Journal of Computer and System Sciences最新文献

The goal of this paper is to open up a new research direction aimed at understanding the power of preprocessing in speeding up algorithms that solve NP-hard problems exactly. We explore this direction for the classic Feedback Vertex Set problem on undirected graphs, leading to a new type of graph structure called antler decomposition, which identifies vertices that belong to an optimal solution. It is an analogue of the celebrated crown decomposition which has been used for Vertex Cover. We develop the graph structure theory around such decompositions and develop fixed-parameter tractable algorithms to find them, parameterized by the number of vertices for which they witness presence in an optimal solution. This reduces the search space of fixed-parameter tractable algorithms parameterized by the solution size that solve Feedback Vertex Set.

We introduce a cops and robbers game with one cop and one robber on a special type of time-varying graphs (TVGs), namely edge-periodic graphs. These are TVGs in which, for each edge e, a binary string $\tau \left(e\right)$ is given such that the edge e is present in time step t if and only if $\tau \left(e\right)$ contains a 1 at position $t\mathrm{mod}\left|\tau \right(e\left)\right|$. This periodicity allows for a compact representation of infinite TVGs. We prove that even for very simple underlying graphs, i.e., directed and undirected cycles, the problem of deciding whether a cop-winning strategy exists is NP-hard and $W\left[1\right]$-hard parameterized by the number of vertices. Furthermore, we show that this decision problem can be solved on general edge-periodic graphs in PSPACE. Finally, we present tight bounds on the minimum length of a directed or undirected cycle that guarantees the cycle to be robber-winning.

In this paper we study the kernelization of the d-Path Vertex Cover (d-PVC) problem. Given a graph G, the problem requires finding whether there exists a set of at most k vertices whose removal from G results in a graph that does not contain a path (not necessarily induced) with d vertices. It is known that d-PVC is NP-complete for $d\ge 2$. Since the problem generalizes to d-Hitting Set, it is known to admit a kernel with $O\left(d{k}^d\right)$ edges. We improve on this by giving better kernels. Specifically, we give kernels with $O\left(k^2\right)$ vertices and edges for the cases when $d=4$ and $d=5$. Further, we give a kernel with $O\left(k^4{d}^{2d+9}\right)$ vertices and edges for general d.

In this paper, we investigate the structure of the most general kind of substitution shifts, including non-minimal ones, and allowing erasing morphisms. We prove the decidability of many properties of these morphisms with respect to the shift space generated by iteration, such as aperiodicity, recognizability and (under an additional assumption) irreducibility, or minimality.

Arc-Disjoint Cycle Packing is a classical NP-complete problem and we study it from two perspectives: (1) by restricting the cycles in the packing to be of a fixed length, and (2) by restricting the inputs to bipartite tournaments. Focusing first on Arc-Disjoint r-Cycle Packing (where the cycles in the packing are required to be of length r), we show NP-completeness in oriented graphs with girth r for each $r\ge 3$ and study the parameterized complexity of the problem with respect to two parameterizations (solution size and vertex cover size) for $r=4$ in oriented graphs. Moving on to Arc-Disjoint Cycle Packing in bipartite tournaments, we show that every bipartite tournament either contains k arc-disjoint cycles or has a feedback arc set of size at most $7(k-1)$. This result adds to the set of Erdös-Pósa-type results known in the combinatorics literature for packing and covering problems.

A set M of vertices of a graph G is a distance-edge-monitoring set if for every edge $e\in G$, there is a vertex $x\in M$ and a vertex $y\in G$ such that e belongs to all shortest paths between x and y. We denote by $\mathrm{dem}\left(G\right)$ the smallest size of such a set in G. In this paper, we prove that $\mathrm{dem}\left(G\right)\le n-\lfloor g\left(G\right)/2\rfloor$ for any connected graph G, which is not a tree, of order n, where $g\left(G\right)$ is the length of a shortest cycle in G, and give the graphs with $\mathrm{dem}\left(G\right)=n-\lfloor g\left(G\right)/2\rfloor$. We also obtain that $\left|V\right(G\left)\right|\ge k+\lfloor g\left(G\right)/2\rfloor$ for every connected graph G with $\mathrm{dem}\left(G\right)=k$ and $g\left(G\right)=g$. Furthermore, the lower bound holds if and only if $g=3$ and $k=n-1$ or $g=4$ and $k=2$. We prove that $\mathrm{dem}\left(G\right)\le 2n/5$ for $g\left(G\right)\ge 5$.

We describe a kernel of size $9k-8$ for the NP-hard problem of computing the Tree Bisection and Reconnection (TBR) distance k between two unrooted binary phylogenetic trees. To achieve this, we extend the existing portfolio of reduction rules with three new reduction rules. Two of these are based on the idea of topologically transforming the trees in a distance-preserving way in order to guarantee execution of earlier reduction rules. The third rule extends the local neighborhood approach introduced in [20] to more global structures, allowing new situations to be identified when the deletion of a leaf definitely reduces the TBR distance by one. The bound on the kernel size is tight up to an additive term. Our results also apply to the equivalent problem of computing a maximum agreement forest between two unrooted binary phylogenetic trees. We anticipate that our results are widely applicable for computing agreement-forest based dissimilarity measures.

For several decades, much effort has been put into identifying classes of CNF formulas whose satisfiability can be decided in polynomial time. Classic results are the linear-time tractability of Horn formulas (Aspvall, Plass, and Tarjan, 1979) and Krom (i.e., 2CNF) formulas (Dowling and Gallier, 1984). Backdoors, introduced by Williams, Gomes and Selman (2003), gradually extend such a tractable class to all formulas of bounded distance to the class. Backdoor size provides a natural but rather crude distance measure between a formula and a tractable class. Backdoor depth, introduced by Mählmann, Siebertz, and Vigny (2021), is a more refined distance measure, which admits the utilization of different backdoor variables in parallel. We propose FPT approximation algorithms to compute backdoor depth into the classes Horn and Krom. This leads to a linear-time algorithm for deciding the satisfiability of formulas of bounded backdoor depth into these classes.

We introduce a coordination index in regulatory Boolean networks and we expose the maximal coordination principle (MCP), according to which a cohesive society reaches the dynamics characterized by the highest coordination index. Based on simple theoretical examples, we show that the MCP can be used to infer the influence graph from opinion dynamics/gene expressions. We provide some algorithms to apply the MCP and we compare the coordination index with existing statistical indexes (likelihood, entropy). The advantage of the coordination approach is its simplicity; in particular, we do not need to impose restrictions on the aggregation functions.

Fair clustering is a constrained clustering problem where we need to partition a set of colored points. The fraction of points of each color in every cluster should be more or less equal to the fraction of points of this color in the dataset. The problem was recently introduced by Chierichetti et al. (2017) [1]. We propose a new construction of coresets for fair clustering for Euclidean and general metrics based on random sampling. For the Euclidean space $R^d$, we provide the first coreset whose size does not depend exponentially on the dimension d. The question of whether such constructions exist was asked by Schmidt et al. (2019) [2] and Huang et al. (2019) [5]. For general metrics, our construction provides the first coreset for fair clustering. New coresets appear to be a handy tool for designing better approximation and streaming algorithms for fair and other constrained clustering variants.