Information Processing Letters最新文献

We reconsider two well-known distributed randomized algorithms computing a maximal independent set, proposed in the seminal work of Luby (1986). We enhance these algorithms such that they become self-stabilizing without sacrificing their run-time, i.e., both stabilize in $O(\log n)$ synchronous rounds with high probability on any n-node graph. The first algorithm gets along with three states, but needs to know an upper bound on the maximum degree. The second does not need any information about the graph, but uses a number of states that is linear in the node degree. Both algorithms use messages of logarithmic size.

For a class $G$ of graphs, the objective of Subgraph Complementation to $G$ is to find whether there exists a subset S of vertices of the input graph G such that modifying G by complementing the subgraph induced by S results in a graph in $G$. We obtain a polynomial-time algorithm for the problem when $G$ is the class of graphs with minimum degree at least k, for a constant k, answering an open problem by Fomin et al. (Algorithmica, 2020). When $G$ is the class of graphs without any induced copies of the star graph on $t+1$ vertices (for any constant $t\ge 3$) and diamond, we obtain a polynomial-time algorithm for the problem. This is in contrast with a result by Antony et al. (Algorithmica, 2022) that the problem is NP-complete and cannot be solved in subexponential-time (assuming the Exponential Time Hypothesis) when $G$ is the class of graphs without any induced copies of the star graph on $t+1$ vertices, for every constant $t\ge 5$.

The problem that we consider is the following: given an $n\times n$ array A of positive numbers and a natural number p, find a tiling using at most p rectangles (which means that each array element must be covered by some rectangle and no two rectangles must overlap) that minimizes the maximum weight of any rectangle (the weight of a rectangle is the sum of elements which are covered by it). We prove that it is NP-hard to approximate this problem to within a factor of 1$\frac{1}{3}$ (the previous best result was $1\frac{1}{4}$).

We robustify PCTL and PCTL^⁎, the most important specification languages for probabilistic systems, and show that robustness does not increase the complexity of their model-checking problems.

A subset S of the vertex set of a graph G is an $F$-isolating set of G if $G-N\left[S\right]$ does not contain a copy of a member of $F$ as a subgraph, where $F$ is a family of connected graphs and $N\left[S\right]$ is the closed neighborhood of S. The $F$-isolation number of G is the minimum cardinality of an $F$-isolating set of G, denoted by $\iota (G,F)$. Given a graph G, $\left\{P_k\right\}$-ISOLATION asks for the size of a smallest $\left\{P_k\right\}$-isolating set of G for a fixed positive integer k, where $P_k$ is a path of order k. In this paper, we first show that the decision version of $\left\{P_k\right\}$-ISOLATION is NP-complete for chordal graphs and planar graphs. Secondly, we propose a linear time algorithm to compute a smallest $\left\{P_k\right\}$-isolating set of a tree. Finally, we solve the problem of characterizing the connected graphs G with $\iota (G,\{P_3\left\}\right)=\frac{2}{7}\left|V\right(G\left)\right|$, proposed by Zhang and Wu [Discrete Appl. Math. 304 (2021) 365-374].

Using the notion of visibility representations, our paper establishes a new property of instances of the Nondeterministic Constraint Logic (NCL) problem (a PSPACE-complete problem that is very convenient to prove the PSPACE-hardness of reversible games with pushing blocks). Direct use of this property introduces an explosion in the number of gadgets needed to show PSPACE-hardness, but we show how to bring that number from 32 down to only three in the general case, and down to two in our specific case! We propose it as a step towards a broader and more general framework for studying games with irreversible gravity, and use this connection to guide an indirect polynomial-time many-one reduction from the NCL problem to the Hanano Puzzle—which is NP-hard—to prove it is PSPACE-complete.

Fair division is a longstanding problem in economics and has recently received substantial interest in computer science. Several applications of fair division involve agents with unequal entitlements represented by weights. We review work on weighted fair division of indivisible items, discuss the range of weighted fairness notions that have been proposed, and highlight a number of open questions.

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.