Cathy Wu

Optimizing the diamond lane: A more tractable carpool problem and algorithms

Carpooling has been long deemed a promising approach to better utilizing existing transportation infrastructure. However, there are several reasons carpooling is still not the preferred mode of commute in the United States: first, complex human factors, including time constraints and not having right incentive structures, discourage the sharing of rides; second, algorithmic and technical barriers inhibit the development of online services for matching riders. In this work, we study algorithms for 3+ high-occupancy vehicle (HOV) lanes, which permit vehicles that hold three or more people. We focus on the technical barriers but also address the aforementioned human factors. We formulate the HOV3 Carpool problem, and show that it is NP-Complete. We thus pose the relaxed problem HOV3- Carpool problem, allowing groups of up to size three, and propose several methods for solving the problem of finding globally optimal carpool groups that may utilize these 3- HOV lanes. Our methods include local search, integer programming, and dynamic programming. Our local search methods include sampling-based (hill-climbing and simulated annealing), classical neighborhood search, and a hybrid random neighborhood search. We assess the methods numerically in terms of objective value and scalability. Our findings show that our sampling-based local search methods scale up to 100K agents, thereby improving upon related previous work (which studies up to 1000 agents). The hill climbing local search method converges significantly closer and faster towards a naive lower bound on cumulative carpooling cost.

Clustering for set partitioning with a case study in ridesharing

By exploring alternative approaches to combinatorial optimization, we propose the first known formal connection between clustering and set partitioning, with the goal of identifying a subclass of set partitioning problems that can be solved efficiently and with optimality guarantees through a clustering approach. We prove the equivalence between classical centroid clustering problems and a special case of set partitioning called metric k-set partitioning. We discuss the implications for k-means and regularized geometric k-medians, and we give several future extensions and applications. Finally, we present a case study in combinatorial optimization for ridesharing, in which we use an efficient Expectation Maximization (EM) style algorithm to achieve a 69% reduction in total vehicle distance, as compared with no ridesharing.

Projected sub-gradient with ℓ1 or simplex constraints via isotonic regression

We consider two classic problems in convex optimization: 1) minimizing a convex objective over the nonnegative orthant of the ℓ1-ball and 2) minimizing a convex objective over the probability simplex. We propose an efficient and simple equality constraint elimination technique which converts the ℓ1 and simplex constraints into order constraints. We formulate the projection onto the feasible set as an isotonic regression problem, which can be solved exactly in O(n) time via the Pool Adjacent Violators Algorithm (PAVA), where n is the dimension of the space. We design a C++ implementation of PAVA up to 25,000 times faster than scikit-learn. Our PAVA-based projection step enables the design of efficient projected subgradient methods which compare well against projected algorithms using direct projections onto the ℓ1-ball and onto the simplex, with projection in O(nlog(n)) exact time and O(n) expected time. Interestingly, our technique is particularly well adapted to learning from sparse, skewed, or aggregated data, by decreasing the cross-correlations between data points.