Journal of Computer and System Sciences最新文献

The NP-hard Multiple Hitting Set problem is the problem of finding a minimum-cardinality set intersecting each of the sets in a given input collection a given number of times. Generalizing a well-known data reduction algorithm due to Weihe, we show a problem kernel for Multiple Hitting Set parameterized by the Dilworth number, a graph parameter introduced by Foldes and Hammer in 1978 yet seemingly so far unexplored in the context of parameterized complexity theory. Using matrix multiplication, we speed up the algorithm to quadratic sequential time and logarithmic parallel time. We experimentally evaluate our algorithms. By implementing our algorithm on GPUs, we show the feasibility of realizing kernelization algorithms on SIMD (Single Instruction, Multiple Data) architectures.

We generalize the tree doubling and Christofides algorithm to parameterized approximations for ATSP (constant factor approximations that invest more runtime with respect to a chosen parameter). The parameters we consider are upper bounded by the number of asymmetric distances, which yields algorithms to efficiently compute good approximations for moderately asymmetric TSP instances. As generalization of the Christofides algorithm, we derive a parameterized 2.5-approximation, with the size of a vertex cover for the subgraph induced by the edges with asymmetric distances as parameter. Our generalization of tree doubling gives a parameterized 3-approximation, where the parameter is the minimum number of asymmetric distances in a minimum spanning arborescence. Further, we combine these with a notion of symmetry relaxation which allows to trade approximation guarantee for runtime. Since the parameters we consider are theoretically incomparable, we present experimental results showing that generalized tree doubling frequently outperforms generalized Christofides with respect to parameter size.

The asynchronous automaton of a Boolean network $f:{\{0,1\}}^n\to {\{0,1\}}^n$, considered in many applications, is the finite deterministic automaton where the set of states is ${\{0,1\}}^n$, the alphabet is $\left[n\right]$, and the action of letter i on a state x consists in either switching the ith component if $f_i\left(x\right)\ne x_i$ or doing nothing otherwise. In this paper, we ask for the existence of synchronizing words for this automaton, and their minimal length, when f is the and-net over an arc-signed digraph G on $\left[n\right]$: for every $i\in \left[n\right]$, $f_i\left(x\right)=1$ if and only if $x_j=1$ ($x_j\ne 0$) for every positive (negative) arc from j to i. Our main result is that if G is strongly connected and has no positive cycles, then either there exists a synchronizing word of length at most $10{(\sqrt{5}+1)}^n$ or G is a cycle and there are no synchronizing words. We also give complexity results showing that the situation is much more complex if one of the two hypothesis made on G is removed.

Unambiguous automata are nondeterministic automata in which every word has at most one accepting run. In this paper we give a polynomial-time algorithm for model checking discrete-time Markov chains against ω-regular specifications represented as unambiguous automata. We furthermore show that the complexity of this model checking problem lies in NC: the subclass of P comprising those problems solvable in poly-logarithmic parallel time. These complexity bounds match the known bounds for model checking Markov chains against specifications given as deterministic automata, notwithstanding the fact that unambiguous automata can be exponentially more succinct than deterministic automata. We report on an implementation of our procedure, including an experiment in which the implementation is used to model check LTL formulas on Markov chains.

We analyze the complexity of two fundamental verification problems within a generalization of the two-handed tile self-assembly model (2HAM) where initial system assemblies are not restricted to be singleton tiles, but may be larger prebuilt assemblies. Within this model we consider the producibility problem, which asks if a given tile system builds, or produces, a given assembly, and the unique assembly verification (UAV) problem, which asks if a given system uniquely produces a given assembly. We show that producibility is NP-complete and UAV is coNP${}^{NP}$-complete even when the initial assembly size and temperature threshold are both bounded by a constant. This is in stark contrast to results in the standard model with singleton input tiles where producibility is in P and UAV is coNP-complete with constant temperature. We further provide preliminary polynomial time results for producibility and UAV in the case of 1-dimensional linear assemblies with pre-built assemblies, as well as extend our results to the abstract Tile Assembly Model (aTAM) with constant-size attachable assemblies.

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.