Tight Bound Research Articles

Tighter bound for generalized multiple discrete logarithm problem via MDS matrix method

Discrete logarithm problem (DLP) is one of the fundamental hard problems used in cryptography. For 1≤k≤n, solving the k-out-of-n DLP instances is an important problem emerging in certain scenarios in public-key cryptography. Ying and Kunihiro (ACNS 2017) pioneered in studying k-out-of-n instance solutions of DLP, which is a generalized version of multiple DLP. By reducing the multiple DLP to the generalized version, they established lower bounds on the computational complexity of k-out-of-n DLP for different parameter values of k.In this paper, we further reduce the reduction complexity presented in Ying and Kunihiro's work and increase the range of k and n for the tight lower bound of k-out-of-n DLP in the generic group model, which has applications in related cryptographic schemes. To achieve the goal, the key technique is to utilize a variant of fast multipoint evaluation. We divide the discussion into two cases. In the special case when n divides p−1, by leveraging Number Theory Transform (NTT) technique, we expand k and n to a larger range. In the general case, by using a variant of fast multipoint evaluation, we increase k and n to a moderately larger range.

Binary distinguishability operation

This paper analyzes a new distinguishability operation for finite deterministic machines and languages. The research was inspired by the “Gedanken experiments” on sequential machines performed by Moore in 1956, and extends the study of the unary distinguishability operation to a binary one. Besides studying the new operation's properties, we give a tight bound of its state complexity on regular languages, including the case for finite languages.

Asymptotically Tight Bounds on the Optimal Pricing Strategy with Patient Customers

In “Asymptotically Tight Bounds on the Optimal Pricing Strategy with Patient Customers,” Qian, Su, and Zhou revisit the problem of a monopolist seller’s optimal pricing strategy when dealing with both patient and impatient customers. The authors give asymptotically tight bounds on the optimal strategy’s cycle period and long-run average revenue when the patient customers’ patience level is high. This result also addresses an open question proposed by others regarding the optimal cycle period.

Probabilistic Unitary Synthesis with Optimal Accuracy

The purpose of unitary synthesis is to find a gate sequence that optimally approximates a target unitary transformation. A new synthesis approach, called probabilistic synthesis, has been introduced, and its superiority has been demonstrated over traditional deterministic approaches with respect to approximation error and gate length. However, the optimality of current probabilistic synthesis algorithms is unknown. We obtain the tight lower bound on the approximation error obtained by the optimal probabilistic synthesis, which guarantees the sub-optimality of current algorithms. We also show its tight upper bound, which improves and unifies current upper bounds depending on the class of target unitaries. These two bounds reveal the fundamental relationship of approximation error between probabilistic approximation and deterministic approximation of unitary transformations. From a computational point of view, we show that the optimal probability distribution can be computed by the semidefinite program (SDP) we construct. We also construct an efficient probabilistic synthesis algorithm for single-qubit unitaries, rigorously estimate its time complexity, and show that it reduces the approximation error quadratically compared with deterministic algorithms.

Tight Bounds for Monotone Minimal Perfect Hashing

Tight Bounds for Monotone Minimal Perfect Hashing

Parallel Covering an Obtuse Triangle with Squares

Suppose that 𝑇 (𝛼, 𝛽) is an obtuse triangle with base length 1 and with base angles 𝛼 and 𝛽 (where 𝛽 > 90◦). In this note a tight lower bound of the sum of the areas of squares that can parallel cover 𝑇 (𝛼, 𝛽) is given. This result complements the previous lower bound obtained for the triangles with the interior angles at the base of the measure not greater than 90◦.

Tight Runtime Bounds for Static Unary Unbiased Evolutionary Algorithms on Linear Functions

Tight Runtime Bounds for Static Unary Unbiased Evolutionary Algorithms on Linear Functions

Tight General Bounds for the Extremal Numbers of 0–1 Matrices

Abstract A zero-one matrix $M$ is said to contain another zero-one matrix $A$ if we can delete some rows and columns of $M$ and replace some $1$-entries with $0$-entries such that the resulting matrix is $A$. The extremal number of $A$, denoted $\operatorname{ex}(n,A)$, is the maximum number of $1$-entries that an $n\times n$ zero-one matrix can have without containing $A$. The systematic study of this function for various patterns $A$ goes back to the work of Füredi and Hajnal from 1992, and the field has many connections to other areas of mathematics and theoretical computer science. The problem has been particularly extensively studied for so-called acyclic matrices, but very little is known about the general case (i.e., the case where $A$ is not necessarily acyclic). We prove the first asymptotically tight general result by showing that if $A$ has at most $t$ $1$-entries in every row, then $\operatorname{ex}(n,A)\leq n^{2-1/t+o(1)}$. This verifies a conjecture of Methuku and Tomon. Our result also provides the first tight general bound for the extremal number of vertex-ordered graphs with interval chromatic number $2$, generalizing a celebrated result of Füredi and Alon, Krivelevich, and Sudakov about the (unordered) extremal number of bipartite graphs with maximum degree $t$ in one of the vertex classes.

Smoothed Analysis of Information Spreading in Dynamic Networks

The best known solutions for k -message broadcast in dynamic networks of size n require Ω ( nk ) rounds. In this article, we see if these bounds can be improved by smoothed analysis. To do so, we study perhaps the most natural randomized algorithm for disseminating tokens in this setting: at every timestep, choose a token to broadcast randomly from the set of tokens you know. We show that with even a small amount of smoothing (i.e., one random edge added per round), this natural strategy solves k -message broadcast in \(\tilde{O}(n+k^3)\) rounds, with high probability, beating the best known bounds for \(k=o(\sqrt {n})\) and matching the Ω ( n + k ) lower bound for static networks for k = O ( n 1/3 ) (ignoring logarithmic factors). In fact, the main result we show is even stronger and more general: Given ℓ-smoothing (i.e., ℓ random edges added per round), this simple strategy terminates in \(O(kn^{2/3}\log ^{1/3}(n)\ell ^{-1/3})\) rounds. We then prove this analysis close to tight with an almost-matching lower bound. To better understand the impact of smoothing on information spreading, we next turn our attention to static networks, proving a tight bound of \(\tilde{O}(k\sqrt {n})\) rounds to solve k -message broadcast, which is better than what our strategy can achieve in the dynamic setting. This confirms the intuition that although smoothed analysis reduces the difficulties induced by changing graph structures, it does not eliminate them altogether. Finally, we apply tools developed to support our smoothed analysis to prove an optimal result for k -message broadcast in so-called well-mixed networks in the absence of smoothing. By comparing this result to an existing lower bound for well-mixed networks, we establish a formal separation between oblivious and strongly adaptive adversaries with respect to well-mixed token spreading, partially resolving an open question on the impact of adversary strength on the k -message broadcast problem.

Tight Lower Bounds for Directed Cut Sparsification and Distributed Min-Cut

In this paper, we consider two fundamental cut approximation problems on large graphs. We prove new lower bounds for both problems that are optimal up to logarithmic factors. The first problem is approximating cuts in balanced directed graphs. In this problem, we want to build a data structure that can provide (1 ± ε)-approximation of cut values on a graph with n vertices. For arbitrary directed graphs, such a data structure requires Ω(n 2 ) bits even for constant ε. To circumvent this, recent works study β-balanced graphs, meaning that for every directed cut, the total weight of edges in one direction is at most β times the total weight in the other direction. We consider the for-each model, where the goal is to approximate each cut with constant probability, and the for-all model, where all cuts must be preserved simultaneously. We improve the previous Ømega(n √β/ε) lower bound in the for-each model to ~Ω (n √β /ε) and we improve the previous Ω(n β/ε) lower bound in the for-all model to Ω(n β/ε 2 ). This resolves the main open questions of (Cen et al., ICALP, 2021). The second problem is approximating the global minimum cut in a local query model, where we can only access the graph via degree, edge, and adjacency queries. We prove an ΩL(min m, m/ε 2 k R) lower bound for this problem, which improves the previous ΩL(m/k R) lower bound, where m is the number of edges, k is the minimum cut size, and we seek a (1+ε)-approximation. In addition, we show that existing upper bounds with minor modifications match our lower bound up to logarithmic factors.

Tight Bounds of Circuits for Sum-Product Queries

In this paper, we ask the following question: given a Boolean Conjunctive Query (CQ), what is the smallest circuit that computes the provenance polynomial of the query over a given semiring? We answer this question by giving upper and lower bounds. Notably, it is shown that any circuit F that computes a CQ over the tropical semiring must have size log |F| ≥ (1-ε) · da-entw for any ε >0, where da-entw is the degree-aware entropic width of the query. We show a circuit construction that matches this bound when the semiring is idempotent. The techniques we use combine several central notions in database theory: provenance polynomials, tree decompositions, and disjunctive Datalog programs. We extend our results to lower and upper bounds for formulas (i.e., circuits where each gate has outdegree one), and to bounds for non-Boolean CQs.

Uncertainty relation and the constrained quadratic programming

The uncertainty relation is a fundamental concept in quantum theory, plays a pivotal role in various quantum information processing tasks. In this study, we explore the additive uncertainty relation pertaining to two or more observables, in terms of their variance, by utilizing the generalized Gell-Mann representation in qudit systems. We find that the tight state-independent lower bound of the variance sum can be characterized as a quadratic programming problem with nonlinear constraints in optimization theory. As illustrative examples, we derive analytical solutions for these quadratic programming problems in lower-dimensional systems, which align with the state-independent lower bounds. Additionally, we introduce a numerical algorithm tailored for solving these quadratic programming instances, highlighting its efficiency and accuracy. The advantage of our approach lies in its potential ability to simultaneously achieve the optimal value of the quadratic programming problem with nonlinear constraints but also precisely identify the extremal state where this optimal value is attained. This enables us to establish a tight state-independent lower bound for the sum of variances, and further identify the extremal state at which this lower bound is realized.

The Latency Performance Analysis and Effective Relay Selection for Visible Light Networks.

In visible light communication (VLC), the precise latency evaluation of wireless access networks and the efficient forwarding strategy of core networks are the crux for end-to-end reliability provisioning. Leveraging martingale theory, an elegant latency-bounded reliability analysis framework is studied for the VLC network. Considering the characteristic that VLC links are easy to block, the martingale of the service process is constructed. Based on the time shift features, the martingale process related to latency is proposed for the VLC system, which models the influence of entanglement between aggregate arrivals and random service on latency. A stopping time event about latency is defined. Renting the stopping time theory, a tight upper bound of the unreliability with regard to latency is derived. In the core network, a dynamic forward backhaul framework is proposed, which uses the relay selection algorithm based on back-pressure theory to improve data transmission quality. The theoretical latency-bounded reliability matches the simulation results well, which verifies the effectiveness of the proposed analysis framework, and the proposed relay selection algorithm can also improve network performance under data-intensive transmission.

Tight Lower Bound on Differential Entropy for Mixed Gaussian Distributions

In this paper, a tight lower bound for the differential entropy of the Gaussian mixture model is presented. First, the probability model of mixed Gaussian distribution that is created by mixing both discrete and continuous random variables is investigated in order to represent symmetric bimodal Gaussian distribution using the hyperbolic cosine function, on which a tighter upper bound is set. Then, this tight upper bound is used to derive a tight lower bound for the differential entropy of the Gaussian mixture model introduced. The proposed lower bound allows to maintain its tightness over the entire range of the model's parameters and shows more tightness when compared with other bounds that lose their tightness over certain parameter ranges. The presented results are then extended to introduce a more general tight lower bound for asymmetric bimodal Gaussian distribution, in which the two modes have a symmetric mean but differ in terms of their weights.

Minimax Optimal Bandits for Heavy Tail Rewards.

Stochastic multiarmed bandits (stochastic MABs) are a problem of sequential decision-making with noisy rewards, where an agent sequentially chooses actions under unknown reward distributions to minimize cumulative regret. The majority of prior works on stochastic MABs assume that the reward distribution of each action has bounded supports or follows light-tailed distribution, i.e., sub-Gaussian distribution. However, in a variety of decision-making problems, the reward distributions follow a heavy-tailed distribution. In this regard, we consider stochastic MABs with heavy-tailed rewards, whose p th moment is bounded by a constant νp for . First, we provide theoretical analysis on sub-optimality of the existing exploration methods for heavy-tailed rewards where it has been proven that existing exploration methods do not guarantee a minimax optimal regret bound. Second, to achieve the minimax optimality under heavy-tailed rewards, we propose a minimax optimal robust upper confidence bound (MR-UCB) by providing tight confidence bound of a p -robust estimator. Furthermore, we also propose a minimax optimal robust adaptively perturbed exploration (MR-APE) which is a randomized version of MR-UCB. In particular, unlike the existing robust exploration methods, both proposed methods have no dependence on νp . Third, we provide the gap-dependent and independent regret bounds of proposed methods and prove that both methods guarantee the minimax optimal regret bound for a heavy-tailed stochastic MAB problem. The proposed methods are the first algorithm that theoretically guarantees the minimax optimality under heavy-tailed reward settings to the best of our knowledge. Finally, we demonstrate the superiority of the proposed methods in simulation with Pareto and Fréchet noises with respect to regrets.

Tight Bounds for the N₂-Chromatic Number of Graphs

Let $G$ be a connected graph. A vertex coloring of $G$ is an $N_2$-vertex coloring if, for every vertex $v$, the number of different colors assigned to the vertices adjacent to $v$ is at most two. The $N_2$-chromatic number of $G$ is the maximum number of colors that can be used in an $N_2$-vertex coloring of $G$. In this paper, we establish tight bounds for the $N_2$-chromatic number of a graph in terms of its maximum degree and its diameter, and characterize those graphs that attain these bounds.

Balancing and scheduling of assembly line with multi-type collaborative robots

Human–robot collaboration (HRC) is a promising production mode that is in line with the vision of human-centered Industry 4.0. HRC contributes to the improvement of productivity as well as the reduction of workers’ ergonomic risk. In this study, we present one of the first attempts to address the assembly line balancing problem with multi-type collaborative robots (cobots), which allows human and robots to perform tasks in parallel or in collaboration. A mixed-integer programming (MIP) model is formulated to minimize the cycle time and a tight lower bound is proposed. We further propose a multi-objective model and an extended model to expand the scope of the study. Due to the complexity, an adaptive neighborhood simulated annealing algorithm (ANSA) is developed with the designed neighborhood operators and structures. Furthermore, an adaptive mechanism is applied to the ANSA to dynamically update the weights of the neighborhood structures based on historical information. Extensive computational experiments and a real case study are conducted to verify the superiority of ANSA. We further compare the application of diverse collaboration modes, i.e., sequential, simultaneous and supportive modes. The results also indicate that a suitable type of robot can improve productivity and the utilization rate of robots.

Tight Toughness, Isolated Toughness and Binding Number Bounds for the [1,n]-Factors and the {K2,Ci≥4}-Factors

Let [Formula: see text] be an integer. The [Formula: see text]-factor of a graph [Formula: see text] is a spanning subgraph [Formula: see text] if [Formula: see text] for all [Formula: see text], and the [Formula: see text]-factor is a subgraph whose each component is either [Formula: see text] or [Formula: see text]. In this paper, we give the lower bounds with regard to tight toughness, isolated toughness and binding number to guarantee the existence of the [Formula: see text]-factors and [Formula: see text]-factors for a graph.

Anticipated Wait and Its Effects on Consumer Choice, Pricing, and Assortment Management

Problem definition: We investigate the effects of waiting time, mainly due to production in a make-to-batch-order (MTBO) system, on consumer choice behavior, pricing, assortment, and model estimation. In an MTBO system, the seller/manufacturer first collects orders placed within a certain period of time into a batch and then starts the production process. After the production of all orders in a batch are complete, the products are then shipped and delivered to individual consumers. Because of batch production and shipping, the disutility of the waiting time exhibits negative externality. Methodology/results: We adopt the widely used multinomial logit (MNL) model as a starting point and incorporate the anticipated wait into consumers’ decision making. The derived model, referred to as the MNL with wait model, is a solution of the rational expectation equilibrium and is capable of capturing the effects of negative externality induced by anticipated wait that may change the substitution patterns dramatically. We characterize the multiproduct price optimization problem under the MNL with wait model by establishing a one-to-one mapping between the price vector and the choice probability vector. We find that firms tend to charge higher prices for time-consuming items and charge lower prices for time-saving items compared with the optimal prices under the standard MNL model. In addition to price competition, we also study the Cournot-type competition, in which the decision is the choice probability for each firm and establish the existence of a Nash equilibrium. For assortment optimization, we identify mild conditions under which the optimality of revenue-ordered assortments still holds. However, the assortment problem under the MNL with wait model is generally NP-hard, so we develop approximation algorithms with performance guarantees and provide an easy-to-compute tight upper bound. Moreover, we develop an efficient maximum likelihood-based algorithm for model calibration and further conduct numerical studies to showcase the importance of incorporating disutility due to wait in estimation, pricing, and assortment planning problems. Managerial implications: The MNL with wait model can increase prediction accuracy for consumers’ choice behavior especially when they are aware of the potential wait. Failure to take into account the effects of anticipated wait in firms’ decision making may lead to substantial losses. Funding: The research of C. Ke is supported in part by the National Natural Science Foundation of China [Grant 72101113]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2020.0346 .

Additional findings on S‐parameter bounds valid for unconditionally stable N‐ports

AbstractIn a recent paper, it has been shown that, if an N‐port network fulfills the condition of (geometrical) unconditional stability at a given frequency, then its scattering parameters will also necessarily satisfy N easily computable bounds, one per port. In order to complete that picture, this contribution investigates whether a tighter bound can be obtained by combining the bounds into just one. The answer is in general negative, except that the 3‐port case does indeed exhibit a peculiar behavior: this can be exploited to reduce the upper bound when the diagonal elements of the scattering matrix are limited in magnitude up to some , and in particular for (simultaneous conjugate match).