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1. An O(n5/4) Time ∊-Approximation Algorithm for RMS Matching in a Plane

   https://doi.org/10.1137/1.9781611976465.55  

   Lahn, Nathaniel; Raghvendra, Sharath (January 2021, Proceedings of the Annual ACMSIAM Symposium on Discrete Algorithms)

   null (Ed.)

   The 2-Wasserstein distance (or RMS distance) is a useful measure of similarity between probability distributions with exciting applications in machine learning. For discrete distributions, the problem of computing this distance can be expressed in terms of finding a minimum-cost perfect matching on a complete bipartite graph given by two multisets of points A, B ⊂ ℝ2, with |A| = |B| = n, where the ground distance between any two points is the squared Euclidean distance between them. Although there is a near-linear time relative ∊-approximation algorithm for the case where the ground distance is Euclidean (Sharathkumar and Agarwal, JACM 2020), all existing relative ∊-approximation algorithms for the RMS distance take Ω(n3/2) time. This is primarily because, unlike Euclidean distance, squared Euclidean distance is not a metric. In this paper, for the RMS distance, we present a new ∊-approximation algorithm that runs in O(n^5/4 poly{log n, 1/∊}) time. Our algorithm is inspired by a recent approach for finding a minimum-cost perfect matching in bipartite planar graphs (Asathulla et al, TALG 2020). Their algorithm depends heavily on the existence of sublinear sized vertex separators as well as shortest path data structures that require planarity. Surprisingly, we are able to design a similar algorithm for a complete geometric graph that is far from planar and does not have any vertex separators. Central components of our algorithm include a quadtree-based distance that approximates the squared Euclidean distance and a data structure that supports both Hungarian search and augmentation in sublinear time. 

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2. Mean‐standard deviation model for minimum cost flow problem

   https://doi.org/10.1002/net.22135  

   Gokalp, Can; Boyles, Stephen D.; Unnikrishnan, Avinash (December 2022, Networks)

   We study the mean-standard deviation minimum cost flow (MSDMCF) problem, where the objective is minimizing a linear combination of the mean and standard deviation of flow costs. Due to the nonlinearity and nonseparability of the objective, the problem is not amenable to the standard algorithms developed for network flow problems. We prove that the solution for the MSDMCF problem coincides with the solution for a particular mean-variance minimum cost flow (MVMCF) problem. Leveraging this result, we propose bisection (BSC), Newton–Raphson (NR), and a hybrid (NR-BSC)—method seeking to find the specific MVMCF problem whose optimal solution coincides with the optimal solution for the given MSDMCF problem. We further show that this approach can be extended to solve more generalized nonseparable parametric minimum cost flow problems under certain conditions. Computational experiments show that the NR algorithm is about twice as fast as the CPLEX solver on benchmark networks generated with NETGEN. 

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3. Learning Robust Distance Metric with Side Information via Ratio Minimization of Orthogonally Constrained L21-Norm Distances

   https://doi.org/10.24963/ijcai.2019/417  

   Liu, Kai; Brand, Lodewijk; Wang, Hua; Nie, Feiping (August 2019, Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence)

   Metric Learning, which aims at learning a distance metric for a given data set, plays an important role in measuring the distance or similarity between data objects. Due to its broad usefulness, it has attracted a lot of interest in machine learning and related areas in the past few decades. This paper proposes to learn the distance metric from the side information in the forms of must-links and cannot-links. Given the pairwise constraints, our goal is to learn a Mahalanobis distance that minimizes the ratio of the distances of the data pairs in the must-links to those in the cannot-links. Different from many existing papers that use the traditional squared L2-norm distance, we develop a robust model that is less sensitive to data noise or outliers by using the not-squared L2-norm distance. In our objective, the orthonormal constraint is enforced to avoid degenerate solutions. To solve our objective, we have derived an efficient iterative solution algorithm. We have conducted extensive experiments, which demonstrated the superiority of our method over state-of-the-art. 

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4. Universal Algorithms for Clustering

   Bruce Maggs, Arun Ganesh (April 2021, Leibniz international proceedings in informatics)

   Bansal, Nikhil and (Ed.)

   his paper presents universal algorithms for clustering problems, including the widely studied k-median, k-means, and k-center objectives. The input is a metric space containing all potential client locations. The algorithm must select k cluster centers such that they are a good solution for any subset of clients that actually realize. Specifically, we aim for low regret, defined as the maximum over all subsets of the difference between the cost of the algorithm’s solution and that of an optimal solution. A universal algorithm’s solution sol for a clustering problem is said to be an (α, β)-approximation if for all subsets of clients C', it satisfies sol(C') ≤ α ⋅ opt(C') + β ⋅ mr, where opt(C') is the cost of the optimal solution for clients C' and mr is the minimum regret achievable by any solution. Our main results are universal algorithms for the standard clustering objectives of k-median, k-means, and k-center that achieve (O(1), O(1))-approximations. These results are obtained via a novel framework for universal algorithms using linear programming (LP) relaxations. These results generalize to other 𝓁_p-objectives and the setting where some subset of the clients are fixed. We also give hardness results showing that (α, β)-approximation is NP-hard if α or β is at most a certain constant, even for the widely studied special case of Euclidean metric spaces. This shows that in some sense, (O(1), O(1))-approximation is the strongest type of guarantee obtainable for universal clustering. 

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5. A global algorithm for AC optimal power flow based on successive linear conic optimization

   https://doi.org/10.1109/PESGM.2017.8273957  

   Barati, Masoud; Kargarian, Amin (July 2017, 2017 IEEE Power & Energy Society General Meeting)

   Newly, there has been significant research interest in the exact solution of the AC optimal power flow (AC-OPF) problem. A semideflnite relaxation solves many OPF problems globally. However, the real problem exists in which the semidefinite relaxation fails to yield the global solution. The appropriation of relaxation for AC-OPF depends on the success or unfulflllment of the SDP relaxation. This paper demonstrates a quadratic AC-OPF problem with a single negative eigenvalue in objective function subject to linear and conic constraints. The proposed solution method for AC-OPF model covers the classical AC economic dispatch problem that is known to be NP-hard. In this paper, by combining successive linear conic optimization (SLCO), convex relaxation and line search technique, we present a global algorithm for AC-OPF which can locate a globally optimal solution to the underlying AC-OPF within given tolerance of global optimum solution via solving linear conic optimization problems. The proposed algorithm is examined on modified IEEE 6-bus test system. The promising numerical results are described. 

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