Information Processing Letters最新文献

The balanced hypercube $B{H}_n$, a variant of the hypercube, is a novel interconnection network topology for massive parallel systems. It is showed in [Theor. Comput. Sci. 947 (2023) 113708] that for any edge subset F of $B{H}_n$ there exists a fault-free Hamiltonian cycle in $B{H}_n-F$ for $n\ge 2$ with $\left|F\right|\le 5n-7$ if the degree of every vertex in $B{H}_n-F$ is at least two and there exist no $f_4$-cycles in $B{H}_n-F$. In this paper, we consider the existence of Hamiltonian cycles of $B{H}_n$ when F is a matching (a set of disjoint edges), and show that each edge $e\notin F$ lies on a fault-free Hamiltonian cycle of $B{H}_n-F$ with $n\ge 2$. The number of faulty edges in F can be up to $2^{2n-1}$, which is exponential to the dimension n.

The seminal work by Pagh [1] proposed a matrix multiplication algorithm for real-valued squared matrices called Compressed Matrix Multiplication (CMM) having a sparse matrix output product. The algorithm is based on a popular sketching technique called Count-Sketch [2] and Fast Fourier Transform (FFT). For input square matrices A and B of order n and the product matrix AB with Frobenius norm $\left|\right|\mathrm{AB}|{|}_F$, the algorithm offers an unbiased estimate for each entry, i.e., ${\left(\mathrm{AB}\right)}_{i,j}$ of the product matrix AB with a variance bounded by $\left|\right|\mathrm{AB}|{|}_F^2/b$, where b is the compressed bucket size. Thus, the variance will eventually become high for a small bucket size. In this work, we address the high variance problem of CMM with the help of a simple and practical technique based on classical variance reduction methods in statistics. Our techniques rely on the Control Variate (CV) method. We suggest rigorous theoretical analysis for variance reduction and complement it via supporting empirical evidence.

In this paper, we derive parameterized Chernoff bounds and show their applications for simplifying the analysis of some well-known probabilistic algorithms and data structures. The parameterized Chernoff bounds we provide give probability bounds that are powers of two, with a clean formulation of the relation between the constant in the exponent and the relative distance from the mean. In addition, we provide new simplified analyses with these bounds for hash tables, randomized routing, and a simplified, non-recursive adaptation of the Floyd-Rivest selection algorithm.

Consider a set of demands, each taking length-k strings as input. The k-restriction problem is to construct a small set of length-m strings, such that given any k positions and any demand, there exists a string in the set satisfying the demand at these positions. The k-restriction problem relates to many problems, such as k-independent sets, covering arrays, and many other combinatorial applications. By considering the VC-dimension of demands, we prove bounds independent of the number of demands with the Lovász Local Lemma. As a result, we can prove better bounds for demands with finite VC-dimension.

The labelling-based approach of abstract argumentation frameworks (AAFs) is beneficial for various applications requiring different levels of decisiveness. For labelling-based semantics, this paper provides an operator so-called reduced meet modulo an ultrafilter, which is inspired by its counterpart over extensions. All criteria involved in the definitions of fundamental labelling-based semantics in AAFs are shown to be closed under this operator. Based on this fact, this paper develops a simple and uniform way for exploring common properties of labelling-based semantics in AAFs, including the compactness of extensibility, the Dcpo and Lindenbaum properties, etc.

We initiate the study of spanners under the Hausdorff and Fréchet distances. Let S be a set of points in $R^d$ and ε a non-negative real number. A subgraph H of the Euclidean graph over S is an ε-Hausdorff-spanner (resp., an ε-Fréchet-spanner) of S, if for any two points $u,v\in S$ there exists a path $P(u,v)$ in H between u and v, such that the Hausdorff distance (resp., the Fréchet distance) between $P(u,v)$ and $\overline{uv}$ is at most ε. We show that any t-spanner of a planar point-set S is a $\frac{\sqrt{t^2-1}}{2}$-Hausdorff-spanner and a $\min \{\frac{t}{2},\frac{\sqrt{5t^2-2t-3}}{4}\}$-Fréchet spanner. We also prove that for any $t>1$, there exist a set of points S and an ${\epsilon }_1$-Hausdorff-spanner of S and an ${\epsilon }_2$-Fréchet-spanner of S, where ${\epsilon }_1$ and ${\epsilon }_2$ are constants, such that neither of them is a t-spanner.

This article treats AND-OR tree computation in terms of query complexity. We are interested in the cases where assignments (inputs) or algorithms are randomized. For the former case, it is known that there is a unique randomized assignment achieving the distributional complexity of balanced trees. On the other hand, the dual problem has the opposite result; the optimal randomized algorithms for balanced trees are not unique. We extend the latter study on randomized algorithms to weakly-balanced trees, and see that the uniqueness still fails.

In this paper, some lower and upper bounds for the subgraph centrality and communicability of a graph are proved. The expected value of the normalized total communicability of a random $G(n,p)$ graph is also considered and asymptotically determined. Moreover, some computational results are presented to compare the bounds obtained in this paper with some other bounds in the literature.

One way to define the Matching Cut problem is: Given a graph G, is there an edge-cut M of G such that M is an independent set in the line graph of G? We propose the more general Conflict-Free Cut problem: Together with the graph G, we are given a so-called conflict graph $\hat{G}$ on the edges of G, and we ask for an edge-cutset M of G that is independent in $\hat{G}$. Since conflict-free settings are popular generalizations of classical optimization problems and Conflict-Free Cut was not considered in the literature so far, we start the study of the problem. We show that the problem is $\mathrm{NP}$-complete even when the maximum degree of G is 5 and $\hat{G}$ is 1-regular. The same reduction implies an exponential lower bound on the solvability based on the Exponential Time Hypothesis. We also give parameterized complexity results: We show that the problem is fixed-parameter tractable with the vertex cover number of G as a parameter, and we show $W\left[1\right]$-hardness even when G has a feedback vertex set of size one, and the clique cover number of $\hat{G}$ is the parameter. Since the clique cover number of $\hat{G}$ is an upper bound on the independence number of $\hat{G}$ and thus the solution size, this implies $W\left[1\right]$-hardness when parameterized by the cut size. We list polynomial-time solvable cases and interesting open problems. At last, we draw a connection to a symmetric variant of SAT.

We consider the differentially private (DP) facility location problem in the so called super-set output setting proposed by Gupta et al. [13]. The current best known expected approximation ratio for an ϵ-DP algorithm is $O\left(\frac{\log n}{\sqrt{ϵ}}\right)$ due to Cohen-Addad et al. [3] where n denote the size of the metric space, meanwhile the best known lower bound is $\Omega (1/\sqrt{ϵ})$ [8].

In this short note, we give a lower bound of $\tilde{\Omega }(\min \{\log n,\sqrt{\frac{\log n}{ϵ}}\})$ on the expected approximation ratio of any ϵ-DP algorithm, which is the first evidence that the approximation ratio has to grow with the size of the metric space.