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The hitting set problem is a well-known NP-hard optimization problem in which, given a set of elements and a collection of subsets, the goal is to find the smallest selection of elements, such that each subset contains at least one element in the selection. Many geometric set systems enjoy improved approximation ratios, which have recently been shown to be tight with respect to the shallow cell complexity of the set system. The algorithms that exploit the cell complexity, however, tend to be involved and computationally intensive. This paper shows that comparable approximation ratios for the hitting set problem can be attained using a much simpler algorithm: solve the linear programming relaxation, take one initial random sample from the set of elements with probabilities proportional to the LP-solution, and, while there is an unhit set, take an additional sample from it proportional to the LP-solution. Our algorithm is based on a generalization of the elegant net-finder algorithm of Nabil Mustafa. To analyze this algorithm for the hitting set problem, we generalize the classic Packing Lemma, and the more recent Shallow-Packing Lemma, to the setting of weighted epsilon nets., Comment: Submitted to WADS2023

There is significant interest in deploying machine learning algorithms for diagnostic radiology, as modern learning techniques have made it possible to detect abnormalities in medical images within minutes. While machine-assisted diagnoses cannot yet reliably replace human reviews of images by a radiologist, they could inform prioritization rules for determining the order by which to review patient cases so that patients with time-sensitive conditions could benefit from early intervention. We study this scenario by formulating it as a learning-augmented online scheduling problem. We are given information about each arriving patient's urgency level in advance, but these predictions are inevitably error-prone. In this formulation, we face the challenges of decision making under imperfect information, and of responding dynamically to prediction error as we observe better data in real-time. We propose a simple online policy and show that this policy is in fact the best possible in certain stylized settings. We also demonstrate that our policy achieves the two desiderata of online algorithms with predictions: consistency (performance improvement with prediction accuracy) and robustness (protection against the worst case). We complement our theoretical findings with empirical evaluations of the policy under settings that more accurately reflect clinical scenarios in the real world.

The American winner-take-all congressional district system empowers politicians to engineer electoral outcomes by manipulating district boundaries. Existing computational solutions mostly focus on drawing unbiased maps by ignoring political and demographic input, and instead simply optimize for compactness. We claim that this is a flawed approach because compactness and fairness are orthogonal qualities, and introduce a scalable two-stage method to explicitly optimize for arbitrary piecewise-linear definitions of fairness. The first stage is a randomized divide-and-conquer column generation heuristic which produces an exponential number of distinct district plans by exploiting the compositional structure of graph partitioning problems. This district ensemble forms the input to a master selection problem to choose the districts to include in the final plan. Our decoupled design allows for unprecedented flexibility in defining fairness-aligned objective functions. The pipeline is arbitrarily parallelizable, is flexible to support additional redistricting constraints, and can be applied to a wide array of other regionalization problems. In the largest ever ensemble study of congressional districts, we use our method to understand the range of possible expected outcomes and the implications of this range on potential definitions of fairness.

The recently introduced online Minimum Peak Appointment Scheduling (MPAS) problem is a variant of the online bin-packing problem that allows for deferred decision making. Specifically, it allows for the problem to be split into an online phase where a stream of appointment requests arrive requiring a scheduled time, followed by an offline phase where those appointments are scheduled into rooms. Similar to the bin-packing problem, the aim is to use the minimum number of rooms in the final configuration. This model more accurately captures scheduling appointments than bin packing. For example, a dialysis patient needs to know what time to arrive for an appointment, but does not need to know the assigned station ahead of time. Previous work developed a randomized algorithm for this problem which achieved an asymptotic competitive ratio of at most 1.5, proving that online MPAS was fundamentally different from the online bin-packing problem. Our main contribution is to develop a new randomized algorithm for the problem that achieves an asymptotic competitive ratio under 1.455, indicating the potential for further progress. This improvement is attained by modifying the process for scheduling appointments to increase the density of the packing in the worst case, along with utilizing the dual of the bin-packing linear programming relaxation to perform the analysis. We also present the first known lower bound of 1.2 on the asymptotic competitive ratio of both deterministic and randomized online MPAS algorithm. These results demonstrate how deferred decision-making can be leveraged to yield improved worst-case performance, a phenomenon which should be investigated in a broader class of settings.

We consider a two-stage robust facility location problem on a metric under an uncertain demand. The decision-maker needs to decide on the (integral) units of supply for each facility in the first stage to satisfy an uncertain second-stage demand, such that the sum of first stage supply cost and the worst-case cost of satisfying the second-stage demand over all scenarios is minimized. The second-stage decisions are only assignment decisions without the possibility of adding recourse supply capacity. This makes our model different from existing work on two-stage robust facility location and set covering problems. We consider an implicit model of uncertainty with an exponential number of demand scenarios specified by an upper bound $k$ on the number of second-stage clients. In an optimal solution, the second-stage assignment decisions depend on the scenario; surprisingly, we show that restricting to a fixed (static) fractional assignment for each potential client irrespective of the scenario gives us an $O(\log k/\log \log k)$-approximation for the problem. Moreover, the best such static assignment can be computed efficiently giving us the desired guarantee.

In the winter of 1976, Alexander Rinnooy Kan and Jan Karel Lenstra defended their PhD theses at the University of Amsterdam. Gene Lawler was on their committees. It was a natural idea to turn the theses into a textbook on scheduling. They set out to compile a survey with Ron Graham (1979), but progress on the book was hampered by the many research opportunities offered by the field. After David Shmoys joined the team in the mid 1980's, several chapters were drafted, and the survey was rewritten (1993). Gene passed away in 1994. Colleagues were asked to contribute chapters or to complete existing drafts. However, by the turn of the century the project was losing its momentum, and finite convergence to completion fell beyond our reach. Over the years, several chapters have been used in the classroom. We continue to receive requests from colleagues who look for a text on the elements of scheduling at an advanced undergraduate or early graduate level. This document is a compilation of what currently exists. We have made a marginal effort in patching it up at some places but is essentially what was written long ago. We did make an attempt to include most of the citations in the bibliography.

This paper has been merged into 1110.4604., Comment: This paper has been merged into 1110.4604