Worst-case Time Research Articles

Fast computation of the centralizer of a permutation group in the symmetric group

Let G be a permutation group acting on a set Ω. Best known algorithms for computing the centralizer of G in the symmetric group on Ω are all based on the same general approach that involves solving the following two fundamental problems: given a G-orbit Δ of size n, compute the centralizer of the restriction of G to Δ in the symmetric group on Δ; and given two G-orbits Δ and Δ′ each of size n, find an equivalence between the action of G restricted to Δ and the action of G restricted to Δ′ when one exists. If G is given by a generating set X, previous solutions to each of these two problems take O(|X|n2) time.In this paper, we first solve each fundamental problem in O(δn+|X|nlog⁡n) time, where δ is the depth of the breadth-first-search Schreier tree for X rooted at some fixed vertex. For the important class of small-base groups G, we improve the theoretical worst-case time bound of our solutions to O(nlogc⁡n+|X|nlog⁡n) for some constant c. Moreover, if ⌈20log2⁡n⌉ uniformly distributed random elements of G are given in advance, our solutions have, with probability at least 1−1/n, a running time of O(nlog2⁡n+|X|nlog⁡n). We then apply our solutions to obtain a more efficient algorithm for computing the centralizer of G in the symmetric group on Ω. In an experimental evaluation we demonstrate that it is substantially faster than previously known algorithms.

Submodularity-based false data injection attack scheme in multi-agent dynamical systems

Consensus in multi-agent dynamical systems is prone to be sabotaged by the adversary, which has attracted much attention due to its key role in broad applications. In this paper, we study a new false data injection (FDI) attack design problem, where the adversary with limited capability aims to select a subset of agents and manipulate their local multi-dimensional states to maximize the consensus convergence error. We first formulate the FDI attack design problem as a combinatorial optimization problem and prove it is NP-hard. Then, based on the submodularity optimization theory, we provide the necessary and sufficient conditions to guarantee that the convergence error is a submodular function of the set of compromised agents, satisfying the property of diminishing marginal returns. In other words, the benefit of adding an extra agent to the compromised set decreases as that set becomes larger. With this property, we exploit the greedy scheme to find the optimal compromised agent set that can produce the maximum convergence error when adding one extra agent to that set each time. Thus, the FDI attack set selection algorithms are developed to obtain the near-optimal subset of the compromised agents. Furthermore, we derive the analytical suboptimality bounds and the worst-case running time under the proposed algorithm. Extensive simulation results are conducted to show the effectiveness of the proposed algorithm.

Efficient Maximum k-Defective Clique Computation with Improved Time Complexity

k-defective cliques relax cliques by allowing up-to k missing edges from being a complete graph. This relaxation enables us to find larger near-cliques and has applications in link prediction, cluster detection, social network analysis and transportation science. The problem of finding the largest k-defective clique has been recently studied with several algorithms being proposed in the literature. However, the currently fastest algorithm KDBB does not improve its time complexity from being the trivial O(2n), and also, KDBB's practical performance is still not satisfactory. In this paper, we advance the state of the art for exact maximum k-defective clique computation, in terms of both time complexity and practical performance. Moreover, we separate the techniques required for achieving the time complexity from others purely used for practical performance consideration; this design choice may facilitate the research community to further improve the practical efficiency while not sacrificing the worst case time complexity. In specific, we first develop a general framework kDC that beats the trivial time complexity of O(2n) and achieves a better time complexity than all existing algorithms. The time complexity of kDC is solely achieved by our newly designed non-fully-adjacent-first branching rule, excess-removal reduction rule and high-degree reduction rule. Then, to make kDC practically efficient, we further propose a new upper bound, two new reduction rules, and an algorithm for efficiently computing a large initial solution. Extensive empirical studies on three benchmark graph collections with 290 graphs in total demonstrate that kDC outperforms the currently fastest algorithm KDBB by several orders of magnitude.

Fast Maximal Quasi-clique Enumeration: A Pruning and Branching Co-Design Approach

Mining cohesive subgraphs from a graph is a fundamental problem in graph data analysis. One notable cohesive structure is γ-quasi-clique (QC), where each vertex connects at least a fraction γ of the other vertices inside. Enumerating maximal γ-quasi-cliques (MQCs) of a graph has been widely studied and used for many applications such as community detection and significant biomolecule structure discovery. One common practice of finding all MQCs is to (1) find a set of QCs containing all MQCs and then (2) filter out non-maximal QCs. While quite a few algorithms have been developed (which are branch-and-bound algorithms) for finding a set of QCs that contains all MQCs, all focus on sharpening the pruning techniques and devote little effort to improving the branching part. As a result, they provide no guarantee on pruning branches and all have the worst-case time complexity of O*(2n), where O* suppresses the polynomials and n is the number of vertices in the graph. In this paper, we focus on the problem of finding a set of QCs containing all MQCs but deviate from further sharpening the pruning techniques as existing methods do. We pay attention to both the pruning and branching parts and develop new pruning techniques and branching methods that would suit each other better towards pruning more branches both theoretically and practically. Specifically, we develop a new branch-and-bound algorithm called FastQC based on newly developed pruning techniques and branching methods, which improves the worst-case time complexity to O*(αkn), where αk is a positive real number strictly smaller than 2. Furthermore, we develop a divide-and-conquer strategy for boosting the performance of FastQC. Finally, we conduct extensive experiments on both real and synthetic datasets, and the results show that our algorithms are up to two orders of magnitude faster than the state-of-the-art on real datasets.

Dynamic probabilistic risk assessment with K-shortest-paths planning for generating discrete dynamic event trees

Conventional risk assessment methods are often not well suited for fast-changing dynamic and complex systems since the results from the analysis may be averages or valid for a short time before the system’s state changes. A response to this problem is dynamic probabilistic risk assessment (DPRA), which considers the ever-changing nature of such systems and how their dynamic behavior affects the likelihood of future accident scenarios. Performing DPRA is difficult for complex systems — i.e., systems with many interconnected subsystems and components, for example, autonomous systems. There is a combinatorial “explosion” when considering how component failures affect one another and the overall system performance, known in the DPRA literature as the “state explosion problem”, causing DPRA methods to have poor computational performance for large-scale systems. Although solutions to state explosion alleviate the average-case performance, most DPRA methods remain computationally expensive, with exponential worst-case time complexities. In this paper, a method for solving DPRA problems with K-Shortest-Paths planning algorithms is proposed. The method, called KPRA, consists in framing a subset of DPRA problems as relaxed versions of the K-Shortest-Paths (KSP) planning problem, allowing these DPRA problems to be solved by a graph search algorithm called K*, which has a theoretical log-linear worst-case complexity. Therefore, in theory, KPRA with K* solves DPRA problems with computational performance better than exponential. KPRA was implemented and applied to a case study of DPRA for an autonomous ship for validation and comparison with conventional DPRA methods. The case study consists of two ships, one of them autonomous, in a crossing encounter with a possible collision risk. The task is to find the most critical situations for the autonomous ship, i.e., the scenarios where its collision risk is the highest. KPRA’s performance in solving the case study is compared with two conventional DPRA methods. The results show that the KPRA implementation in this work can solve the case study, i.e., produce an output equivalent to the other methods, with a polynomial worst-case computational complexity, i.e., more efficiently than the other methods with exponential complexities.

Improving Worst-case TSN Communication Times of Large Sensor Data Samples by Exploiting Synchronization

Higher levels of automated driving also require a more sophisticated environmental perception. Therefore, an increasing number of sensors transmit their data samples as frame bursts to other applications for further processing. As a vehicle has to react to its environment in time, such data is subject to safety-critical latency constraints. To keep up with the resulting data rates, there is an ongoing transition to a Time-Sensitive Networking (TSN)-based communication backbone. However, the use of TSN-related industry standards does not match the automotive requirements of large timely sensor data transmission, nor it offers benefits on time-critical transmissions of single control data packets. By using the full data rate of prioritized IEEE 802.1Q Ethernet, giving time guarantees on large data samples is possible, but with strongly degraded results due to data collision. Resolving such collisions with time-aware shaping comes with significant overhead. Hence, rather than optimizing the parameters of the existing protocol, we propose a system design that synchronizes the transmission times of sensor data samples. This limits network protocol complexity and hardware requirements by avoiding tight time synchronization and time-aware shaping. We demonstrate that individual sensor data samples are transmitted without significant interference, exclusively at full Ethernet data rate. We provide a synchronous event model together with a straightforward response time analysis for synchronous multi-frame sample transmissions. The results show that worst-case latencies of such sample communication, in contrast to non-synchronized approaches, are close to their theoretical minimum as well as to simulative results while keeping the overall network utilization high.

Deterministic Dynamic Matching in Worst-Case Update Time

We present deterministic algorithms for maintaining a (3/2 + epsilon ) and (2 + epsilon )-approximate maximum matching in a fully dynamic graph with worst-case update times {hat{O}}(sqrt{n}) and {tilde{O}}(1) respectively. The fastest known deterministic worst-case update time algorithms for achieving approximation ratio (2 - delta ) (for any delta > 0) and (2 + epsilon ) were both shown by Roghani et al. (Beating the folklore algorithm for dynamic matching, 2021) with update times O(n^{3/4}) and O_epsilon (sqrt{n}) respectively. We close the gap between worst-case and amortized algorithms for the two approximation ratios as the best deterministic amortized update times for the problem are O_epsilon (sqrt{n}) and {tilde{O}}(1) which were shown in Bernstein and Stein (in: Proceedings of the twenty-seventh annual ACM-SIAM symposium on discrete algorithms, 2016) and Bhattacharya and Kiss (in: 48th international colloquium on automata, languages, and programming, ICALP 2021, 12–16 July, Glasgow, 2021) respectively. The algorithm achieving (3/2 + epsilon ) approximation builds on the EDCS concept introduced by the influential paper of Bernstein and Stein (in: International colloquium on automata, languages, and programming, Springer, Berlin, 2015). Say that H is a (alpha , delta )-approximate matching sparsifier if at all times H satisfies that mu (H) cdot alpha + delta cdot n ge mu (G) (define (alpha , delta )-approximation similarly for matchings). We show how to maintain a locally damaged version of the EDCS which is a (3/2 + epsilon , delta )-approximate matching sparsifier. We further show how to reduce the maintenance of an alpha -approximate maximum matching to the maintenance of an (alpha , delta )-approximate maximum matching building based on an observation of Assadi et al. (in: Proceedings of the twenty-seventh annual (ACM-SIAM) symposium on discrete algorithms, (SODA) 2016, Arlington, VA, USA, January 10–12, 2016). Our reduction requires an update time blow-up of {hat{O}}(1) or {tilde{O}}(1) and is deterministic or randomized against an adaptive adversary respectively. To achieve (2 + epsilon )-approximation we improve on the update time guarantee of an algorithm of Bhattacharya and Kiss (in: 48th International colloquium on automata, languages, and programming, ICALP 2021, 12–16 July, Glasgow, 2021). In order to achieve both results we explicitly state a method implicitly used in Nanongkai and Saranurak (in: Proceedings of the twenty-seventh annual ACM symposium on theory of computing, 2017) and Bernstein et al. (Fully-dynamic graph sparsifiers against an adaptive adversary, 2020) which allows to transform dynamic algorithms capable of processing the input in batches to a dynamic algorithms with worst-case update time.

On the approximation of Euclidean SL via geometric method

Single-linkage with distance-r stopping condition (SL) is a classical clustering technique, which can discover arbitrary shaped clusters. However, its O(n2) worst-case time complexity is still a challenge, where n is the number of points in the dataset. In this paper, we present a relaxed version of the problem in Euclidean space, called the ρ-approximate SL problem. We propose a grid-based algorithm, called ρ-SL, to solve this problem in O(n+ν) expected time, where ν is an integer associated with the maximum and minimum coordinates of each dimension. ρ-SL first partitions the dataset into non-empty grids and constructs a grid tree to organize them for efficient non-empty neighboring grid queries. Then, an initial cluster structure is established to keep track of all pairs of grids that may be mergeable. ρ-SL proceeds to identify the pairs of grids that are mergeable by a fast merging algorithm which takes into account the spatial relationships among points to reduce unnecessary distance calculations. Experiments were conducted on synthetic and real-world datasets to evaluate the performance of ρ-SL. The results show that the proposed algorithm is more efficient than SL and its fastest variants.

Trade-offs in Static and Dynamic Evaluation of Hierarchical Queries

We investigate trade-offs in static and dynamic evaluation of hierarchical queries with arbitrary free variables. In the static setting, the trade-off is between the time to partially compute the query result and the delay needed to enumerate its tuples. In the dynamic setting, we additionally consider the time needed to update the query result under single-tuple inserts or deletes to the database. Our approach observes the degree of values in the database and uses different computation and maintenance strategies for high-degree (heavy) and low-degree (light) values. For the latter it partially computes the result, while for the former it computes enough information to allow for on-the-fly enumeration. We define the preprocessing time, the update time, and the enumeration delay as functions of the light/heavy threshold. By appropriately choosing this threshold, our approach recovers a number of prior results when restricted to hierarchical queries. We show that for a restricted class of hierarchical queries, our approach achieves worst-case optimal update time and enumeration delay conditioned on the Online Matrix-Vector Multiplication Conjecture.

An optimal algorithm for the minimum-width cubic shell problem

Given a set of n points in R3, the minimum-width cubic shell problem asks to find a thinnest cubic shell that encloses the input points, where a cubic shell refers to as a closed volume between two concentric axis-aligned cubes. In this paper, we improve the previous O(nlog2⁡n)-time algorithm presented in Bae (2019) [6] to O(nlog⁡n) worst-case time. This is the first optimal-time algorithm to the problem.

Direct solution of the normal equations in interior point methods for convex transportation problems

Recent research has shown that very large convex transportation problems can be solved efficiently with interior point methods by finding a good pre-conditioner for an iterative linear-equation solver. In this article, it is demonstrated that the specific structure of the constraints allows use of a direct method for solving those linear equations with the same order of worst-case time complexity.

Sound static analysis of regular expressions for vulnerabilities to denial of service attacks

Modern programming languages often provide functions to manipulate regular expressions in standard libraries. If they offer support for advanced features, the matching algorithm has an exponential worst-case time complexity: for some so-called vulnerable regular expressions, an attacker can craft ad hoc strings to force the matcher to exhibit an exponential behavior and perform a Regular Expression Denial of Service (ReDoS) attack. In this paper, we introduce a framework based on a tree semantics to statically identify ReDoS vulnerabilities. In particular, we put forward an algorithm to extract an overapproximation of the set of words that are dangerous for a regular expression, effectively catching all possible attacks. We have implemented the analysis in a tool called rat, and testing it on a dataset of 74,669 regular expressions, we observed that in 99.78% of the instances the analysis terminates in less than one second. We compared rat to seven other ReDoS detectors, and we found that our tool is faster, often by orders of magnitude, than most other tools. While raising a low number of false positives, rat is the only ReDoS detector that does not report false negatives.

Hereditary Cohesive Subgraphs Enumeration on Bipartite Graphs: The Power of Pivot-based Approaches

Finding cohesive subgraphs from a bipartite graph is a fundamental operator in bipartite graph analysis. In this paper, we focus on the problem of mining cohesive subgraphs from a bipartite graph that satisfy a hereditary property. Here a cohesive subgraph meets the hereditary property if all of its subgraphs satisfy the same property as itself. We show that several important cohesive subgraph models, such as maximal biclique and maximal k-biplex, satisfy the hereditary property. The problem of enumerating all maximal hereditary subgraphs was known to be NP-hard. To solve this problem, we first propose a novel and general pivot-based enumeration framework to efficiently enumerate all maximal hereditary subgraphs in a bipartite graph. Then, based on our general framework, we develop a new pivot-based algorithm with several pruning techniques to enumerate all maximal bicliques. We prove that the worst-case time complexity of our pivot-based maximal biclique enumeration algorithm is O(m x 2n/2 ) (or O(m\times 1.414^n)) which is near optimal since there exist up to O(2n/2 ) maximal bicliques in a bipartite graph with n vertices and m edges. Moreover, we also show that our algorithm can achieve polynomial-delay time complexity with a slight modification. Third, on the basis of our general framework, we also devise a novel pivot-based algorithm with several non-trivial pruning techniques to enumerate maximal k-biplexes in a bipartite graph. Finally, we conduct extensive experiments using 11 real-world bipartite graphs to evaluate the proposed algorithms. The results show that our pivot-based solutions can achieve one order of magnitude (three orders of magnitude) faster than the state-of-the-art maximal biclique enumeration algorithms (maximal k-biplex enumeration algorithms).

Backward Pattern Matching on Elastic-Degenerate Strings

The concept of elastic-degenerate strings (EDS) was introduced as a way of representing a sequenced population of the same species. Several online elastic-degenerate string matching (EDSM) algorithms were presented so far. Some of them provide a practical implementation. We propose a new on-line EDSM algorithm BNDM-EDS. Our algorithm combines two traditional algorithms BNDM and Shift-And that were adapted to the specificities needed by Elastic-Degenerate Strings. BNDM-EDS is running in {mathcal {O}}(Nmlceil frac{m}{w}rceil ) worst-case time. This implies {mathcal {O}}(Nm) time for small patterns, where m is the length of the searched pattern, N is the size of EDS, and w is the size of the computer word. The algorithm uses {mathcal {O}}(N + n) space, where n is the length of EDS. BNDM-EDS requires a simple preprocessing step with time and space {mathcal {O}}(m). We further present an extended version of the algorithm called BNDM-EDS-AA, which allows searching for proteins (amino acid sequences) encoded in DNA EDS. Though BNDM-EDS-AA considers more complex patterns than BNDM-EDS, it keeps its time and space efficiency. Experimental results on real genomic data show the superiority of BNDM-EDS over state-of-the-art algorithms. We also experimentally evaluate the efficiency of BNDM-EDS-AA for different pattern complexities, and evaluate the scaling of both BNDM-EDS and BNDM-EDS-AA with varying attributes of EDS datasets.

Friction-adaptive stochastic nonlinear model predictive control for autonomous vehicles

This paper addresses the trajectory-tracking problem under uncertain road-surface conditions for autonomous vehicles. We propose a stochastic nonlinear model predictive controller (SNMPC) that learns a tyre–road friction model online using standard automotive-grade sensors. Learning the entire tyre–road friction model in real time requires driving in the nonlinear, potentially unstable regime of the vehicle dynamics, using a prediction model that may not have fully converged. To handle this, we formulate the tyre-friction model learning in a Bayesian framework and propose two estimators that learn different aspects of the tyre–road friction. The estimators output the estimate of the tyre-friction model as well as the uncertainty of the estimate, which expresses the confidence in the model for different driving regimes. The SNMPC exploits the uncertainty estimate in its prediction model to take proper action when the uncertainty is large. We validate the approach in an extensive Monte Carlo study using real vehicle parameters and in CarSim. The results when comparing to various MPC approaches indicate a substantial reduction in constraint violations, as well as a reduction in closed-loop cost. We also demonstrate the real-time feasibility in automotive-grade processors using a dSPACE MicroAutoBox-II rapid prototyping unit, showing a worst-case computation time of roughly 40 ms.

Fast Density-Based Clustering: Geometric Approach

DBSCAN is a fundamental density-based clustering algorithm with extensive applications. However, a bottleneck of DBSCAN is its O(n2) worst-case time complexity. In this paper, we propose an algorithm called GAP-DBC, which exploits the geometric relationships between points to solve this problem. GAP-DBC introduces an efficient partitioning algorithm to partition the data set with a limited number of range queries and then establishes an initial cluster structure based on the partition. GAP-DBC proceeds to iteratively refine the cluster structure by additional range queries. Finally, the cluster structure is accomplished using an iterative algorithm that utilizes the spatial relationships among points to reduce unnecessary distance calculations. We further demonstrate theoretically that GAP-DBC has an excellent guarantee in terms of computational efficiency. We conducted experiments on both synthetic and real-world data sets to evaluate the performance of GAP-DBC. The results show that our algorithm is competitive with other state-of-the-art algorithms.

Maximum k-Biplex Search on Bipartite Graphs: A Symmetric-BK Branching Approach

Enumerating maximal k-biplexes (MBPs) of a bipartite graph has been used for applications such as fraud detection. Nevertheless, there usually exists an exponential number of MBPs, which brings up two issues when enumerating MBPs, namely the effectiveness issue (many MBPs are of low values) and the efficiency issue (enumerating all MBPs is not affordable on large graphs). Existing proposals of tackling this problem impose constraints on the number of vertices of each MBP to be enumerated, yet they are still not sufficient (e.g., they require to specify the constraints, which is often not user-friendly, and cannot control the number of MBPs to be enumerated directly). Therefore, in this paper, we study the problem of finding K MBPs with the most edges called MaxBPs, where K is a positive integral user parameter. The new proposal well avoids the drawbacks of existing proposals (i.e., the number of MBPs to be enumerated is directly controlled and the MBPs to be enumerated tend to have high values since they have more edges than the majority of MBPs). We formally prove the NP-hardness of the problem. We then design two branch-and-bound algorithms, among which, the better one called FastBB improves the worst-case time complexity to O*(γkn), where O* suppresses the polynomials, γk is a real number that relies on k and is strictly smaller than 2, and n is the number of vertices in the graph. For example, for k=1, γk is equal to 1.754. We further introduce three techniques for boosting the performance of the branch-and-bound algorithms, among which, the best one called PBIE can further improve the time complexity to O*(γkd3) for large sparse graphs, where d is the maximum degree of the graph (note that d< <n for sparse graphs). We conduct extensive experiments on both real and synthetic datasets, and the results show that our algorithm is up to four orders of magnitude faster than all baselines and finding MaxBPs works better than finding all MBPs for a fraud detection application.

Multi-agent deep reinforcement learning for user association and resource allocation in integrated terrestrial and non-terrestrial networks

Integrating the terrestrial network with non-terrestrial networks to provide radio access as anticipated in the beyond 5G networks calls for efficient user association and resource allocation strategies. In this work, a weighted sum single objective optimization problem that maximizes the total network data rate while minimizing the mobility-induced handoffs and prioritizing service provisioning of mission-critical users in the integrated terrestrial and non-terrestrial network is formulated. The problem’s complexity is reduced by solving the defined problem in two phases; the user association followed by the resource distribution phase. Several proposed approaches to the user association sub-problem utilize a central node that requires nearly-complete information, which may not be available in real time. On the contrary, this paper proposes a centralized training and distributed execution multi-agent dueling double deep Q network (MA3DQN) solution. Each user collects the channel state and access node loading information in this approach and makes an association decision that considers its quality of service requirements. The algorithm’s performance is validated through comparison with the genetic algorithm (GA), the integer linear programming (ILP), a heuristic approximation-based solution, the greedy approach, and the random user association (RUA) algorithm. Moreover, the paper simulates the multi-agent deep Q network solution as an additional benchmark algorithm. Simulation results reveal that as the number of users in the network increases, the acquired data rate of the MA3DQN is within 0.48% and 0.4% of that achieved by the GA and ILP, respectively, and outperforms all other algorithms. Notably, the proposed MA3DQN algorithm presents the best running time, attaining a gain of 99.9% over the GA algorithm, which performs the poorest among the algorithms characterized by polynomial worst-case time complexity. Besides, the MA3DQN approach maintains a handoff probability of zero, unlike the ILP, the approximation-based, greedy, and RUA solutions.

GriT-DBSCAN: A spatial clustering algorithm for very large databases

DBSCAN is a fundamental spatial clustering algorithm with numerous practical applications. However, a bottleneck of DBSCAN is its O(n2) worst-case time complexity. To address this limitation, we propose a new grid-based algorithm for exact DBSCAN in Euclidean space called GriT-DBSCAN, which is based on the following two techniques. First, we introduce grid tree to organize the non-empty grids for the purpose of efficient non-empty neighboring grids queries. Second, by utilizing the spatial relationships among points, we propose a technique that iteratively prunes unnecessary distance calculations when determining whether the minimum distance between two sets is less than or equal to a certain threshold. We theoretically demonstrate that GriT-DBSCAN has excellent reliability in terms of time complexity. In addition, we obtain two variants of GriT-DBSCAN by incorporating heuristics, or by combining the second technique with an existing algorithm. Experiments are conducted on both synthetic and real-world data sets to evaluate the efficiency of GriT-DBSCAN and its variants. The results show that our algorithms outperform existing algorithms.

Joint Localization and Communication Study for Intelligent Reflecting Surface Aided Wireless Communication System

The intelligent reflecting surface (IRS) is promising in assisting user localization and wireless communication in the future wireless networks. In this paper, a novel IRS-aided joint localization and communication (L&C) scheme is designed in a millimeter-wave transmission system. For the proposed scheme, the user position/orientation estimation error bound (POEB) and the effective achievable data rate (EADR) are derived in closed-form as L&C performance metrics, which reveal the inherent trade-off between L&C capabilities. To achieve the joint optimal point of the POEB and EADR in consideration of the localization errors, a worst-case robust beamforming and time allocation optimization problem is formulated. To solve the original non-convex problem, a novel joint optimization approach is developed. Specifically, from an equivalent minimax problem, the local optimal solutions of the transceiver beamformers, the IRS phase-shift matrix, and the time allocation ratio between user localization stage (ULS) and effective data transmission stage (EDTS), are obtained in closed-form with respect to the localization errors. Then, the worst-case localization error is iteratively found by a dedicated majorize-minimization (MM) based algorithm. Subsequently, potential extensions to general wireless channels and discrete phase-shift models are discussed in detail. Finally, simulations are carried out to show the optimization results and the L&C performance trade-off. In comparison with the conventional non-robust method, the proposed approach is validated to be robust against the user localization uncertainty.