Journal of Computer and System Sciences最新文献

Codd's rule of entity integrity stipulates that every table has a primary key. Key sets can control entity integrity when primary keys do not exist. While key set validation is quadratic, update maintenance for unary key sets is efficient when incomplete values only occur in few key columns. We establish a binary axiomatization for the implication problem, and prove its coNP-completeness. However, the implication of unary by arbitrary key sets has better properties. The fragment enjoys a unary axiomatization and is decidable in quadratic time. Hence, we can minimize overheads before validating key sets. While Armstrong relations do not always exist, we show how to compute them for any instance of our fragment. Similarly, we show how unary keys sets can be mined from relations using hypergraph transversals. Finally, we establish an axiomatization and computational complexity for the implication problem of key sets combined with NOT NULL constraints.

The problem of deletion of vertices to a hereditary graph class is a well-studied problem in parameterized complexity. Recently, a natural extension of the problem was initiated where we are given a finite set of hereditary graph classes and we determine whether k vertices can be deleted from a given graph so that the connected components of the resulting graph belong to one of the given hereditary graph classes. The problem is shown to be fixed parameter tractable (FPT) when the deletion problem to each of the given hereditary graph classes is fixed-parameter tractable, and the property of being in any of the graph classes is expressible in the counting monodic second order (CMSO) logic. This paper focuses on pairs of specific graph classes (${\Pi }_1,{\Pi }_2$) in which we would like the connected components of the resulting graph to belong to, and design simpler and more efficient FPT algorithms.

In the Categorical Clustering problem, we are given a set of vectors (matrix) $A=\{a_1,\dots ,a_n\}$ over ${\Sigma }^m$, where Σ is a finite alphabet, and integers k and B. The task is to partition A into k clusters such that the median objective of the clustering in the Hamming norm is at most B. Fomin, Golovach, and Panolan [ICALP 2018] proved that the problem is fixed-parameter tractable for the binary case $\Sigma =\{0,1\}$. We extend this algorithmic result to a popular capacitated clustering model, where in addition the sizes of the clusters are lower and upper bounded by integer parameters p and q, respectively. Our main theorem is that the problem is solvable in time $2^{O(B\log B)}|\Sigma {|}^B\cdot {\left(mn\right)}^{O\left(1\right)}$.

Over the past few decades, a large body of works studied the (in)tractability of various computational problems on grid graphs, which often yield substantially faster algorithms than general graphs. Unfortunately, the recognition of a grid graph is hard—it was shown to be NP-hard already in 1987. In this paper, we provide several positive results in this regard in the framework of parameterized complexity. Specifically, our contribution is threefold. First, we show that the problem is FPT parameterized by $k+\mathrm{mcc}$ where $\mathrm{mcc}$ is the maximum size of a connected component of G. Second, we present a new parameterization, denoted $a_G$, relating graph distance to geometric distance. We show that the problem is para-NP-hard parameterized by $a_G$, but FPT parameterized by $a_G$ on trees, as well as FPT parameterized by $k+a_G$. Third, we show that the recognition of $k\times r$ grid graphs is NP-hard on graphs of pathwidth 2 where $k=3$.

We aim at investigating the solvability/insolvability of nondeterministic logarithmic-space (NL) decision, search, and optimization problems parameterized by natural size parameters using simultaneously polynomial time and sub-linear space. We are particularly focused on ${\mathrm{2SAT}}_3$—a restricted variant of the 2CNF Boolean (propositional) formula satisfiability problem in which each variable of a given 2CNF formula appears at most 3 times in the form of literals—parameterized by the total number $m_{vbl}\left(\varphi \right)$ of variables of each given Boolean formula ϕ. We propose a new, practical working hypothesis, called the linear space hypothesis (LSH), which asserts that $({\mathrm{2SAT}}_3,m_{vbl})$ cannot be solved in polynomial time using only “sub-linear” space (i.e., $m_{vbl}{\left(x\right)}^{\epsilon }polylog\left(\right|x\left|\right)$ space for a constant $\epsilon \in [0,1)$) on all instances x. Immediate consequences of LSH include $L\ne \mathrm{NL}$, $\mathrm{LOGDCFL}\ne \mathrm{LOGCFL}$, and $\mathrm{SC}\ne \mathrm{NSC}$. For our investigation, we fully utilize a key notion of “short reductions”, under which the class PsubLIN of all parameterized polynomial-time sub-linear-space solvable problems is indeed closed.

Given a graph G on n vertices and two integers k and d, the Contraction(vc) problem asks whether one can contract at most k edges to reduce the vertex cover number of G by at least d. Recently, Lima et al. [JCSS 2021] proved that Contraction(vc) admits an XP algorithm running in time $f\left(d\right)\cdot n^{O\left(d\right)}$. They asked whether this problem is FPT under this parameterization. In this article, we prove that: (i) Contraction(vc) is W[1]-hard parameterized by $k+d$. Moreover, unless the ETH fails, the problem does not admit an algorithm running in time $f(k+d)\cdot n^{o(k+d)}$ for any function f. This answers negatively the open question stated in Lima et al. [JCSS 2021]. (ii) Contraction(vc) is NP-hard even when $k=d$. (iii) Contraction(vc) can be solved in time $2^{O\left(d\right)}\cdot n^{k-d+O\left(1\right)}$. This improves the algorithm of Lima et al. [JCSS 2021], and shows that when $k=d$, Contraction(vc) is FPT parameterized by d (or by k).

Given a graph G and two independent sets $I_s$ and $I_t$ of size k, the Independent Set Reconfiguration problem asks whether there exists a sequence of independent sets that transforms $I_s$ to $I_t$ such that each independent set is obtained from the previous one using a so-called reconfiguration step. Viewing each independent set as a collection of k tokens placed on the vertices of a graph G, the two most studied reconfiguration steps are token jumping and token sliding. Over a series of papers, it was shown that the Token Jumping problem is fixed-parameter tractable (for parameter k) when restricted to sparse graph classes, such as planar, bounded treewidth, and nowhere dense graphs. As for the Token Sliding problem, almost nothing is known. We remedy this situation by showing that Token Sliding is fixed-parameter tractable on graphs of bounded degree, planar graphs, and chordal graphs of bounded clique number.

We contribute to the refined understanding of language-logic-algebra interplay in a recent algebraic framework over countable words. Algebraic characterizations of the one variable fragment of FO as well as the boolean closure of the existential fragment of FO are established. We develop a seamless integration of the block product operation in the countable setting, and generalize well-known decompositional characterizations of FO and its two variable fragment. We propose an extension of FO admitting infinitary quantifiers to reason about inherent infinitary properties of countable words, and obtain a natural hierarchical block-product based characterization of this extension. Properties expressible in this extension can be simultaneously expressed in the classical logical systems such as WMSO and FO[cut]. We also rule out the possibility of a finite-basis for a block-product based characterization of these logical systems. Finally, we report algebraic characterizations of one variable fragments of the hierarchies of the new extension.

We consider Internet of Things (IoT) sensors deployed inside an area to be monitored. Drones can be used to collect the data from the sensors, but they are constrained in energy and storage. Therefore, all drones need to select a subset of sensors whose data are the most relevant to be acquired, modeled by assigning a reward. We present an optimization problem called Multiple-drone Data-collection Maximization Problem (MDMP) whose objective is to plan a set of drones' missions aimed at maximizing the overall reward from the collected data, and such that each individual drone's mission energy cost and total collected data are within the energy and storage limits, respectively. We optimally solve MDMP by proposing an Integer Linear Programming based algorithm. Since MDMP is NP-hard, we devise suboptimal algorithms for single- and multiple-drone scenarios. Finally, we thoroughly evaluate our algorithms on the basis of random generated synthetic data.

Floyd and Knuth investigated in 1990 register machines which can add, subtract and compare integers as primitive operations. They asked whether their current bound on the number of registers for multiplying and dividing fast (running in time linear in the size of the input) can be improved and whether one can output fast the powers of two summing up to a positive integer in subquadratic time. Both questions are answered positively. Furthermore, it is shown that every function computed by only one register is automatic and that the automatic functions with one input can be computed with four registers in linear time; automatic functions with a larger number of inputs can be computed with 5 registers in linear time. There is a nonautomatic function with one input which can be computed with two registers in linear time.