Time Complexity Research Articles

A novel HP-Growth model for enhancing community policing in South Africa

South Africa (SA) grapples with a rising crime rate, which poses challenges to the safety and economic growth of the country. Despite the limited literature on pattern-based models in SA, frequent pattern-based models, particularly Frequent Pattern Growth (FP-Growth) and Hyper Structure Mining (Hmine), have demonstrated utility in various research fields. This paper introduces a novel model, Hybrid Pattern-Growth (HP-Growth), which combines the strengths of FP-Growth and Hmine. A comparative analysis of the South African crime statistics (Stats SA crime) dataset’s computational time complexity, scalability, and memory usage revealed that HP-Growth and Hmine outperform FP-Growth. This study establishes association rule thresholds and emphasizes the importance of selecting the most appropriate pattern-based model for generating crime patterns. According to the study, HP-Growth outperformed the other two models on sparse datasets, whereas Hmine excelled on dense datasets. FP-Growth uses more memory and has a greater time complexity than HP-Growth and Hmine. The most suitable model was then integrated into the developed crime support system. The research outcome has practical implications for law enforcement in aiding strategic resource allocation and proactive policing for crime reduction and control.

Temporal betweenness centrality on shortest walks variants

Betweenness centrality has been extensively studied since its introduction in 1977 as a measure of node importance in graphs. This measure has found use in various applications and has been extended to temporal graphs with time-labeled edges. Recent research by Buß et al. and Rymar et al. has shown that it is possible to compute the shortest walks betweenness centrality of all nodes in a temporal graph in On3T2 and On2mT2 time, respectively, where T is the maximum time, m is the number of temporal edges, and n is the number of nodes. These approaches considered walks that do not take into account contributions from intermediate temporal nodes. In this paper, we study the temporal betweenness centrality on classical walks that we call passive, as well as on a variant that we call active walks, which takes into account contributions from all temporal nodes. We present an improved analysis of the running time of the classical algorithm for computing betweenness centrality of all nodes, reducing the time complexity to OnmT+n2T. Furthermore, for active walks, we show that the betweenness centrality can be computed in OnmT+n2T2. We also show that our results hold for different shortest walks variants. Finally, we provide an open-source implementation of our algorithms and conduct experiments on several real-world datasets of cities and contact traces. We compare the results of the two variants on both the node and time dimensions of the temporal graph, and we also compare the temporal betweenness centrality to its static counterpart. Our experiments suggest that for the shortest foremost variant looking only at the first 10% of the temporal interaction is a good approximation for the overall top ranked nodes.

Lot scheduling on a single machine to minimize total weighted completion time

Lot scheduling is one of the important production patterns in modern manufacturing systems. Each lot processes simultaneously one or more customer orders with a total size no more than the fixed lot capacity, consuming a uniform lot processing time. In this work we consider the single machine lot scheduling environment where any order can be split and processed in consecutive lots. The objective is to minimize total weighted completion time. We first prove the NP-hardness of the considered problem, and propose a polynomial-time algorithm with approximation ratio equal to the ratio of the largest to the smallest order weight. Moreover, we explore two special cases. For the first case where the total size of all the orders is at most twice of the lot capacity, a dynamic programming algorithm is provided. In the last special case, the sizes and weights of orders are in reverse-agreeable, \ie, a larger size of an order implies an equal or smaller weight. We prove that processing orders in the non-increasing sequence of their weights results in an optimal schedule, implying that the case is solvable in polynomial time. Finally, experimental results demonstrate the effectiveness of the approximation algorithm.

Influence-Based Community Partition with DegreeRank Label Propagation (DRLP) Algorithm for Social Networks

Community detection is increasingly important in social networks with the rapid growth of big data, which provides a deep understanding of the mesoscopic structure of social networks. In this article, we propose a label improvement algorithm, DegreeRank Label Propagation (DRLP), which is based on the degree centrality of nodes and adopts a PageRank optimization strategy. We present a damping factor reflecting the affinity between nodes, which can be adjusted to affect the change of affinity between nodes caused by unexpected events, aiming to simulate interpersonal communication in real networks. Next, a novel importance index is designed for nodes to solve the random problem of existing similar algorithms by globalizing the local characteristics of nodes. We also develop an update algorithm with low time complexity during the label selection process to ensure the sum of influence propagation is maximized within each community. Experimental results verify that the algorithm achieves stable and excellent community partitioning results on real network datasets and artificial synthetic networks. Especially in large and medium-sized networks, our method demonstrates higher accuracy and better performance in terms of normalized mutual information (NMI) and modularity than other methods.

Multi-Apartment Rent Division

Rent division is the well-studied problem of fairly assigning rooms and dividing rent among a set of roommates within a single apartment. A shortcoming of existing solutions is that renters are assumed to be considering apartments in isolation, whereas in reality, renters can choose among multiple apartments. In this paper, we generalize the rent division problem to the multi-apartment setting, where the goal is to both fairly choose an apartment among a set of alternatives and fairly assign rooms and rents within the chosen apartment. Our main contribution is a generalization of envy-freeness called *negotiated envy-freeness*. We show that a solution satisfying negotiated envy-freeness is guaranteed to exist and that it is possible to optimize over all negotiated envy-free solutions in polynomial time. We also define an even stronger fairness notion called *universal envy-freeness* and study its existence when values are drawn randomly.

Balanced and Fair Partitioning of Friends

In the recently introduced model of fair partitioning of friends, there is a set of agents located on the vertices of an underlying graph that indicates the friendships between the agents. The task is to partition the graph into k balanced-sized groups, keeping in mind that the value of an agent for a group is equal to the number of edges they have in that group. The goal is to construct partitions that are "fair", i.e., no agent would like to replace an agent in a different group. We generalize the standard model by considering utilities for the agents that are beyond binary and additive. Having this as our foundation, our contribution is threefold: (a) we adapt several fairness notions that have been developed in the fair division literature to our setting; (b) we give several existence guarantees supported by polynomial-time algorithms; (c) we initiate the study of the computational (and parameterized) complexity of the model and provide an almost complete landscape of the (in)tractability frontier for our fairness concepts.

Towards Learnable Anchor for Deep Multi-View Clustering

Deep multi-view clustering incorporating graph learning has presented tremendous potential. Most methods encounter costly square time consumption w.r.t. data size. Theoretically, anchor-based graph learning can alleviate this limitation, but related deep models mainly rely on manual discretization approaches to select anchors, which indicates that 1) the anchors are fixed during model training and 2) they may deviate from the true cluster distribution. Consequently, the unreliable anchors may corrupt clustering results. In this paper, we propose the Deep Multi-view Anchor Clustering (DMAC) model that performs clustering in linear time. Concretely, the initial anchors are intervened by the positive-incentive noise sampled from Gaussian distribution, such that they can be optimized with a newly designed anchor learning loss, which promotes a clear relationship between samples and anchors. Afterwards, anchor graph convolution is devised to model the cluster structure formed by the anchors, and the mutual information maximization loss is built to provide cross-view clustering guidance. In this way, the learned anchors can better represent clusters. With the optimal anchors, the full sample graph is calculated to derive a discriminative embedding for clustering. Extensive experiments on several datasets demonstrate the superior performance and efficiency of DMAC compared to state-of-the-art competitors.

Fast Computing of Dung Semantics in Acyclic Probabilistic Argumentation Frameworks

This paper presents fast and exact methods for computing the probability of an argument’s acceptance using Dung’s semantics in the Constellation paradigm of Abstract Argumentation. For (directed) Singly-Connected Graphs (SCGs), the problem can now be solved in linearithmic time instead of being exponential in the number of attacks, as reported in the literature. Moreover, in the more general case of Directed Acyclic Graphs (DAGs), we provide an algorithm whose time complexity is linearithmic in the product of the out-degree of dependent arguments, i.e., arguments reaching the argument considered for acceptance through multiple paths in the graph. We theoretically show that this complexity is lower than the lower bound of the (exact) Constellation method, which is also supported by empirical results. Our approach to DAGs is also compared with the (approximate) Monte-Carlo method, which is stopped when exact results are obtained. Within this time constraint, Monte-Carlo still outputs significant errors, underlying the fast computation of our approach.

Scalable Decentralized Algorithms for Online Personalized Mean Estimation

In numerous settings, agents lack sufficient data to learn a model directly. Collaborating with other agents may help, but introduces a bias-variance trade-off when local data distributions differ. A key challenge is for each agent to identify clients with similar distributions while learning the model, a problem that remains largely unresolved. This study focuses on a particular instance of the overarching problem, where each agent collects samples from a real-valued distribution over time to estimate its mean. Existing algorithms face impractical per-agent space and time complexities (linear in the number of agents |A|). To address scalability challenges, we propose a framework where agents self-organize into a graph, allowing each agent to communicate with only a selected number of peers r. We propose two collaborative mean estimation algorithms: one employs a consensus-based approach, while the other uses a message-passing scheme, with complexity O(r) and O(r log |A|), respectively. We establish conditions for both algorithms to yield asymptotically optimal estimates and we provide a theoretical characterization of their performance.

The Role of Energy Loss in Achieving Ultralow Dark Current Density in Organic Photodetectors

AbstractIn this study, organic photodetectors (OPDs) are fabricated using the polymer donor D18 and four Y6 derivatives (Y6, L8‐BO, BTP‐eC9, and Y18), each exhibiting distinct surface energies and HOMO/LUMO energy levels. This allows the investigation of the impact of morphological variations and energy level alignment within the blend films on energy loss (Eloss), a key factor influencing near‐infrared (NIR) OPD performance. Grazing‐incidence wide‐angle X‐ray scattering (GIWAXS) analysis reveals that the D18/Y18 blend film exhibits an optimized unique 3D‐packing, which significantly reduces trap density and minimizes non‐radiative recombination. As a result, OPDs based on D18/Y18 demonstrate the lowest Eloss, primarily due to their higher charge transfer energy (ECT) and suppressed non‐radiative recombination. This effectively inhibits thermal carrier generation in which D18/Y18‐based OPDs, demonstrate an ultralow dark current density (Jd) of 3.95 × 10−10 A cm−2, a high responsivity (R) of 0.48 A W−1, and an exceptionally high detectivity (Dshot*) of 4.24 × 1013 Jones at a bias of −2 V. D18/Y18‐based OPDs demonstrate superior performance in key metrics such as linear dynamic range, cutoff frequency, and response time, making them highly competitive in the field of OPDs.

Tensorized Label Learning Based Fast Fuzzy Clustering

Multi-view graph clustering methods have been widely concerned due to the ability of dealing with arbitrarily shaped datasets. However, many methods with higher time and space complexity make them challenging to deal with large-scale datasets. Besides, many fuzzy clustering methods needs additional regularization terms or hyper-parameters to obtain the membership matrix or avoid trivial solutions, which weakens the model generalization ability. Furthermore, inconsistent clustering labels can arise when there are significant discrepancies between views, making it challenging to effectively leverage the complementary information from different views. To this end, we propose Tensorized Label Learning based Fast Fuzzy Clustering (TLLFFC). Specifically, we design a novel balanced regularization term to reduce pressure of tuning regularization parameters for fuzzy clustering. The label transmission strategy with the anchor graph makes TLLFFC suitable for large-scale datasets. Moreover, incorporating the Schatten p-norm regularization on the label matrices can effectively unearth the complementary information distributed among views, thereby align the labels across views more consistently. Extensive experiments verify the superiority of TLLFFC.

FROC: Building Fair ROC from a Trained Classifier

This paper considers the problem of fair probabilistic binary classification with binary protected groups. The classifier assigns scores, and a practitioner predicts labels using a certain cut-off threshold based on the desired trade-off between false positives vs. false negatives. It derives these thresholds from the ROC of the classifier. The resultant classifier may be unfair to one of the two protected groups in the dataset. It is desirable that no matter what threshold the practitioner uses, the classifier should be fair to both the protected groups; that is, the ℒₚ norm between FPRs and TPRs of both the protected groups should be at most ε. We call such fairness on ROCs of both the protected attributes εₚ-Equalized ROC. Given a classifier not satisfying ε₁-Equalized ROC, we aim to design a post-processing method to transform the given (potentially unfair) classifier's output (score) to a suitable randomized yet fair classifier. That is, the resultant classifier must satisfy ε₁-Equalized ROC. First, we introduce a threshold query model on the ROC curves for each protected group. The resulting classifier is bound to face a reduction in AUC. With the proposed query model, we provide a rigorous theoretical analysis of the minimal AUC loss to achieve ε₁-Equalized ROC. To achieve this, we design a linear time algorithm, namely FROC, to transform a given classifier's output to a probabilistic classifier that satisfies ε₁-Equalized ROC. We prove that under certain theoretical conditions, FROC achieves the theoretical optimal guarantees. We also study the performance of our FROC on multiple real-world datasets with many trained classifiers.

Multi Agent Reinforcement Learning for Sequential Satellite Assignment Problems

Assignment problems are a classic combinatorial optimization problem in which a group of agents must be assigned to a group of tasks such that maximum utility is achieved while satisfying assignment constraints. Given the utility of each agent completing each task, polynomial-time algorithms exist to solve a single assignment problem in its simplest form. However, in many modern-day applications such as satellite constellations, power grids, and mobile robot scheduling, assignment problems unfold over time, with the utility for a given assignment depending heavily on the state of the system. We apply multi-agent reinforcement learning to this problem, learning the value of assignments by bootstrapping from the known polynomial-time greedy solver and then learning from further experience. We then choose assignments using a distributed optimal assignment mechanism rather than by selecting them directly. We demonstrate that this algorithm is theoretically justified and avoids pitfalls experienced by other RL algorithms in this setting. Finally, we show that our algorithm significantly outperforms other methods in the literature, even while scaling to realistic scenarios with hundreds of agents and tasks.

Scenario-Based Robust Optimization of Tree Structures

We initiate the study of tree structures in the context of scenario-based robust optimization. Specifically, we study Binary Search Trees (BSTs) and Huffman coding, two fundamental techniques for efficiently managing and encoding data based on a known set of frequencies of keys. Given a number of distinct scenarios, each defined by a frequency distribution over the keys, our objective is to compute a single tree of best-possible performance, relative to any scenario. We consider, as performance metrics, the competitive ratio, which compares multiplicatively the cost of the solution to the tree of least cost among all scenarios, as well as the regret, which induces a similar, but additive comparison. For BSTs, we show that the problem is NP-hard across both metrics. We also obtain an optimal competitive ratio that is logarithmic in the number of scenarios. For Huffman Trees, we likewise prove NP-hardness, and we present an algorithm with logarithmic regret, which we prove to be near-optimal by showing a corresponding lower bound. Last, we give a polynomial-time algorithm for computing Pareto-optimal BSTs with respect to their regret, assuming scenarios defined by uniform distributions over the keys. This setting captures, in particular, the first study of fairness in the context of data structures. We provide an experimental evaluation of all algorithms. To this end, we also provide mixed integer linear program formulation for computing optimal trees.

Eliminating Majority Illusion Is Easy

Majority illusion is a phenomenon in social networks wherein the decision by the majority of the network is not the same as one's personal social circle's majority, leading to an incorrect perception of the majority in a large network. We present polynomial-time algorithms which completely eliminate majority illusion by altering as few connections in the network as possible. Eliminating majority illusion ensures each neighbourhood in the network has at least a 1/2-fraction of the majority winner. This result is surprising as partially eliminating majority illusion is NP-hard. We generalize the majority illusion problem to an arbitrary fraction p and show that the problem of ensuring all neighbourhoods in the network contain at least a p-fraction of nodes consistent with a given preference is NP-hard, for nearly all values of p.

Improved Maximin Share Approximations for Chores by Bin Packing

We study fair division of indivisible chores among n agents with additive cost functions using the popular fairness notion of maximin share (MMS). Since MMS allocations do not always exist for more than two agents, the goal has been to improve its approximations and identify interesting special cases where MMS allocations exist. We show the existence of · 1-out-of-9n/11 MMS allocations, which improves the state-of-the-art factor of 1-out-of-3n/4. · MMS allocations for factored instances, which resolves an open question posed by Ebadian et al. (2021). · 15/13-MMS allocations for personalized bivalued instances, improving the state-of-the-art factor of 13/11. We achieve these results by leveraging the HFFD algorithm of Huang and Lu (2021). Our approach also provides polynomial-time algorithms for computing an MMS allocation for factored instances and a 15/13-MMS allocation for personalized bivalued instances.

Trade-Offs Between Information and Crowding in Sequential Decisions (Student Abstract)

A rich line of theoretical work has modeled scenarios in which a set of agents make decisions sequentially, based on observing a growing mix of public and private signals that are revealed as these decisions occur. Here, we study a second crucial dimension, which is the way in which strategies can depend on crowding. In particular, consider a setting in which agents must sequentially decide which of several options to invest in, each based on a public signal that they receive. One of these options will ultimately be revealed to be valuable; but crucially, all the agents who selected this option must divide the value that comes from it. As a result, when a given agent j goes to make a decision among the options, the decisions of earlier agents convey information about the payoff that j will receive in any eventual division of the value. When many earlier agents have chosen a specific option, the greater crowding on this option means it must be divided more finely, resulting in lower payoffs. To simulate large games when signals are public, we define a polynomial-time algorithm to compute equilibrium strategies. We show that even in this case of public signals, the interaction of crowding with informational effects leads to complex non-monotonicities in the resulting sequential decisions, with agents sometimes choosing options with lower expected levels of crowding --- and hence a better split of the potential value --- over options with better informational or current crowding properties.

Unlocking the Game: Estimating Games in Möbius Representation for Explanation and High-Order Interaction Detection

Shapley value-based explanations are widely utilized to demystify predictions made by opaque models. Approaches to estimating Shapley values often approximate explanation games as inessential and estimate the Shapley value directly as feature attribution with a limited capacity to quantify feature interactions. This paper introduces a new approach for calculating Shapley values that relaxes the assumption of inessential games and is proven to provide additive feature attribution. The initial formulation of the proposed approach includes the estimation of game values in their Möbius representation with exponentially many parameters, but we put forward a polynomial-time algorithm designed to manage the game's numerous values and achieve an efficient linear-time computation of the Shapley value. Moreover, this formulation uniquely enables identifying only the significant high-order feature interactions amidst a potentially exponential set. Through experiments, we demonstrate the robust performance of our methodology in game estimation and in providing explanations for multiple black-box models.

Symbolic Functional Decomposition: A Reconfiguration Approach

Functional decomposition is the process of breaking down a function f into a composition f=g(f_1,...,f_k) of simpler functions f_1,...,f_k belonging to some class F. This fundamental notion can be used to model applications arising in a wide variety of contexts, ranging from machine learning to formal language theory. In this work, we study functional decomposition by leveraging on the notion of functional reconfiguration. In this setting, constraints are imposed not only on the factor functions f_1,...,f_k but also on the intermediate functions arising during the composition process. We introduce a symbolic framework to address functional reconfiguration and decomposition problems. In our framework, functions arising during the reconfiguration process are represented symbolically, using ordered binary decision diagrams (OBDDs). The function g used to specify the reconfiguration process is represented by a Boolean circuit C. Finally, the function class F is represented by a second-order finite automaton A. Our main result states that functional reconfiguration, and hence functional decomposition, can be solved in fixed-parameter linear time when parameterized by the width of the input OBDD, by structural parameters associated with the reconfiguration circuit C, and by the size of the second-order finite automaton A.

Category Prompt Mamba Network for Nuclei Segmentation and Classification

Nuclei segmentation and classification provide an essential basis for tumor immune microenvironment analysis. The previous nuclei segmentation and classification models require splitting large images into smaller patches for training, leading to two significant issues. First, nuclei at the borders of adjacent patches often misalign during inference. Second, this patch-based approach significantly increases the model's training and inference time. Recently, Mamba has garnered attention for its ability to model large-scale images with linear time complexity and low memory consumption. It offers a promising solution for training nuclei segmentation and classification models on full-sized images. However, the Mamba orientation-based scanning method lacks account for category-specific features, resulting in suboptimal performance in scenarios with imbalanced class distributions. To address these challenges, this paper introduces a novel scanning strategy based on category probability sorting, which independently ranks and scans features for each category according to confidence from high to low. This approach enhances the feature representation of uncertain samples and mitigates the issues caused by imbalanced distributions. Extensive experiments conducted on four public datasets demonstrate that our method outperforms state-of-the-art approaches, delivering superior performance in nuclei segmentation and classification tasks.