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During the COVID-19 pandemic, safely implementing in-person indoor instruction was a high priority for universities nationwide. To support this effort at the University, we developed a mathematical model for estimating the risk of SARS-CoV-2 transmission in university classrooms. This model was used to evaluate combinations of feasible interventions for classrooms at the University during the pandemic and optimize the set of interventions that would allow higher occupancy levels, matching the pre-pandemic numbers of in-person courses. Importantly, we determined that requiring masking in dense classrooms with unrestricted seating with more than 90% of students vaccinated was easy to implement, incurred little logistical or financial cost, and allowed classes to be held at full capacity. A retrospective analysis at the end of the semester confirmed the model's assessment that the proposed classroom configuration would be safe. Our framework is generalizable and was used to support reopening decisions at Stanford University. In addition, our framework is flexible and applies to a wide range of indoor settings. It was repurposed for large university events and gatherings and could be used to support planning indoor space use to avoid transmission of infectious diseases across various industries, from secondary schools to movie theaters and restaurants.

Understanding how subsets of items are chosen from offered sets is critical to assortment planning, wireless network planning, and many other applications. There are two seemingly unrelated subset choice models that capture dependencies between items: intuitive and interpretable random utility models; and tractable determinantal point processes (DPPs). This paper connects the two. First, all DPPs are shown to be random utility models. Next, a determinantal choice model that enjoys the best of both worlds is specified; the model is shown to subsume logistic regression when dependence is minimal, and MNL when dependence is maximally negative. This makes the model interpretable, while retaining the tractability of DPPs. A simulation study verifies that the model can learn a continuum of negative dependencies from data, and an applied study using original experimental data produces novel insights on wireless interference in LoRa networks., Comment: submitted to ICML2023

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 a slightly improved asymptotic approximation ratio 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 a simple generalization of the elegant net-finder algorithm by 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: Accepted by WAOA2023

The Simplex algorithm for solving linear programs-one of Computing in Science & Engineering's top 10 most influential algorithms of the 20th century-is an important topic in many algorithms courses. While the Simplex algorithm relies on intuitive geometric ideas, the computationally-involved mechanics of the algorithm can obfuscate a geometric understanding. In this paper, we present gilp, an easy-to-use Simplex algorithm visualization tool designed to explicitly connect the mechanical steps of the algorithm with their geometric interpretation. We provide an extensive library with example visualizations, and our tool allows an instructor to quickly produce custom interactive HTML files for students to experiment with the algorithm (without requiring students to install anything!). The tool can also be used for interactive assignments in Jupyter notebooks, and has been incorporated into a forthcoming Data Science and Decision Making interactive textbook. In this paper, we first describe how the tool fits into the existing literature on algorithm visualizations: how it was designed to facilitate student engagement and instructor adoption, and how it substantially extends existing algorithm visualization tools for Simplex. We then describe the development and usage of the tool, and report feedback from its use in a course with roughly 100 students. Student feedback was overwhelmingly positive, with students finding the tool easy to use: it effectively helped them link the algebraic and geometrical views of the Simplex algorithm and understand its nuances. Finally, gilp is open-source, includes an extension to visualizing linear programming-based branch and bound, and is readily amenable to further extensions., Comment: ACM SIGCSE 2023 Manuscript, 12 pages, 6 figures

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.

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.

We present the first nontrivial approximation algorithm for the bottleneck asymmetric traveling salesman problem. Given an asymmetric metric cost between n vertices, the problem is to find a Hamiltonian cycle that minimizes its bottleneck (or maximum-length edge) cost. We achieve an O(log n / log log n) approximation performance guarantee by giving a novel algorithmic technique to shortcut Eulerian circuits while bounding the lengths of the shortcuts needed. This allows us to build on a related result of Asadpour, Goemans, M\k{a}dry, Oveis Gharan, and Saberi to obtain this guarantee. Furthermore, we show how our technique yields stronger approximation bounds in some cases, such as the bounded orientable genus case studied by Oveis Gharan and Saberi. We also explore the possibility of further improvement upon our main result through a comparison to the symmetric counterpart of the problem., Comment: 16 pages, 3 figures

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.

We consider the following single-machine scheduling problem, which is often denoted $1||\sum f_{j}$: we are given $n$ jobs to be scheduled on a single machine, where each job $j$ has an integral processing time $p_j$, and there is a nondecreasing, nonnegative cost function $f_j(C_{j})$ that specifies the cost of finishing $j$ at time $C_{j}$; the objective is to minimize $\sum_{j=1}^n f_j(C_j)$. Bansal \& Pruhs recently gave the first constant approximation algorithm with a performance guarantee of 16. We improve on this result by giving a primal-dual pseudo-polynomial-time algorithm based on the recently introduced knapsack-cover inequalities. The algorithm finds a schedule of cost at most four times the constructed dual solution. Although we show that this bound is tight for our algorithm, we leave open the question of whether the integrality gap of the LP is less than 4. Finally, we show how the technique can be adapted to yield, for any $\epsilon >0$, a $(4+\epsilon )$-approximation algorithm for this problem., Comment: 26 pages. A preliminary version appeared in APPROX 2011. arXiv admin note: text overlap with arXiv:1403.0298

The growing popularity of bike-sharing systems around the world has motivated recent attention to models and algorithms for their effective operation. Most of this literature focuses on their daily operation for managing asymmetric demand. In this work, we consider the more strategic question of how to (re-)allocate dock-capacity in such systems. We develop mathematical formulations for variations of this problem (either for service performance over the course of one day or for a long-run-average) and exhibit discrete convex properties in associated optimization problems. This allows us to design a practically fast polynomial-time allocation algorithm to compute an optimal solution for this problem, which can also handle practically motivated constraints, such as a limit on the number of docks moved in the system. We apply our algorithm to data sets from Boston, New York City, and Chicago to investigate how different dock allocations can yield better service in these systems. Recommendations based on our analysis have led to changes in the system design in Chicago and New York City. Beyond optimizing for improved quality of service through better allocations, our results also provide a metric to compare the impact of strategically reallocating docks and the rebalancing of bikes.

We present a deterministic (1+sqrt(5))/2-approximation algorithm for the s-t path TSP for an arbitrary metric. Given a symmetric metric cost on n vertices including two prespecified endpoints, the problem is to find a shortest Hamiltonian path between the two endpoints; Hoogeveen showed that the natural variant of Christofides' algorithm is a 5/3-approximation algorithm for this problem, and this asymptotically tight bound in fact has been the best approximation ratio known until now. We modify this algorithm so that it chooses the initial spanning tree based on an optimal solution to the Held-Karp relaxation rather than a minimum spanning tree; we prove this simple but crucial modification leads to an improved approximation ratio, surpassing the 20-year-old barrier set by the natural Christofides' algorithm variant. Our algorithm also proves an upper bound of (1+sqrt(5))/2 on the integrality gap of the path-variant Held-Karp relaxation. The techniques devised in this paper can be applied to other optimization problems as well: these applications include improved approximation algorithms and improved LP integrality gap upper bounds for the prize-collecting s-t path problem and the unit-weight graphical metric s-t path TSP., Comment: 31 pages, 5 figures

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