears to mainly depend on $C$ and since $C=1$ in almost all benchmark instances, we propose the Ob benchmark set, simulating cities that are divided into several components by a river., Comment: A preliminary version of this article appeared in the Proceedings of the 15th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems (ATMOS'15). This version describes several algorithmic enhancements, contains an experimental evaluation of our algorithm, and provides a new benchmark data set

We consider the Minimum Feasible Tileset problem: Given a set of symbols and subsets of these symbols (scenarios), find a smallest possible number of pairs of symbols (tiles) such that each scenario can be formed by selecting at most one symbol from each tile. We show that this problem is \(\mathsf {NP}\)-complete even if each scenario contains at most three symbols. Our main result is a 4/3-approximation algorithm for the general case. In addition, we show that the Minimum Feasible Tileset problem is fixed-parameter tractable both when parameterized with the number of scenarios and with the number of symbols.

In the Vector Connectivity problem we are given an undirected graph $G=(V,E)$, a demand function $\phi\colon V\to\{0,\ldots,d\}$, and an integer $k$. The question is whether there exists a set $S$ of at most $k$ vertices such that every vertex $v\in V\setminus S$ has at least $\phi(v)$ vertex-disjoint paths to $S$; this abstractly captures questions about placing servers or warehouses relative to demands. The problem is \NP-hard already for instances with $d=4$ (Cicalese et al., arXiv '14), admits a log-factor approximation (Boros et al., Networks '14), and is fixed-parameter tractable in terms of~$k$ (Lokshtanov, unpublished '14). We prove several results regarding kernelization and approximation for Vector Connectivity and the variant Vector $d$-Connectivity where the upper bound $d$ on demands is a fixed constant. For Vector $d$-Connectivity we give a factor $d$-approximation algorithm and construct a vertex-linear kernelization, i.e., an efficient reduction to an equivalent instance with $f(d)k=O(k)$ vertices. For Vector Connectivity we have a factor $\text{opt}$-approximation and we can show that it has no kernelization to size polynomial in $k$ or even $k+d$ unless $\mathsf{NP\subseteq coNP/poly}$, making $f(d)\operatorname{poly}(k)$ optimal for Vector $d$-Connectivity. Finally, we provide a write-up for fixed-parameter tractability of Vector Connectivity($k$) by giving an alternative FPT algorithm based on matroid intersection., Comment: Non-constructive Kernelization argument, improved technical details of signatures

International audience; The NP-hard Subset Interconnection Design problem, also known as Minimum Topic-Connected Overlay, is motivated by numerous applications including the design of scalable overlay networks and vacuum systems. It has as input a finite set V and a collection of subsets V 1 , V 2 ,. .. , Vm ⊆ V , and asks for a minimum-cardinality edge set E such that for the graph G = (V, E) all induced subgraphs G[V 1 ], G[V 2 ],. .. , G[Vm] are connected. We study Subset In-terconnection Design in the context of polynomial-time data reduction rules that preserve the possibility to construct optimal solutions. Our contribution is threefold: First, we show the incor-rectness of earlier polynomial-time data reduction rules. Second, we show linear-time solvability in case of a constant number m of subsets, implying fixed-parameter tractability for the parameter m. Third, we provide a fixed-parameter tractability result for small subset sizes and tree-like output graphs. To achieve our results, we elaborate on polynomial-time data reduction rules which also may be of practical use in solving Subset Interconnection Design.

The NP-hard Distinct Vectors problem asks to delete as many columns as possible from a matrix such that all rows in the resulting matrix are still pairwise distinct. Our main result is that, for binary matrices, there is a complexity dichotomy for Distinct Vectors based on the maximum (H) and the minimum (h) pairwise Hamming distance between matrix rows: Distinct Vectors can be solved in polynomial time if H, Accepted for publication in Journal of Computer and System Sciences (Elsevier)

A popular way of formalizing clusters in networks are highly connected subgraphs, that is, subgraphs of k vertices that have edge connectivity larger than k/2 (equivalently, minimum degree larger than k/2). We examine the computational complexity of finding highly connected subgraphs. We first observe that this problem is NP-hard. Thus, we explore possible parameterizations, such as the solution size, number of vertices in the input, the size of a vertex cover in the input, and the number of edges outgoing from the solution (edge isolation), and expose their influence on the complexity of this problem. For some parameters, we find strong intractability results; among the parameters yielding tractability, the edge isolation seems to provide the best trade-off between running time bounds and a small parameter value in relevant instances.

Given an undirected graph with edge costs and edge demands, the Capacitated Arc Routing problem ( CARP ) asks for minimum-cost routes for equal-capacity vehicles so as to satisfy all demands. Constant-factor polynomial-time approximation algorithms were proposed for CARP with triangle inequality, while CARP was claimed to be NP-hard to approximate within any constant factor in general. Correcting this claim, we show that any factor α approximation for CARP with triangle inequality yields a factor α approximation for the general CARP .

A balanced partition is a clustering of a graph into a given number of equal-sized parts. For instance, the Bisection problem asks to remove at most k edges in order to partition the vertices into two equal-sized parts. We prove that Bisection is FPT for the distance to constant cliquewidth if we are given the deletion set. This implies FPT algorithms for some well-studied parameters such as cluster vertex deletion number and feedback vertex set. However, we show that Bisection does not admit polynomial-size kernels for these parameters. For the Vertex Bisection problem, vertices need to be removed in order to obtain two equal-sized parts. We show that this problem is FPT for the number of removed vertices k if the solution cuts the graph into a constant number c of connected components. The latter condition is unavoidable, since we also prove that Vertex Bisection is W[1]-hard w.r.t. (k,c). Our algorithms for finding bisections can easily be adapted to finding partitions into d equal-sized parts, which entails additional running time factors of n^{O(d)}. We show that a substantial speed-up is unlikely since the corresponding task is W[1]-hard w.r.t. d, even on forests of maximum degree two. We can, however, show that it is FPT for the vertex cover number., Comment: This version of the article is to appear in Theory of Computing Systems

We examine the algorithmic tractability of NP-hard combinatorial feature selection problems in terms of parameterized complexity theory. In combinatorial feature selection, one seeks to discard dimensions from high-dimensional data such that the resulting instances fulfill a desired property. In parameterized complexity analysis, one seeks to identify relevant problem-specific quantities and tries to determine their influence on the computational complexity of the considered problem. In this paper, for various combinatorial feature selection problems, we identify parameterizations and reveal to what extent these govern computational complexity. We provide tractability as well as intractability results; for example, we show that the Distinct Vectors problem on binary points is polynomial-time solvable if each pair of points differs in at most three dimensions, whereas it is NP-hard otherwise.