Approximation Algorithm Research Articles

Approximate bregman proximal gradient algorithm for relatively smooth nonconvex optimization

AbstractIn this paper, we propose the approximate Bregman proximal gradient algorithm (ABPG) for solving composite nonconvex optimization problems. ABPG employs a new distance that approximates the Bregman distance, making the subproblem of ABPG simpler to solve compared to existing Bregman-type algorithms. The subproblem of ABPG is often expressed in a closed form. Similarly to existing Bregman-type algorithms, ABPG does not require the global Lipschitz continuity for the gradient of the smooth part. Instead, assuming the smooth adaptable property, we establish the global subsequential convergence under standard assumptions. Additionally, assuming that the Kurdyka–Łojasiewicz property holds, we prove the global convergence for a special case. Our numerical experiments on the $$\ell _p$$ ℓ p regularized least squares problem, the $$\ell _p$$ ℓ p loss problem, and the nonnegative linear system show that ABPG outperforms existing algorithms especially when the gradient of the smooth part is not globally Lipschitz or even locally Lipschitz continuous.

Hardness of Approximate Diameter: Now for Undirected Graphs

Approximating the graph diameter is a basic task of both theoretical and practical interest. A simple folklore algorithm can output a 2-approximation to the diameter in linear time by running BFS from an arbitrary vertex. It has been open whether a better approximation is possible in near-linear time. A series of papers on fine-grained complexity have led to strong hardness results for diameter in directed graphs, culminating in a recent tradeoff curve independently discovered by [Li, STOC’21] and [Dalirrooyfard and Wein, STOC’21], showing that under the Strong Exponential Time Hypothesis (SETH), for any integer k ≥ 2 and δ > 0, a \(2-\frac{1}{k}-\delta \) approximation for diameter in directed m -edge graphs requires m 1 + 1/( k − 1) − o (1) time. In particular, the simple linear time 2-approximation algorithm is optimal for directed graphs. In this paper we prove that the same tradeoff lower bound curve is possible for undirected graphs as well, extending results of [Roditty and Vassilevska W., STOC’13], [Li’20] and [Bonnet, ICALP’21] who proved the first few cases of the curve, k = 2, 3 and 4, respectively. Our result shows in particular that the simple linear time 2-approximation algorithm is also optimal for undirected graphs. To obtain our result, we extract the core ideas in known reductions and introduce a unification and generalization that could be useful for proving SETH-based hardness for other problems in undirected graphs related to distance computation.

Effects of carrier structure parameters on diesel particulate filter capture performance based on grey relational analysis

To reduce the pressure drop of diesel particulate filter (DPF) as well as improve its collection efficiency. A mathematical model of DPF pressure drop and capture efficiency was established. This research systematically investigates the effects of channels per square inch (CPSI), wall thickness, and carrier ratio on DPF performance through grey relational analysis (GRA) and polynomial approximation algorithm (PAA). The findings reveal that DPF exhibits lower pressure drop and higher collection efficiency in the case where the carrier ratio, wall thickness, and CPSI are within the ranges of 1.40 to 1.53, 9.8 to 10.5 mil, and 200 to 300, respectively. Particularly, at a carrier ratio of 1.52 and a wall thickness of 10.38 mil, the pressure drop reaches a minimum value of 4.23 kPa, with a collection efficiency of 90.28%. Meanwhile, at a carrier ratio of 1.52 and a CPSI of 200, the pressure drop reaches a minimum value of 4.17 kPa, with a collection efficiency of 90.21%. The model expressions for pressure drop and collection efficiency derived from the PAA are: y 1 = 51.19 + 3.815 x 1 − 8.726 x 2 + 3.794 x 1 2 − 1.735 x 1 x 2 + 0.524 x 2 2 , and y 2 = 0.383 + 0.105 x 1 + 0.085 x 2 + 0.032 x 1 2 − 0.016 x 1 x 2 − 0.0032 x 2 2 . The foregoing models exhibit excellent predictive accuracy and goodness of fit for the DPF performance. Under the interactive influence of carrier ratio and wall thickness, the root mean square errors (RMSE) for pressure drop and collection efficiency are 0.0110 and 0.0002, respectively, with corresponding goodness of fit values of 0.9996 and 0.9985. As revealed by the GRA results, wall thickness exerts the most significant impact on DPF performance, presenting a GRG of 0.84 for filtration efficiency and 0.79 for pressure drop. The overall influence on both collection efficiency and pressure drop follows the descending order: wall thickness > carrier ratio > CPSI. This work has important engineering significance for optimizing the DPF capture performance and extending its working time.

Exact and approximation algorithms for routing a convoy through a graph

Exact and approximation algorithms for routing a convoy through a graph

Probabilistic Answer Set Programming with Discrete and Continuous Random Variables

Abstract Probabilistic Answer Set Programming under the credal semantics extends Answer Set Programming with probabilistic facts that represent uncertain information. The probabilistic facts are discrete with Bernoulli distributions. However, several real-world scenarios require a combination of both discrete and continuous random variables. In this paper, we extend the PASP framework to support continuous random variables and propose Hybrid Probabilistic Answer Set Programming. Moreover, we discuss, implement, and assess the performance of two exact algorithms based on projected answer set enumeration and knowledge compilation and two approximate algorithms based on sampling. Empirical results, also in line with known theoretical results, show that exact inference is feasible only for small instances, but knowledge compilation has a huge positive impact on performance. Sampling allows handling larger instances but sometimes requires an increasing amount of memory.

Algorithmic Persuasion with Evidence

In a game of persuasion with evidence, a sender has private information. By presenting evidence on the information, the sender wishes to persuade a receiver to take a single action (e.g., hire a job candidate, or convict a defendant). The sender’s utility depends solely on whether the receiver takes the action. The receiver’s utility depends on both the action and the sender’s private information. We study three natural variations. First, we consider the problem of computing an equilibrium of the game without commitment power. Second, we consider a persuasion variant, where the sender commits to a signaling scheme and the receiver, after seeing the evidence, takes the action or not. Third, we study a delegation variant, where the receiver first commits to taking the action if being presented certain evidence, and the sender presents evidence to maximize the probability the action is taken. We study these variants through the computational lens, and give hardness results, optimal approximation algorithms, and polynomial-time algorithms for special cases. Among our results is an approximation algorithm that rounds a semidefinite program that might be of independent interest, since, to the best of our knowledge, it is the first such approximation algorithm in algorithmic economics.

Machine Learning-enhanced Signature of Metastasis-related T Cell Marker Genes for Predicting Overall Survival in Malignant Melanoma.

In this study, we aimed to investigate disparities in the tumor immune microenvironment (TME) between primary and metastatic malignant melanoma (MM) using single-cell RNA sequencing (scRNA-seq) and to identify metastasis-related T cell marker genes (MRTMGs) for predicting patient survival using machine learning techniques. We identified 6 distinct T cell clusters in 10×scRNA-seq data utilizing the Uniform Manifold Approximation and Projection (UMAP) algorithm. Four machine learning algorithms highlighted SRGN, PMEL, GPR143, EIF4A2, and DSP as pivotal MRTMGs, forming the foundation of the MRTMGs signature. A high MRTMGs signature was found to be correlated with poorer overall survival (OS) and suppression of antitumor immunity in MM patients. We developed a nomogram that combines the MRTMGs signature with the T stage and N stage, which accurately predicts 1-year, 3-year, and 5-year OS probabilities. Furthermore, in an immunotherapy cohort, a high MRTMG signature was associated with an unfavorable response to anti-programmed death 1 (PD-1) therapy. In conclusion, primary and metastatic MM display distinct TME landscapes with different T cell subsets playing crucial roles in metastasis. The MRTMGs signature, established through machine learning, holds potential as a valuable biomarker for predicting the survival of MM patients and their response to anti-PD-1 therapy.

Application of Bivariate Reproducing Kernel-Based Best Interpolation Method in Electrical Tomography

Electrical Tomography (ET) technology is widely used in multiphase flow detection due to its advantages of low cost, visualization, fast response, non-radiation, and non-invasiveness. However, ill-posed solutions lead to low image reconstruction resolution, which limits its practical engineering applications. Although existing interpolation approximation algorithms can alleviate the effects of the ill-posed solutions to some extent, the imaging results remain suboptimal due to the limited approximation capability of these methods. This paper proposes a Bivariate Reproducing Kernel-Based Best Interpolation (BRKBI) method, which offers smaller approximation errors and clearer image reconstruction quality compared to existing methods. The effectiveness of the BRKBI method is validated through theoretical analysis and experimental comparisons.

Edge Manipulations for the Maximum Vertex-Weighted Bipartite \(b\) -matching

In this paper, we explore the Mechanism Design aspects of the Maximum Vertex-weighted \(b\) -Matching (MVbM) problem on bipartite graphs \((A\cup T,E)\) . The set \(A\) comprises agents, while \(T\) represents tasks. The set \(E\) , which connects \(A\) and \(T\) , is the private information of either agents or tasks. In this framework, we investigate three mechanisms – \(\mathbb{M}_{BFS}\) , \(\mathbb{M}_{DFS}\) , and \(\mathbb{M}_{G}\) . We examine scenarios in which either agents or tasks are strategic and report their adjacent edges to one of the three mechanisms. In both cases, we assume that the strategic entities are bounded by their statements: they can hide edges, but they cannot report edges that do not exist. First, we consider the case in which agents can manipulate. In this framework, \(\mathbb{M}_{BFS}\) and \(\mathbb{M}_{DFS}\) are optimal but not truthful. By characterizing the Nash Equilibria induced by \(\mathbb{M}_{BFS}\) and \(\mathbb{M}_{DFS}\) , we reveal that both mechanisms have a Price of Anarchy ( \(PoA\) ) and Price of Stability ( \(PoS\) ) of \(2\) . These efficiency guarantees are tight; no deterministic mechanism can achieve a lower \(PoA\) or \(PoS\) . In contrast, the third mechanism, \(\mathbb{M}_{G}\) , is not optimal, but truthful and its approximation ratio is \(2\) . We demonstrate that this ratio is optimal; no deterministic and truthful mechanism can outperform it. We then shift our focus to scenarios where tasks can exhibit strategic behaviour. In this case, \(\mathbb{M}_{BFS}\) , \(\mathbb{M}_{DFS}\) , and \(\mathbb{M}_{G}\) all maintain truthfulness, making \(\mathbb{M}_{BFS}\) and \(\mathbb{M}_{DFS}\) truthful and optimal mechanisms. In conclusion, we investigate the manipulability of \(\mathbb{M}_{BFS}\) and \(\mathbb{M}_{DFS}\) through experiments on randomly generated graphs. We observe that (i) \(\mathbb{M}_{BFS}\) is less prone to be manipulated by the first agent than \(\mathbb{M}_{DFS}\) , and (ii) \(\mathbb{M}_{BFS}\) is more manipulable on instances in which the total capacity of the agents is equal to the number of tasks. 1

Scaling whole-chip QAOA for higher-order ising spin glass models on heavy-hex graphs

We show that the quantum approximate optimization algorithm (QAOA) for higher-order, random coefficient, heavy-hex compatible spin glass Ising models has strong parameter concentration across problem sizes from 16 up to 127 qubits for p = 1 up to p = 5, which allows for computationally efficient parameter transfer of QAOA angles. Matrix product state (MPS) simulation is used to compute noise-free QAOA performance. Hardware-compatible short-depth QAOA circuits are executed on ensembles of 100 higher-order Ising models on noisy IBM quantum superconducting processors with 16, 27, and 127 qubits using QAOA angles learned from a single 16-qubit instance using the JuliQAOA tool. We show that the best quantum processors find lower energy solutions up to p = 2 or p = 3, and find mean energies that are about a factor of two off from the noise-free distribution. We show that p = 1 QAOA energy landscapes remain very similar as the problem size increases using NISQ hardware gridsearches with up to a 414 qubit processor.

A Novel Noise-Aware Classical Optimizer for Variational Quantum Algorithms

A key component of variational quantum algorithms (VQAs) is the choice of classical optimizer employed to update the parameterization of an ansatz. It is well recognized that quantum algorithms will, for the foreseeable future, necessarily be run on noisy devices with limited fidelities. Thus, the evaluation of an objective function (e.g., the guiding function in the quantum approximate optimization algorithm (QAOA) or the expectation of the electronic Hamiltonian in variational quantum eigensolver (VQE)) required by a classical optimizer is subject not only to stochastic error from estimating an expected value but also to error resulting from intermittent hardware noise. Model-based derivative-free optimization methods have emerged as popular choices of a classical optimizer in the noisy VQA setting, based on empirical studies. However, these optimization methods were not explicitly designed with the consideration of noise. In this work we adapt recent developments from the “noise-aware numerical optimization” literature to these commonly used derivative-free model-based methods. We introduce the key defining characteristics of these novel noise-aware derivative-free model-based methods that separate them from standard model-based methods. We study an implementation of such noise-aware derivative-free model-based methods and compare its performance on demonstrative VQA simulations to classical solvers packaged in scikit-quant. History: Accepted by Giacomo Nannicini, Area Editor for Quantum Computing and Operations Research. Accepted for Special Issue. Funding: This material is based upon work supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers and the Office of Advanced Scientific Computing Research, Accelerated Research for Quantum Computing program under contract number DE-AC02-06CH11357. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0177 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0177 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Fixed-Ratio Approximation Algorithm for the Minimum Cost Cover of a Digraph by Bounded Number of Cycles

We consider polynomial time approximation for the minimum cost cycle cover problem of an edge-weighted digraph, where feasible covers are restricted to have at most k disjoint cycles. In the literature this problem is referred to as Min-k-SCCP. The problem is closely related to classic Traveling Salesman Problem (TSP) and Vehicle Routing Problem (VRP) and has many important applications in algorithms design and operations research. Unlike its unconstrained variant, the Min-k-SCCP is strongly NP-hard even on undirected graphs and remains intractable in very specific settings. For any metric, the problem can be approximated in polynomial time within ratio 2, while in fixed-dimensional Euclidean spaces it admits Polynomial Time Approximation Schemes (PTAS). In the same time, approximation of the more general asymmetric Min-k-SCCP still remains weakly studied. In this paper, we propose the first fixed-ratio approximation algorithm for this problem, which extends the recent breakthrough Svensson-Tarnawski-Vegh and Traub-Vygen results for the Asymmetric Traveling Salesman Problem.

Efficiently Constructing Convex Approximation Sets in Multiobjective Optimization Problems

Convex approximation sets for multiobjective optimization problems are a well-studied relaxation of the common notion of approximation sets. Instead of approximating each image of a feasible solution by the image of some solution in the approximation set up to a multiplicative factor in each component, a convex approximation set only requires this multiplicative approximation to be achieved by some convex combination of finitely many images of solutions in the set. This makes convex approximation sets efficiently computable for a wide range of multiobjective problems: even for many problems for which (classic) approximations sets are hard to compute. In this article, we propose a polynomial-time algorithm to compute convex approximation sets that builds on an exact or approximate algorithm for the weighted sum scalarization and is therefore applicable to a large variety of multiobjective optimization problems. The provided convex approximation quality is arbitrarily close to the approximation quality of the underlying algorithm for the weighted sum scalarization. In essence, our algorithm can be interpreted as an approximate version of the dual variant of Benson’s outer approximation algorithm. Thus, in contrast to existing convex approximation algorithms from the literature, information on solutions obtained during the approximation process is utilized to significantly reduce both the practical running time and the cardinality of the returned solution sets while still guaranteeing the same worst-case approximation quality. We underpin these advantages by the first comparison of all existing convex approximation algorithms on several instances of the triobjective knapsack problem and the triobjective symmetric metric traveling salesman problem. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This research was supported by the German Research Foundation [Project 398572517]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0220 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0220 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Efficient quantum circuits based on the quantum natural gradient

Efficient preparation of arbitrary entangled quantum states is crucial for quantum computation. This is particularly important for noisy intermediate-scale quantum simulators relying on variational hybrid quantum-classical algorithms. To that end, we propose symmetry-conserving modified quantum approximate optimization algorithm (SCom-QAOA) circuits. The depths of these circuits depend not only on the desired fidelity to the target state but also on the amount of entanglement the state contains. The parameters of the SCom-QAOA circuits are optimized using the quantum natural gradient method based on the Fubini-Study metric. The SCom-QAOA circuit transforms an unentangled state into a ground state of a gapped one-dimensional Hamiltonian with a circuit depth that depends not on the system size but rather on the finite correlation length. In contrast, the circuit depth grows proportionally to the system size for preparing low-lying states of critical one-dimensional systems. Even in the latter case, SCom-QAOA circuits with depth less than the system size were sufficient to generate states with fidelity in excess of 99%, which is relevant for near-term applications. The proposed scheme enlarges the set of the initial states accessible for variational quantum algorithms and widens the scope of investigation of nonequilibrium phenomena in quantum simulators. Published by the American Physical Society 2024

Improved Approximation Algorithms for Relational Clustering

Improved Approximation Algorithms for Relational Clustering

Estimating Asset Pricing Models in the Presence of Cross-Sectionally Correlated Pricing Errors

In this study, we propose an adversarial learning approach to the asset pricing model estimation problem which aims to find estimates of factors and loadings that capture time-series covariations while minimizing the worst-case cross-sectional pricing errors. The proposed estimator is defined by a novel min-max optimization problem in which finding a solution is known to be difficult. This contrasts with other related estimators that admit a well-defined analytic solution but do not effectively account for correlations among the pricing errors. To this end, we propose an approximate algorithm based on the alternating optimization procedure and empirically demonstrate that our proposed adversarial estimation framework outperforms other existing factor models, especially when the explanatory power of the pricing model is limited.

An approximation algorithm for the prize-collecting connected dominating set problem

An approximation algorithm for the prize-collecting connected dominating set problem

Multi-objective Route Planning of an Unmanned Air Vehicle in Continuous Terrain: An Exact and an Approximation Algorithm

Unmanned Aerial Vehicles (UAVs) are widely used for military and civilian purposes. Effective route planning is an important component of their successful missions. In this study, we address the route planning problem of a UAV tasked with collecting information from various target locations in a protected terrain. We consider multiple targets, three objectives, and time-dependent information availability. Modeling the movement of UAVs in a continuous terrain in the presence of multiple objectives is complex. Conflicting objectives typically lead to a continuum of efficient trajectory options between two targets. We formulate the routing problem as a mixed-integer programming (MIP) model that captures the movement in the continuous terrain. We demonstrate the superiority of the continuous terrain formulation over the simplified discretized terrain formulation. We also develop an approximation algorithm that reduces the computational requirements of the MIP model substantially while ensuring a desired level of precision.

Algorithms for solving the inverse scattering problem for the Manakov model

The paper considers algorithms for solving inverse scattering problems based on the discretization of the Gelfand–Levitan–Marchenko integral equations, associated with the system of nonlinear Schrödinger equations of the Manakov model. The numerical algorithm of the first order approximation for solving the scattering problem is reduced to the inversion of a series of nested block Toeplitz matrices using the Levinson-type bordering method. Increasing the approximation accuracy violates the Toeplitz structure of block matrices. Two algorithms are described that solve this problem for second order accuracy. One algorithm uses a block version of the Levinson bordering algorithm, which recovers the Toeplitz structure of the matrix by moving some terms of the systems of equations to the right-hand side. Another algorithm is based on the Toeplitz decomposition of an almost block-Toeplitz matrix and the Tyrtyshnikov bordering algorithm. The speed and accuracy of calculations using the presented algorithms are compared on an exact solution (the Manakov vector soliton).

Continuous Approximate Dynamic Programming Algorithm to Promote Multiple Battery Energy Storage Lifespan Benefit in Real-Time Scheduling

Continuous Approximate Dynamic Programming Algorithm to Promote Multiple Battery Energy Storage Lifespan Benefit in Real-Time Scheduling