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1. Robust Algorithms for TSP and Steiner Tree

   https://doi.org/10.4230/LIPIcs.WABI.2020.17  

   Ganesh, Arun; Maggs, Bruce; Panigrahi, Debmalya (January 2020, Leibniz international proceedings in informatics)

   Czumaj, Arturr; Dawar, Anuj; Merelli, Emanuela (Ed.)

   Robust optimization is a widely studied area in operations research, where the algorithm takes as input a range of values and outputs a single solution that performs well for the entire range. Specifically, a robust algorithm aims to minimize regret, defined as the maximum difference between the solution’s cost and that of an optimal solution in hindsight once the input has been realized. For graph problems in P, such as shortest path and minimum spanning tree, robust polynomial-time algorithms that obtain a constant approximation on regret are known. In this paper, we study robust algorithms for minimizing regret in NP-hard graph optimization problems, and give constant approximations on regret for the classical traveling salesman and Steiner tree problems. 

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2. Barriers for the performance of graph neural networks (GNN) in discrete random structures

   https://doi.org/10.1073/pnas.2314092120  

   Gamarnik, David (November 2023, Proceedings of the National Academy of Sciences)

   Recently, graph neural network (GNN)-based algorithms were proposed to solve a variety of combinatorial optimization problems [M. J. Schuetz, J. K. Brubaker, H. G. Katzgraber,Nat. Mach. Intell.4, 367–377 (2022)]. GNN was tested in particular on randomly generated instances of these problems. The publication [M. J. Schuetz, J. K. Brubaker, H. G. Katzgraber,Nat. Mach. Intell.4, 367–377 (2022)] stirred a debate whether the GNN-based method was adequately benchmarked against best prior methods. In particular, critical commentaries [M. C. Angelini, F. Ricci-Tersenghi,Nat. Mach. Intell.5, 29–31 (2023)] and [S. Boettcher,Nat. Mach. Intell.5, 24–25 (2023)] point out that a simple greedy algorithm performs better than the GNN. We do not intend to discuss the merits of arguments and counterarguments in these papers. Rather, in this note, we establish a fundamental limitation for running GNN on random instances considered in these references, for a broad range of choices of GNN architecture. Specifically, these barriers hold when the depth of GNN does not scale with graph size (we note that depth 2 was used in experiments in [M. J. Schuetz, J. K. Brubaker, H. G. Katzgraber,Nat. Mach. Intell.4, 367–377 (2022)]), and importantly, these barriers hold regardless of any other parameters of GNN architecture. These limitations arise from the presence of the overlap gap property (OGP) phase transition, which is a barrier for many algorithms, including importantly local algorithms, of which GNN is an example. At the same time, some algorithms known prior to the introduction of GNN provide best results for these problems up to the OGP phase transition. This leaves very little space for GNN to outperform the known algorithms, and based on this, we side with the conclusions made in [M. C. Angelini, F. Ricci-Tersenghi,Nat. Mach. Intell.5, 29–31 (2023)] and [S. Boettcher,Nat. Mach. Intell.5, 24–25 (2023)]. 

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3. Structured Robust Submodular Maximization: Offline and Online Algorithms

   Anari, N; Haghtalab, N.; Naor, S; Pokutta, S.; Singh, M.; Torrico, A. (May 2019, 22nd International Conference on Artificial Intelligence and Statistics (AISTATS) 2019)

   Constrained submodular function maximization has been used in subset selection problems such as selection of most informative sensor locations. While these models have been quite popular, the solutions obtained via this approach are unstable to perturbations in data defining the submodular functions. Robust submodular maximization has been proposed as a richer model that aims to overcome this discrepancy as well as increase the modeling scope of submodular optimization. In this work, we consider robust submodular maximization with structured combinatorial constraints and give efficient algorithms with provable guarantees. Our approach is applicable to constraints defined by single or multiple matroids, knapsack as well as distributionally robust criteria. We consider both the offline setting where the data defining the problem is known in advance as well as the online setting where the input data is revealed over time. For the offline setting, we give a nearly optimal bi-criteria approximation algorithm that relies on new extensions of the classical greedy algorithm. For the online version of the problem, we give an algorithm that returns a bi-criteria solution with sub-linear regret. 

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4. Universal Algorithms for Clustering Problems

   https://doi.org/10.1145/3572840  

   Ganesh, Arun; Maggs, Bruce M; Panigrahi, Debmalya (April 2023, ACM Transactions on Algorithms)

   This article presentsuniversalalgorithms for clustering problems, including the widely studiedk-median,k-means, andk-center objectives. The input is a metric space containing allpotentialclient locations. The algorithm must selectkcluster centers such that they are a good solution foranysubset of clients that actually realize. Specifically, we aim for lowregret, 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 solutionSolfor a clustering problem is said to be an α , β-approximation if for all subsets of clientsC^′, it satisfiessol(C^′) ≤ α ċopt(C′) + β ċmr, whereopt(C′ is the cost of the optimal solution for clients (C′) andmris the minimum regret achievable by any solution. Our main results are universal algorithms for the standard clustering objectives ofk-median,k-means, andk-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 arefixed. 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. Joinable Parallel Balanced Binary Trees

   https://doi.org/10.1145/3512769  

   Blelloch, Guy; Ferizovic, Daniel; Sun, Yihan (June 2022, ACM Transactions on Parallel Computing)

   In this article, we show how a single function,join, can be used to implement parallelbalanced binary search trees(BSTs) simply and efficiently. Based onjoin, our approach applies to multiple balanced tree data structures, and a variety of functions for ordered sets and maps. We describe our technique as an algorithmic framework calledjoin-based algorithms. We show that thejoinfunction fully captures what is needed for rebalancing trees for a variety of tree algorithms, as long as the balancing scheme satisfies certain properties, which we refer to asjoinabletrees. We discuss four balancing schemes that are joinable: AVL trees, red-black trees, weight-balanced trees, and treaps. We present a variety of tree algorithms that apply to joinable trees, includinginsert,delete,union,intersection,difference,split,range,filter, and so on, most of them also parallel. These algorithms are generic across balancing schemes. Many algorithms are optimal in the comparison model, and we provide a general proof to show the efficiency in work for joinable trees. The algorithms are highly parallel, all with polylogarithmic span (parallel dependence). Specifically, the set-set operationsunion,intersection, anddifferencehave work\( O(m\log (\frac{n}{m}+1)) \)and polylogarithmic span for input set sizes\( n \)and\( m\le n \). We implemented and tested our algorithms on the four balancing schemes. In general, all four schemes have quite similar performance, but the weight-balanced tree slightly outperforms the others. They have the same speedup characteristics, getting around 73\( \times \)speedup on 72 cores (144 hyperthreads). Experimental results also show that our implementation outperforms existing parallel implementations, and our sequential version achieves close or much better performance than the sequential merging algorithm in C++ Standard Template Library (STL) on various input sizes. 

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