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Nonadaptive Stochastic Score Classification Sequential testing problems involve a system with several components, each of which is working with some independent probability. The working/failed status of each component can be determined by performing a test, which is usually expensive. So, the goal is to perform tests in a carefully chosen sequence until the overall system status can be evaluated. These problems arise in a variety of applications, such as healthcare, manufacturing, and telecommunication. A common task in these applications is to categorize the system into one of several classes that correspond to the system status being poor, fair, good, excellent, etc. In “Nonadaptive Stochastic Score Classification and Explainable Half-Space Evaluation,” Ghuge, Gupta, and Nagarajan provide the first constant-factor approximation algorithm for this problem. Moreover, the resulting policy is nonadaptive, which results in significant savings in computational time. The authors also validate their theoretical results via computational experiments, where they observe that their algorithm’s cost is on average at most 50% more than an information-theoretic lower bound. 

Adaptivity in Stochastic Submodular Cover Solutions to stochastic optimization problems are typically sequential decision processes that make decisions one by one, waiting for (and using) the feedback from each decision. Whereas such “adaptive” solutions achieve the best objective, they can be very time-consuming because of the need to wait for feedback after each decision. A natural question is are there solutions that only adapt (i.e., wait for feedback) a few times whereas still being competitive with the fully adaptive optimal solution? In “The Power of Adaptivity for Stochastic Submodular Cover,” Ghuge, Gupta, and Nagarajan resolve this question in the context of stochastic submodular cover, which is a fundamental stochastic covering problem. They provide algorithms that achieve a smooth trade-off between the number of adaptive “rounds” and the solution quality. The authors also demonstrate via experiments on real-world and synthetic data sets that, even for problems with more than 1,000 decisions, about six rounds of adaptivity suffice to obtain solutions nearly as good as fully adaptive solutions. 

We consider the following general network design problem. The input is an asymmetric metric (V, c), root [Formula: see text], monotone submodular function [Formula: see text], and budget B. The goal is to find an r-rooted arborescence T of cost at most B that maximizes f(T). Our main result is a simple quasi-polynomial time [Formula: see text]-approximation algorithm for this problem, in which [Formula: see text] is the number of vertices in an optimal solution. As a consequence, we obtain an [Formula: see text]-approximation algorithm for directed (polymatroid) Steiner tree in quasi-polynomial time. We also extend our main result to a setting with additional length bounds at vertices, which leads to improved [Formula: see text]-approximation algorithms for the single-source buy-at-bulk and priority Steiner tree problems. For the usual directed Steiner tree problem, our result matches the best previous approximation ratio but improves significantly on the running time. For polymatroid Steiner tree and single-source buy-at-bulk, our result improves prior approximation ratios by a logarithmic factor. For directed priority Steiner tree, our result seems to be the first nontrivial approximation ratio. Under certain complexity assumptions, our approximation ratios are the best possible (up to constant factors). 

We study the assortment optimization problem when customer choices are governed by the paired combinatorial logit model. We study unconstrained, cardinality-constrained, and knapsack-constrained versions of this problem, which are all known to be NP-hard. We design efficient algorithms that compute approximately optimal solutions, using a novel relation to the maximum directed cut problem and suitable linear-program rounding algorithms. We obtain a randomized polynomial time approximation scheme for the unconstrained version and performance guarantees of 50% and [Formula: see text] for the cardinality-constrained and knapsack-constrained versions, respectively. These bounds improve significantly over prior work. We also obtain a performance guarantee of 38.5% for the assortment problem under more general constraints, such as multidimensional knapsack (where products have multiple attributes and there is a knapsack constraint on each attribute) and partition constraints (where products are partitioned into groups and there is a limit on the number of products selected from each group). In addition, we implemented our algorithms and tested them on random instances available in prior literature. We compared our algorithms against an upper bound obtained using a linear program. Our average performance bounds for the unconstrained, cardinality-constrained, knapsack-constrained, and partition-constrained versions are over 99%, 99%, 96%, and 99%, respectively.