Efficient Polynomial Time Research Articles

Statistical and computational efficiency for smooth tensor estimation with unknown permutations

We consider the problem of structured tensor denoising in the presence of unknown permutations. Such data problems arise commonly in recommendation systems, neuroimaging, community detection, and multiway comparison applications. Here, we develop a general family of smooth tensor models up to arbitrary index permutations; the model incorporates the popular tensor block models and Lipschitz hypergraphon models as special cases. We show that a constrained least-squares estimator in the block-wise polynomial family achieves the minimax error bound. A phase transition phenomenon is revealed with respect to the smoothness threshold needed for optimal recovery. In particular, we find that a polynomial of degree up to ( m − 2 ) ( m + 1 ) / 2 is sufficient for accurate recovery of order-m tensors, whereas higher degrees exhibit no further benefits. This phenomenon reveals the intrinsic distinction for smooth tensor estimation problems with and without unknown permutations. Furthermore, we provide an efficient polynomial-time Borda count algorithm that provably achieves the optimal rate under monotonicity assumptions. The efficacy of our procedure is demonstrated through both simulations and Chicago crime data analysis.

MDSA: A Dynamic and Greedy Approach to Solve the Minimum Dominating Set Problem

The graph theory is one of the fundamental structures in computer science used to model various scientific and engineering problems. Many problems within the graph theory are categorized as NP-hard and NP-complete. One such problem is the minimum dominating set (MDS) problem, which seeks to identify the minimum possible subsets in a graph such that every other node in the subset is directly connected to a node in this subset. Due to its inherent complexity, developing an efficient polynomial-time method to address the MDS problem remains a significant challenge in graph theory. This paper introduces a novel algorithm that utilizes a centrality measure known as the Malatya Centrality to effectively address the MDS problem. The proposed algorithm, called the Malatya Dominating Set Algorithm (MDSA), leverages centrality values to identify dominating sets within a graph. It extends the Malatya centrality by incorporating a second-level centrality measure, which enhances the identification of dominating nodes. Through a systematic and algorithmic approach, these centrality values are employed to pinpoint the elements of the dominating set. The MDSA uniquely integrates greedy and dynamic programming strategies. At each step, the algorithm selects the most optimal (or near-optimal) node based on the centrality values (greedy approach) while updating the neighboring nodes’ criteria to influence subsequent decisions (dynamic programming). The proposed algorithm demonstrates efficient performance, particularly in large-scale graphs, with time and space requirements scaling proportionally with the size of the graph and its average degree. Experimental results indicate that our algorithm outperforms existing methods, especially in terms of time complexity when applied to large datasets, showcasing its effectiveness in addressing the MDS problem.

Efficient polynomial-time approximation scheme for the genus of dense graphs

The main results of this paper provide an Efficient Polynomial-Time Approximation Scheme (EPTAS) for approximating the genus (and non-orientable genus) of dense graphs. By dense we mean that | E ( G )| ≥ α | V ( G )| 2 for some fixed α > 0. While a constant-factor approximation is trivial for this class of graphs, approximations with factor arbitrarily close to 1 need a sophisticated algorithm and complicated mathematical justification. More precisely, we provide an algorithm that for a given (dense) graph G of order n and given ε > 0, returns an integer g such that G has an embedding in a surface of genus g , and this is ε-close to a minimum genus embedding in the sense that the minimum genus \(\mathsf {g}(G) \) of G satisfies: \(\mathsf {g}(G)\le g\le (1+\varepsilon)\mathsf {g}(G) \) . The running time of the algorithm is O ( f (ε) n 2 ), where f (·) is an explicit function. Next, we extend this algorithm to also output an embedding (rotation system) whose genus is g . This second algorithm is an Efficient Polynomial-time Randomized Approximation Scheme (EPRAS) and runs in time O ( f 1 (ε) n 2 ). Our algorithms are based on the analysis of minimum genus embeddings of quasirandom graphs. We use a general notion of quasirandom graphs [25]. We start with a regular partition obtained via an algorithmic version of the Szemerédi Regularity Lemma (due to Frieze and Kannan [17] and to Fox, Lovász, and Zhao [14, 15]). We then partition the input graph into a bounded number of quasirandom subgraphs, which are preselected in such a way that they admit embeddings using as many triangles and quadrangles as faces as possible. Here we provide an ε-approximation ν ( G ) for the maximum number of edge-disjoint triangles in G . The value ν ( G ) can be computed by solving a linear program whose size is bounded by certain value f 2 (ε) depending only on ε. After solving the linear program, the genus can be approximated (see Corollary). The proof of this result is long and will be of independent interest in topological graph theory.

On submodular prophet inequalities and correlation gap

Prophet inequalities and secretary problems have been extensively studied in recent years due to their elegance, connections to online algorithms, stochastic optimization, and mechanism design problems in game theoretic settings. Rubinstein and Singla [31] developed a notion of combinatorial prophet inequalities in order to generalize the standard prophet inequality setting to combinatorial valuation functions such as submodular and subadditive functions. For non-negative submodular functions they demonstrated a constant factor prophet inequality for matroid constraints. Along the way they showed a variant of the correlation gap for non-negative submodular functions.In this paper we revisit their notion of correlation gap as well as the standard notion of correlation gap and prove much tighter and cleaner bounds. Via these bounds and other insights we obtain substantially improved constant factor combinatorial prophet inequalities for both monotone and non-monotone submodular functions over any constraint that admits an Online Contention Resolution Scheme. In addition to improved bounds we describe efficient polynomial-time algorithms that achieve these bounds.

The Continuous-Time Joint Replenishment Problem: ϵ-Optimal Policies via Pairwise Alignment

The main contribution of this paper resides in developing a new algorithmic approach for addressing the continuous-time joint replenishment problem, termed [Formula: see text]-pairwise alignment. The latter mechanism, through which we synchronize multiple economic order quantity models, allows us to devise a purely combinatorial algorithm for efficiently approximating optimal policies within any degree of accuracy. As a result, our work constitutes the first quantitative improvement over power-of-2 policies, which have been state-of-the-art in this context since the mid-1980s. Moreover, in light of recent intractability results, by proposing an efficient polynomial-time approximation scheme for the joint replenishment problem, we resolve the long-standing open question regarding the computational complexity of this classical setting. This paper was accepted by Chung Piaw Teo, optimization. Funding: This work was supported by the Israel Science Foundation [Grant 1407/20].

Anti Tai mapping for unordered labeled trees

The well-studied Tai mapping between two rooted labeled trees T1=(V1,E1) and T2=(V2,E2) defines a one-to-one mapping between nodes in T1 and T2 that preserves ancestor relationship [1]. For unordered trees the problem of finding a maximum-weight Tai mapping is known to be NP-complete [2]. In this work, we define an anti Tai mapping M⊆V1×V2 as a binary relation between two unordered labeled trees such that any two (x,y),(x′,y′)∈M violate ancestor relationship and thus cannot be part of the same Tai mapping, i.e. (x≤x′⇔y≰y′)∨(x′≤x⇔y′≰y), given an ancestor order x<x′ meaning that x is an ancestor of x′. Finding a maximum-weight anti Tai mapping arises in the cutting plane method for solving the maximum-weight Tai mapping problem via integer programming. We give an efficient polynomial-time algorithm for finding a maximum-weight anti Tai mapping for the case when one of the two trees is a path and further show how to extend this result in order to provide a polynomially computable lower bound on the optimal anti Tai mapping for two unordered labeled trees. The latter result stems from the special class of anti Tai mappings defined by the more restricted condition x∼x′⇔y≁y′, where ∼ denotes that two nodes belong to the same root-to-leaf path. For this class, we give an efficient algorithm that solves the problem exactly on two unordered trees in O(|V1|2|V2|2).

Peak Demand Minimization via Sliced Strip Packing

We study the Non-preemptive Peak Demand Minimization (NPDM) problem, where we are given a set of jobs, specified by their processing times and energy requirements. The goal is to schedule all jobs within a fixed time period such that the peak load (the maximum total energy requirement at any time) is minimized. This problem has recently received significant attention due to its relevance in smart-grids. Theoretically, the problem is related to the classical strip packing problem (SP). In SP, a given set of axis-aligned rectangles must be packed into a fixed-width strip, such that the height of the strip is minimized. NPDM can be modeled as strip packing with slicing and stacking constraint: each rectangle may be cut vertically into multiple slices and the slices may be packed into the strip as individual pieces. The stacking constraint forbids solutions where two slices of the same rectangle are intersected by the same vertical line. Non-preemption enforces the slices to be placed in contiguous horizontal locations (but may be placed at different vertical locations). We obtain a (5/3+varepsilon )-approximation algorithm for the problem. We also provide an asymptotic efficient polynomial-time approximation scheme (AEPTAS) which generates a schedule for almost all jobs with energy consumption (1+varepsilon ) {textrm{OPT}}. The remaining jobs fit into a thin container of height 1. This AEPTAS is used as a subroutine to acquire the (5/3+varepsilon )-approximation algorithm. The previous best result for NPDM was a 2.7-approximation based on FFDH (Ranjan et al., in: 2015 IEEE symposium on computers and communication (ISCC), pp 758–763, IEEE, 2015). One of our key ideas is providing several new lower bounds on the optimal solution of a geometric packing, which could be useful in other related problems. These lower bounds help us to obtain approximative solutions based on Steinberg’s algorithm in many cases. In addition, we show how to split schedules generated by the AEPTAS into few segments and to rearrange the corresponding jobs to insert the thin container mentioned above, such that it does not exceed the bound of (5/3+varepsilon ) {textrm{OPT}}.

Efficient Graph-based Signal Motif Discovery with Performance Bounds

In this paper, we investigate the problem of signal motif discovery. In the formulation considered, the goal is to find an unknown pattern (or motif) repeated across multiple data sets. This problem formulation can be applied to various domains, such as DNA sequence alignment and matching, motif discovery in time-series, and object detection and localization in images. We take the approach of minimizing an objective that is the sum over a measure of the difference between a candidate instance in each signal collection (or set) and the unknown pattern. Additionally, a non-negative instance dependent penalty is introduced. The proposed general objective can be used to capture well-known problems (e.g., blind joint time-delay estimation). Due to the non-convex nature of the problem and often the integer programming flavor of the approach, brute-force solution is non-polynomial and computationally prohibitive for large scale problems. We propose an efficient polynomial time (quadratic in the number of instances) bipartite graph based approximation to solve the problem. We provide a theoretical analysis for the proposed solution including bounds on the gap from the optimal solution and conditions for optimality. In particular, we show that the objective value for the proposed solution is no more than twice the objective value of the optimal solution. To illustrate the merit of the proposed approach, we present qualitative and quantitative empirical analysis of the proposed approach on several applications and compare our method to appropriate alternatives.

Models and Algorithms of Abstract Flows in Evacuation Planning

Flows over time generalize classical network flows by introducing a notion of time. Each arc is equipped with a transit time that specifies how long flow takes to traverse it, while flow rates may vary over time within the given edge capacities. Ford and Fulkerson’s original 1956 max flow/min cut paper formulated max flow in terms of flows on paths. In 1974, Hoffman pointed out that Ford and Fulkerson’s original proof was quite abstract, and applied to a wide range of max flow-like problems. In this abstract model we have capacitated elements and linearly ordered subsets of elements called paths that satisfy switching property. When two paths P and Q cross at an element (node) then there must be a path that is a subset of the first path up to the crossing element and a subset of the second path after the crossing element. Contraflow is a widely accepted solution approach that increases the flow and decreases the evacuation time making the traffic smooth during evacuation by reversing the required road directions from the risk areas to the safe places. In this paper, we integrate the abstract flow with contraflow, give mathematical formulations of these models and present efficient polynomial time algorithms for solving the abstract contraflow problems.

Multiway Spherical Clustering Via Degree-Corrected Tensor Block Models

We consider the problem of multiway clustering in the presence of unknown degree heterogeneity. Such data problems arise commonly in applications such as recommendation system, neuroimaging, community detection, and hypergraph partitions in social networks. The allowance of degree heterogeneity provides great flexibility in clustering models, but the extra complexity poses significant challenges in both statistics and computation. Here, we develop a degree-corrected tensor block model with estimation accuracy guarantees. We present the phase transition of clustering performance based on the notion of angle separability, and we characterize three signal-to-noise regimes corresponding to different statistical-computational behaviors. In particular, we demonstrate that an intrinsic statistical-to-computational gap emerges only for tensors of order three or greater. Further, we develop an efficient polynomial-time algorithm that provably achieves exact clustering under mild signal conditions. The efficacy of our procedure is demonstrated through two data applications, one on human brain connectome project, and another on Peru Legislation network dataset.

Machine learning assisted delay minimization in full duplex energy constrained cooperative communication network

Cooperative communication is key in attaining high network throughput and better coverage area for a wireless powered communication network. The conventional solution methodologies proposed in the literature for these networks suffer from high run-time complexity. This paper presents an efficient polynomial-time machine learning approach for solving the total transmission time minimization problem through relay selection and scheduling of the users in a full duplex wireless powered cooperative communication network. Exploiting the simplicity offered by the tree ensemble models, a distributed gradient-boosted decision tree based model (XGBoost) is incorporated. The inputs of the model are the uplink channel coefficients and energy harvesting rates of the relays and the users, and outputs are the optimal completion time of the relays and users, relay selection and schedule of the users. Simulation results indicate that the proposed model very well approximates the true outputs, i.e., performs very close to the optimal solution of the optimization problem, while maintaining a very low run-time computational complexity. Specifically, for a network of 9 users, the run-time of the proposed model is just 25% of the optimal solution, and this ratio grows as the number of users increases in the network.

The Electric Dial-a-Ride Problem on a Fixed Circuit

Shared mobility services involving electric autonomous shuttles have increasingly been implemented in recent years. Because of various restrictions, these services are currently offered on fixed circuits and operated with fixed schedules. This study introduces a service variant with flexible stopping patterns and schedules. Specifically, in the electric dial-a-ride problem on a fixed circuit (eDARP-FC), a fleet of capacitated electric shuttles operates on a given circuit consisting of a recharging depot and a sequence of stations where passengers can be picked up and dropped off. The shuttles may perform multiple laps, between which they may need to recharge. The goal of the problem is to determine the vehicles’ stopping sequences and schedules, including recharging plans, so as to minimize a weighted sum of the total passenger excess time and the total number of laps. The eDARP-FC is formulated as a nonstandard lap-based mixed integer linear programming and is shown to be NP-Hard. Efficient polynomial time algorithms are devised for two special scheduling subproblems. These algorithms and several heuristics are then applied as subroutines within a large neighborhood search metaheuristic. Experiments on instances derived from a real-life system demonstrate that the flexible service results in a 32%–75% decrease in the excess time at the same operational costs. Funding: This work was supported by the Fonds Wetenschappelijk Onderzoek [Project Data-Driven Logistics: Grant S007318N; Project Optimizing the Design of a Hybrid Urban Mobility System: Grant G020222N; and Grant OR4Logistics]. Y. Molenbruch is partially funded by the Fonds Wetenschappelijk Onderzoek [Grant 1202719N]. The computational resources and services used in this work were provided by the Flemish Supercomputer Center funded by the Fonds Wetenschappelijk Onderzoek and the Flemish Government. Supplemental Material: The electronic companion is available at https://doi.org/10.1287/trsc.2023.1208 .

Parallel-batch scheduling with rejection: Structural properties and approximation algorithms

In this paper, we consider a parallel-batch machine scheduling (PBMS) model that unifies several existing scheduling models in the extant literature. In our scheduling model, a given set of jobs with different release dates are scheduled onto a machine that can process multiple jobs simultaneously in a batch. However, the number of jobs in a batch is limited. Each job can be rejected subject to a job-dependent penalty cost, but the total penalty cost of the jobs rejected is required to be no greater than a given bound. The objective is to minimize the weighted sum of the makespan of accepted jobs and the total rejection penalty of rejected jobs. We provide some important structural properties and develop fast approximation algorithms. We also present an efficient polynomial time approximation scheme (EPTAS) with near-linear-time complexity. Our results significantly improve both the time complexities and performance bounds of several existing algorithms and approximation schemes.

Statistical matching and subclassification with a continuous dose: Characterization, algorithm, and application to a health outcomes study

Subclassification and matching are often used in empirical studies to adjust for observed covariates; however, they are largely restricted to relatively simple study designs with a binary treatment and less developed for designs with a continuous exposure. Matching with exposure doses is particularly useful in instrumental variable designs and in understanding the dose-response relationships. In this article we propose two criteria for optimal subclassification based on subclass homogeneity, in the context of having a continuous exposure dose, and propose an efficient polynomial-time algorithm that is guaranteed to find an optimal subclassification with respect to one criterion and serves as a 2-approximation algorithm for the other criterion. We discuss how to incorporate dose and use appropriate penalties to control the number of subclasses in the design. Via extensive simulations, we systematically compare our proposed design to optimal nonbipartite pair matching and demonstrate that combining our proposed subclassification scheme with regression adjustment helps to reduce model dependence for parametric causal inference with a continuous dose. We apply the new design and associated randomization-based inferential procedure to study the effect of transesophageal echocardiography (TEE) monitoring during coronary artery bypass graft (CABG) surgery on patients’ postsurgery clinical outcomes, using Medicare and Medicaid claims data, and find evidence that TEE monitoring lowers patients’ all-cause 30-day mortality rate.

Revisit the scheduling problem with assignable or generalized due dates to minimize total weighted late work

ABSTRACT We revisit the single-machine scheduling for minimising the total weighted late work with assignable due dates (ADD-scheduling) and generalised due dates (GDD-scheduling). In particular, we consider the following three problems: (i) the GDD-scheduling problem for minimising the total weighted late work, (ii) the ADD-scheduling problem for minimising the total weighted late work, and (iii) the ADD-scheduling problem for minimising the total late work. In the literature, the above three problems are proved to be NP-hard, but their exact complexity (unary NP-hardness or pseudo-polynomial-time solvability) are unknown. In this paper, we address these open problems by showing that the first two problems are unary NP-hard and the third problem admits pseudo-polynomial-time algorithms. For the third problem, we also present a 2-approximation solution and a fully polynomial-time approximation scheme. Computational experiments show that our algorithms and solutions are efficient. When the jobs have identical processing times, we further present more efficient polynomial-time algorithms.

Learning to Solve Task-Optimized Group Search for Social Internet of Things

With the maturity and popularity of Internet of Things (IoT), the notion of Social Internet of Things (SIoT) has been proposed to support novel applications and networking services for the IoT in more effective and efficient ways. Although there are many works for SIoT, they focus on designing the architectures and protocols for SIoT under the specific schemes. How to efficiently utilize the collaboration capability of SIoT to complete complex tasks remains unexplored. Therefore, we propose a new problem family, namely, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Task-Optimized SIoT Selection (TOSS)</i> , to find the best group of IoT objects for a given set of tasks in the task pool. TOSS aims to select the target SIoT group such that the target SIoT group is able to easily communicate with each other while maximizing the accuracy of performing the given tasks. We propose two problem formulations, named <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Bounded Communication-loss TOSS (BC-TOSS)</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Robustness Guaranteed TOSS (RG-TOSS)</i> , for different scenarios and prove that they are both NP-hard and inapproximable. We propose a polynomial-time algorithm with a performance guarantee for BC-TOSS, and an efficient polynomial-time algorithm to obtain good solutions for RG-TOSS. Moreover, as RG-TOSS is NP-hard and inapproximable within any factor, we further propose <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Structure-Aware Reinforcement Learning (SARL)</i> to leverage the Graph Convolutional Networks (GCN) and Deep Reinforcement Learning (DRL) to effectively solve RG-TOSS. Further, since we use graph models to simulate the problem instance for DRL, which is different from the real ones, we propose <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Structure-Aware Meta Reinforcement Learning (SAMRL)</i> for fast adapting to new domains. Experimental results on multiple real datasets indicate that our proposed algorithms outperform the other deterministic and learning-based baseline approaches.

A new model and algorithms in firefighting theory

Continuous and discrete models as well as algorithms Bressan (2007) and Finbow and MacGillivray (2009) for firefighting problems are well-studied in Theoretical Computer Science. We introduce a new, discrete, and more general framework based on a graph to study firefighting problems in various terrains. In the context of this model, we present two different firefighting problems called village protection problems, where the goal is to protect a specific set of locations and provide efficient polynomial time algorithms for their solutions. We also discuss extensions of the model.

When Can the Defender Effectively Deceive Attackers in Security Games?

This paper studies defender patrol deception in general Stackelberg security games (SSGs), where a defender attempts to alter the attacker's perception of the defender's patrolling intensity so as to influence the attacker's decision making. We are interested in understanding the complexity and effectiveness of optimal defender deception under different attacker behavior models. Specifically, we consider three different attacker strategies of response (to the defender's deception) with increasing sophistication, and design efficient polynomial-time algorithms to compute the equilibrium for each. Moreover, we prove formal separation for the effectiveness of patrol deception when facing an attacker of increasing sophistication, until it becomes even harmful to the defender when facing the most intelligent attacker we consider. Our results shed light on when and how deception should be used in SSGs. We conduct extensive experiments to illustrate our theoretical results in various game settings.

Achieving QoS for bursty uRLLC applications over passive optical networks

Emerging real-time applications such as those classified under ultra-reliable low latency (uRLLC) generate bursty traffic and have strict quality of service (QoS) requirements. The passive optical network (PON) is a popular access network technology, which is envisioned to handle such applications at the access segment of the network. However, the existing standards cannot handle strict QoS constraints for such applications. The available solutions rely on instantaneous heuristic decisions and maintain QoS constraints (mostly bandwidth) in an average sense. Existing proposals in generic networks with optimal strategies are computationally complex and are, therefore, not suitable for uRLLC applications. This paper presents a novel computationally efficient, far-sighted bandwidth allocation policy design for facilitating bursty uRLLC traffic in a PON framework while satisfying strict QoS (age of information/delay and bandwidth) requirements. To this purpose, first we design a delay-tracking mechanism, which allows us to model the resource allocation problem from a control-theoretic viewpoint as a model predictive control (MPC) problem. MPC helps in making far-sighted decisions regarding resource allocations and captures the time-varying dynamics of the network. We provide computationally efficient polynomial time solutions and show their implementation in the PON framework. Compared to existing approaches, MPC can improve delay violations by 15% and 45% at loads of 0.8 and 0.9, respectively, for delay-constrained applications of 1 ms and 4 ms. Our approach is also robust to varying traffic arrivals.

Comparison between a quantum annealer and a classical approximation algorithm for computing the ground state of an Ising spin glass.

Finding the ground state of an Ising spin glass on general graphs belongs to the class of NP-hard problems, widely believed to have no efficient polynomial-time algorithms to solve them. An approach developed in computer science for dealing with such problems is to devise approximation algorithms; these are algorithms, whose run time scales polynomially with the input size, that provide solutions with provable guarantees on their quality in terms of the optimal unknown solution. Recently, several algorithms for the Ising spin-glass problem on a bounded degree graph that provide different approximation guarantees were introduced. D-Wave, a Canadian-based company, has constructed a physical realization of a quantum annealer and has enabled researchers and practitioners to access it via their cloud service. D-Wave is particularly suited for computing an approximation for the ground state of an Ising spin glass on its Chimera and Pegasus graphs-both with a bounded degree. To assess the quality of D-Wave's solution, it is natural to compare it to classical approximation algorithms specifically designed to solve the same problem. In this work, we compare the performance of a recently developed approximation algorithm to solve the Ising spin-glass problem on graphs of bounded degree against the performance of the D-Wave computer. We also compared the performance of D-Wave's computer in the Chimera architecture against the performance of a heuristic tailored specifically to handle the Chimera graph. We found that the D-Wave computer was able to find better approximations for all the random instances of the problem we studied-Gaussian weights, uniform weights, and discrete binary weights. Furthermore, the convergence times of D-Wave's computer were also significantly better. These results indicate the merit of D-Wave's computer under certain specific instances. More broadly, our method is relevant to a wider class of performance comparison studies, and we suggest that it is important to compare the performance of quantum computers not only against exact classical algorithms with exponential run-time scaling, but also against approximation algorithms with polynomial run-time scaling and a provable guarantee of performance.