Parameterized Complexity of Stabbing Rectangles and Squares in the Plane

A well known result in graph algorithms, due to Edmonds, states that given a digraph D and a positive integer $$\ell $$l, we can test whether D contains $$\ell $$l arc-disjoint out-branchings in polynomial time. However, if we ask whether there exists an out-branching and an in-branching which are arc-disjoint, then the problem becomes NP-complete. In fact, even deciding whether a digraph D contains an out-branching which is arc-disjoint from some spanning tree in the underlying undirected graph remains NP-complete. In this paper we formulate some natural optimization questions around these problems and initiate its study in the realm of parameterized complexity. More precisely, the problems we study are the following: Arc-Disjoint Branchings and Non-Disconnecting Out-Branching. In Arc-Disjoint Branchings (Non-Disconnecting Out-Branching), a digraph D and a positive integer k are given as input and the goal is to test whether there exist an out-branching and in-branching (respectively, a spanning tree in the underlying undirected graph) that differ on at least k arcs. We obtain the following results for these problems. Non-Disconnecting Out-Branching is fixed parameter tractable (FPT) and admits a linear vertex kernel. Arc-Disjoint Branchings is FPT on strong digraphs. The algorithm for Non-Disconnecting Out-Branching runs in time $$2^{\mathcal {O}(k)}n^{\mathcal {O}(1)}$$2O(k)nO(1) and the approach we use to obtain this algorithms seems useful in designing other moderately exponential time algorithms for edge/arc partitioning problems.

For fixed integers r,l >= 0, a graph G is called an (r,l)-graph if the vertex set V(G) can be partitioned into r independent sets and l cliques. Such a graph is also said to have cochromatic number r+l. The class of (r,l) graphs generalizes r-colourable graphs (when l=0) and hence not surprisingly, determining whether a given graph is an (r,l)-graph is NP-hard even when r >= 3 or l >= 3 in general graphs. When r and ell are part of the input, then the recognition problem is NP-hard even if the input graph is a perfect graph (where the Chromatic Number problem is solvable in polynomial time). It is also known to be fixed-parameter tractable (FPT) on perfect graphs when parameterized by r and l. I.e. there is an f(r+l) n^O(1) algorithm on perfect graphs on n vertices where f is a function of r and l. Observe that such an algorithm is unlikely on general graphs as the problem is NP-hard even for constant r and l. In this paper, we consider the parameterized complexity of the following problem, which we call Vertex Partization. Given a perfect graph G and positive integers r,l,k decide whether there exists a set S subset or equal to V(G) of size at most k such that the deletion of S from G results in an (r,l)-graph. This problem generalizes well studied problems such as Vertex Cover (when r=1 and l=0), Odd Cycle Transversal (when r=2, l=0) and Split Vertex Deletion (when r=1=l). 1. Vertex Partization on perfect graphs is FPT when parameterized by k+r+l. 2. The problem, when parameterized by k+r+l, does not admit any polynomial sized kernel, under standard complexity theoretic assumptions. In other words, in polynomial time, the input graph cannot be compressed to an equivalent instance of size polynomial in k+r+l. In fact, our result holds even when k=0. 3. When r,ell are universal constants, then Vertex Partization on perfect graphs, parameterized by k, has a polynomial sized kernel.

This study explored the mechanisms that underlie asymmetries for the horizontal vertical illusion (HVI), which deceives length perception, so that a vertical line is perceived as longer than a horizontal line of equivalent length. In Experiment 1, university students (n=14) made length judgements for vertical and horizontal lines. The vertical line was shifted in eight steps from the far left of the horizontal line (⌊) to the far right (⌋). An HVI was observed for the medial positions (⊥), which diminished towards the lateral positions. The HVI was also stronger when the vertical line was on the left. Because the left/right asymmetry changed as a function of lateral/medial position, the asymmetry within the HVI stimulus is most likely the result of pseudoneglect, which affects judgements of horizontal length. In Experiment 2, participants (n=15) made judgements for HVI stimuli presented to the left- and right-hemispace and the midline. The HVI was stronger in the left hemispace. Because the asymmetry between the left- and right-hemispaces did not interact with the asymmetry within the stimuli, it was concluded that the asymmetry between hemispatial positions was the result of right hemisphere susceptibility to illusory geometrical effects whereas the asymmetry within the stimulus is related to an object-centred attentional asymmetry. The HVI is affected by asymmetries in length judgements and susceptibility to illusions and may provide interesting insights into attentional disorders in clinical populations, such as neglect.

In many practical settings it is useful to find a small unsatisfiable subset of a given unsatisfiable set of constraints. We study this problem from a parameterized complexity perspective, taking the size of the unsatisfiable subset as the natural parameter where the set of constraints is either (i) given a set of clauses, i.e., a formula in conjunctive normal Form (CNF), or (ii) as an instance of the Constraint Satisfaction Problem (CSP). In general, the problem is fixed-parameter in tractable. For an instance of the propositional satisfiability problem (SAT), it was known to be W[1]-complete. We establish A[2]-completeness for CSP instances, where A[2]-hardness prevails already for the Boolean case. With these fixed-parameter intractability results for the general case in mind, we consider various restricted classes of inputs and draw a detailed complexity landscape. It turns out that often Boolean CSP and CNF formulas behave similarly, but we also identify notable exceptions to this rule. The main part of this article is dedicated to classes of inputs that are induced by Boolean constraint languages that Schaefer [1978] identified as the maximal constraint languages with a tractable satisfiability problem. We show that for the CSP setting, the problem of finding small unsatisfiable subsets remains fixed-parameter intractable for all Schaefer languages for which the problem is non-trivial. We show that this is also the case for CNF formulas with the exception of the class of bijunctive (Krom) formulas, which allows for an identification of a small unsatisfiable subset in polynomial time. In addition, we consider various restricted classes of inputs with bounds on the maximum number of times that a variable occurs (the degree), bounds on the arity of constraints, and bounds on the domain size. For the case of CNF formulas, we show that restricting the degree is enough to obtain fixed-parameter tractability, whereas for the case of CSP instances, one needs to restrict the degree, the arity, and the domain size simultaneously to establish fixed-parameter tractability. Finally, we relate the problem of finding small unsatisfiable subsets of a set of constraints to the problem of identifying whether a given variable-value assignment is entailed or forbidden already by a small subset of constraints. Moreover, we use the connection between the two problems to establish similar parameterized complexity results also for the latter problem.

We consider the parameterized complexity of the UNIQUE COVERAGE problem: given a family of sets and a parameter k, we ask whether there exists a subfamily that covers at least k elements exactly once. This NP-complete problem has applications in wireless networks and radio broadcasting and is also a natural generalization of the well-known MAX CUT problem. We show that this problem is fixed-parameter tractable with respect to the parameter k. That is, for every fixed k, there exists a polynomial-time algorithm for it. One way to prove a problem fixed-parameter tractable is to show that it is kernelizable. To this end, we show that if no two sets in the input family intersect in more than c elements there exists a problem kernel of size kc+1. This yields a kk kernel for the UNIQUE COVERAGE problem, proving fixed-parameter tractability. Subsequently, we show a 4k kernel for this problem. However a more general weighted version, with costs associated with each set and profits with each element, turns out to be much harder. The question here is whether there exists a subfamily with total cost at most a prespecified budget B such that the total profit of uniquely covered elements is at least k, where B and k are part of the input. In the most general setting, assuming real costs and profits, the problem is not fixed-parameter tractable unless P = NP. Assuming integer costs and profits we show the problem to be W[1]-hard with respect to B as parameter (that is, it is unlikely to be fixed-parameter tractable). However, under some reasonable restriction, the problem becomes fixed-parameter tractable with respect to both B and k as parameters.

A binary matrix M has the Consecutive Ones Property (COP) if there exists a permutation of columns that arranges the ones consecutively in all the rows. We consider the parameterized complexity of d-COS-R (Consecutive Ones Submatrix by Row deletions) problem [8]: Given a matrix M and a positive integer d, decide whether there exists a set of at most d rows of M whose deletion results in a matrix with the COP. The closely related Interval Deletion problem has recently been shown to be FPT [5]. In this work, we describe a recursive depth-bounded search tree algorithm in which the problems at the leaf-level of the recursion tree are solved as instances of Interval Deletion. Therefore, we show that d-COS-R is fixed-parameter tractable and has the current best run-time of O *(10 d ), which is associated with the Interval Deletion problem. We then consider a closely related optimization problem, called Min-ICPIA, and prove that it is computationally equivalent to the Vertex Cover problem.

We investigate the parameterized complexity of Generalized Red Blue Set Cover (Gen-RBSC), a generalization of the classic Set Cover problem and the more recently studied Red Blue Set Cover problem. Given a universe U containing b blue elements and r red elements, positive integers \(k_\ell \) and \(k_r\), and a family \(\mathcal F \) of \(\ell \) sets over U, the Gen-RBSC problem is to decide whether there is a subfamily \(\mathcal F '\subseteq \mathcal F \) of size at most \(k_\ell \) that covers all blue elements, but at most \(k_r\) of the red elements. This generalizes Set Cover and thus in full generality it is intractable in the parameterized setting, when parameterized by \(k_\ell +k_r\). In this paper, we study Gen-RBSC-lines, where the elements are points in the plane and sets are defined by lines. We study this problem for the parameters \(k_\ell , k_r\), and \(k_\ell +k_r\). For all these cases, we either prove that the problem is W-hard or show that the problem is fixed parameter tractable (FPT). Finally, for the parameter \(k_\ell +k_r\), for which Gen-RBSC-lines admits FPT algorithms, we show that the problem does not have a polynomial kernel unless \(\text { co-NP}\subseteq \text { NP}/\mathrm{poly}\). Further, we show that the FPT algorithm does not generalize to higher dimensions.

The Neural Basis of Vertical and Horizontal Line Bisection Judgments: An fMRI Study of Normal Volunteers

In the Cycle Packing problem, we are given an undirected graph G, a positive integer r, and the task is to check whether there exist r vertex-disjoint cycles. In this paper, we study Cycle Packing with respect to a structural parameter, namely, distance to proper interval graphs (indifference graphs). In particular, we show that Cycle Packing is fixed-parameter tractable (FPT) when parameterized by t, the size of a proper interval deletion set. For this purpose, we design an algorithm with \(\mathcal {O}(2^{\mathcal {O}(t \log t)} n^{\mathcal {O}(1)})\) running time. Several structural parameterizations for Cycle Packing have been studied in the literature and our FPT algorithm fills a gap in the ecology of such parameterizations. We combine color coding, greedy strategy and dynamic programming based on structural properties of proper interval graphs in a non-trivial fashion to obtain the FPT algorithm.

A classical result by Edwards states that every connected graph G on n vertices and m edges has a cut of size at least \(\frac{m}{2}+\frac{n-1}{4}\). We generalize this result to r-hypergraphs, with a suitable notion of connectivity that coincides with the notion of connectivity on graphs for r = 2. More precisely, we show that for every “partition connected” r-hypergraph (every hyperedge is of size at most r) H over a vertex set V(H), and edge set E(H) = {e 1,e 2,…e m }, there always exists a 2-coloring of V(H) with {1, − 1} such that the number of hyperedges that have a vertex assigned 1 as well as a vertex assigned − 1 (or get “split”) is at least \(\mu_H+\frac{n-1}{r2^{r-1}}\). Here \(\mu_H=\sum_{i=1}^{m}(1- 2/2^{|e_i|})=\sum_{i=1}^{m}(1- 2^{1-|e_i|})\). We use our result to show that a version of r -Set Splitting, namely, Above Average r -Set Splitting (AA- r -SS), is fixed parameter tractable (FPT). Observe that a random 2-coloring that sets each vertex of the hypergraph H to 1 or − 1 with equal probability always splits at least μ H hyperedges. In AA- r -SS, we are given an r-hypergraph H and a positive integer k and the question is whether there exists a 2-coloring of V(H) that splits at least μ H + k hyperedges. We give an algorithm for AA- r -SS that runs in time f(k)n O(1), showing that it is FPT, even when r = c 1 logn, for every fixed constant c 1 < 1. Prior to our work AA- r -SS was known to be FPT only for constant r. We also complement our algorithmic result by showing that unless NP ⊆ DTIME(n loglogn), AA-⌈logn ⌉-SS is not in XP.KeywordsBoolean FunctionConnected GraphReduction RulePrimal GraphOrdinary GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

For fixed integers r,ell geq 0, a graph G is called an (r,ell)-graph if the vertex set V(G) can be partitioned into r independent sets and ell cliques. This brings us to the following natural parameterized questions: Vertex (r,ell)-Partization and Edge (r,ell)-Partization. An input to these problems consist of a graph G and a positive integer k and the objective is to decide whether there exists a set S subseteq V(G) (S subseteq E(G)) such that the deletion of S from G results in an (r,ell)-graph. These problems generalize well studied problems such as Odd Cycle Transversal, Edge Odd Cycle Transversal, Split Vertex Deletion and Split Edge Deletion. We do not hope to get parameterized algorithms for either Vertex (r, ell)-Partization or Edge (r,ell)-Partization when either of r or ell is at least 3 as the recognition problem itself is NP-complete. This leaves the case of r,ell in {1,2}. We almost complete the parameterized complexity dichotomy for these problems by obtaining the following results: - We show that Vertex (r,ell)-Partization is fixed parameter tractable (FPT) for r,ell in {1,2}. Then we design an Oh(sqrt log n)-factor approximation algorithms for these problems. These approximation algorithms are then utilized to design polynomial sized randomized Turing kernels for these problems. - Edge (r,ell)-Partization is FPT when (r,ell)in{(1,2),(2,1)}. However, the parameterized complexity of Edge (2,2)-Partization remains open. For our approximation algorithms and thus for Turing kernels we use an interesting finite forbidden induced graph characterization, for a class of graphs known as (r,ell)-split graphs, properly containing the class of (r,ell)-graphs. This approach to obtain approximation algorithms could be of an independent interest.

To facilitate obtaining the mean ordinate of a 60‐minute interval on the magnetograms obtained at the magnetic observatories of the Department of Terrestrial Magnetism, scales have been constructed in the Department's shops somewhat similar to the type introduced by the Potsdam Magnetic Observatory and as used by the United States Coast and Geodetic Survey. The only difference in our glass scale from the Potsdam one arises from the fact that the Potsdam type makes use of a double scale, the actual mean ordinate in millimeters being obtained from the sum of two readings, whereas we scale the mean ordinate in millimeters direct.The scale is a piece of plate glass 12.5 cm. wide by 20 cm. long, on which are ruled two sets of lines at right angles to each other. The vertical lines parallel to the long sides are 20 mm. apart, corresponding to the distance between hour‐breaks on the Eschenhagen magnetograms, there being however for direct scaling of ordinates an extra vertical line half‐way between the vertical lines inclosing the right‐hand 20 mm. of the scale. The horizontal lines parallel to the short sides are 10 mm. apart. The first space at the bottom is subdivided to millimeters by short horizontal lines midway between the vertical lines.

Given a graph G and a positive integer k, the 2-Load Coloring problem is to check whether there is a 2-coloring \(f:V(G) \rightarrow \{r,b\}\) of G such that for every \(i \in \{r,b\}\), there are at least k edges with both end vertices colored i. It is known that the problem is \(\mathsf {NP} \)-complete even on special classes of graphs like regular graphs. Gutin and Jones (Inf Process Lett 114:446-449, 2014) showed that the problem is fixed-parameter tractable by giving a kernel with at most 7k vertices. Barbero et al. (Algorithmica 79:211-229, 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 \(f(w)n^{o(w)}\), unless ETH fails and it can be solved in time \(n^{O(w)}\), 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 \(\mathsf {NP} \)-complete even on bipartite graphs and split graphs.

Long directed detours: Reduction to 2-Disjoint Paths

List-coloring – Parameterizing from triviality