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1. A General Framework for Approximating Min Sum Ordering Problems

   Happach, Felix; Hellerstein, Lisa; Lidbetter, Thomas (July 2020, ArXivorg)

   We consider a large family of problems in which an ordering (or, more precisely, a chain of subsets) of a finite set must be chosen to minimize some weighted sum of costs. This family includes variations of Min Sum Set Cover (MSSC), several scheduling and search problems, and problems in Boolean function evaluation. We define a new problem, called the Min Sum Ordering Problem (MSOP) which generalizes all these problems using a cost and a weight function on subsets of a finite set. Assuming a polynomial time α-approximation algorithm for the problem of finding a subset whose ratio of weight to cost is maximal, we show that under very minimal assumptions, there is a polynomial time 4α-approximation algorithm for MSOP. This approximation result generalizes a proof technique used for several distinct problems in the literature. We apply this to obtain a number of new approximation results. 

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2. A Local Search Algorithm for the Min-Sum Submodular Cover Problem

   https://doi.org/10.4230/LIPIcs.ISAAC.2022.3  

   Hellerstein, Lisa; Lidbetter, Thomas; Witter, R Teal (January 2022, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Bae, Sang Won; Park, Heejin (Ed.)

   We consider the problem of solving the Min-Sum Submodular Cover problem using local search. The Min-Sum Submodular Cover problem generalizes the NP-complete Min-Sum Set Cover problem, replacing the input set cover instance with a monotone submodular set function. A simple greedy algorithm achieves an approximation factor of 4, which is tight unless P=NP [Streeter and Golovin, NeurIPS, 2008]. We complement the greedy algorithm with analysis of a local search algorithm. Building on work of Munagala et al. [ICDT, 2005], we show that, using simple initialization, a straightforward local search algorithm achieves a (4+ε)-approximate solution in time O(n³log(n/ε)), provided that the monotone submodular set function is also second-order supermodular. Second-order supermodularity has been shown to hold for a number of submodular functions of practical interest, including functions associated with set cover, matching, and facility location. We present experiments on two special cases of Min-Sum Submodular Cover and find that the local search algorithm can outperform the greedy algorithm on small data sets. 

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3. Optimal algorithms for scheduling under time-of-use tariffs

   https://doi.org/10.1007/s10479-021-04059-3  

   Chen, Lin; Megow, Nicole; Rischke, Roman; Stougie, Leen; Verschae, José (September 2021, Annals of Operations Research)

   null (Ed.)

   Abstract We consider a natural generalization of classical scheduling problems to a setting in which using a time unit for processing a job causes some time-dependent cost, the time-of-use tariff, which must be paid in addition to the standard scheduling cost. We focus on preemptive single-machine scheduling and two classical scheduling cost functions, the sum of (weighted) completion times and the maximum completion time, that is, the makespan. While these problems are easy to solve in the classical scheduling setting, they are considerably more complex when time-of-use tariffs must be considered. We contribute optimal polynomial-time algorithms and best possible approximation algorithms. For the problem of minimizing the total (weighted) completion time on a single machine, we present a polynomial-time algorithm that computes for any given sequence of jobs an optimal schedule, i.e., the optimal set of time slots to be used for preemptively scheduling jobs according to the given sequence. This result is based on dynamic programming using a subtle analysis of the structure of optimal solutions and a potential function argument. With this algorithm, we solve the unweighted problem optimally in polynomial time. For the more general problem, in which jobs may have individual weights, we develop a polynomial-time approximation scheme (PTAS) based on a dual scheduling approach introduced for scheduling on a machine of varying speed. As the weighted problem is strongly NP-hard, our PTAS is the best possible approximation we can hope for. For preemptive scheduling to minimize the makespan, we show that there is a comparably simple optimal algorithm with polynomial running time. This is true even in a certain generalized model with unrelated machines. 

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4. Approximation Scheme for Weighted Metric Clustering via Sherali-Adams

   https://doi.org/10.1609/aaai.v38i8.28629  

   Avdiukhin, Dmitrii; Chatziafratis, Vaggos; Makarychev, Konstantin; Yaroslavtsev, Grigory (March 2024, Proceedings of the AAAI Conference on Artificial Intelligence)

   Motivated by applications to classification problems on metric data, we study Weighted Metric Clustering problem: given a metric d over n points and a k x k symmetric matrix A with non-negative entries, the goal is to find a k-partition of these points into clusters C1,...,Ck, while minimizing the sum of A[i,j] * d(u,v) over all pairs of clusters Ci and Cj and all pairs of points u from Ci and v from Cj. Specific choices of A lead to Weighted Metric Clustering capturing well-studied graph partitioning problems in metric spaces, such as Min-Uncut, Min-k-Sum, Min-k-Cut, and more.Our main result is that Weighted Metric Clustering admits a polynomial-time approximation scheme (PTAS). Our algorithm handles all the above problems using the Sherali-Adams linear programming relaxation. This subsumes several prior works, unifies many of the techniques for various metric clustering objectives, and yields a PTAS for several new problems, including metric clustering on manifolds and a new family of hierarchical clustering objectives. Our experiments on the hierarchical clustering objective show that it better captures the ground-truth structural information compared to the popular Dasgupta's objective. 

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5. Optimal Algorithms for Continuous Non-monotone Submodular and DR-Submodular Maximization

   Niazadeh, Rad; Roughgarden, Tim; Wang, Joshua R. (April 2020, Journal of machine learning research)

   Kraus, Andreas (Ed.)

   In this paper we study the fundamental problems of maximizing a continuous nonmonotone submodular function over the hypercube, both with and without coordinate-wise concavity. This family of optimization problems has several applications in machine learning, economics, and communication systems. Our main result is the first 1 2 -approximation algorithm for continuous submodular function maximization; this approximation factor of 1 2 is the best possible for algorithms that only query the objective function at polynomially many points. For the special case of DR-submodular maximization, i.e. when the submodular function is also coordinate-wise concave along all coordinates, we provide a different 1 2 -approximation algorithm that runs in quasi-linear time. Both these results improve upon prior work (Bian et al., 2017a,b; Soma and Yoshida, 2017). Our first algorithm uses novel ideas such as reducing the guaranteed approximation problem to analyzing a zero-sum game for each coordinate, and incorporates the geometry of this zero-sum game to fix the value at this coordinate. Our second algorithm exploits coordinate-wise concavity to identify a monotone equilibrium condition sufficient for getting the required approximation guarantee, and hunts for the equilibrium point using binary search. We further run experiments to verify the performance of our proposed algorithms in related machine learning applications. 

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