Recent advances in natural language processing highlight two key factors for improving reasoning in large language models (LLMs): (i) allocating more test-time compute tends to help on harder problems but often introduces redundancy in the reasoning trace, and (ii) compute is most effective when reasoning is systematic and incremental, forming structured chains of thought (CoTs) akin to human problem-solving. To study these factors in isolation, we introduce a controlled setting based on shortest-path tasks in layered graphs. We train decoder-only transformers on question-trace-answer triples using a custom tokenizer, comparing models trained on optimal bottom-up dynamic programming traces with those trained on longer, valid traces involving backtracking. Surprisingly, with the same training-token budget, models trained on inefficient traces generalize better to unseen graphs. This benefit is not due to length alone-injecting arbitrary redundancy into reasoning traces fails to help and can even hurt performance. Instead, we find that generalization correlates with the model's confidence in next-token prediction, suggesting that long, coherent, and locally incremental traces make the training signal easier to optimize.

A canonical problem in computer science is to find the shortest route to every point in a network. A new approach beats the classic algorithm taught in textbooks. If you want to solve a tricky problem, it often helps to get organized. You might, for example, break the problem into pieces and tackle the easiest pieces first. But this kind of sorting has a cost.

We introduce a semidefinite relaxation for optimal control of linear systems with time scaling. These problems are inherently nonconvex, since the system dynamics involves bilinear products between the discretization time step and the system state and controls. The proposed relaxation is closely related to the standard second-order semidefinite relaxation for quadratic constraints, but we carefully select a subset of the possible bilinear terms and apply a change of variables to achieve empirically tight relaxations while keeping the computational load light. We further extend our method to handle piecewise-affine (PWA) systems by formulating the PWA optimal-control problem as a shortest-path problem in a graph of convex sets (GCS). In this GCS, different paths represent different mode sequences for the PWA system, and the convex sets model the relaxed dynamics within each mode. By combining a tight convex relaxation of the GCS problem with our semidefinite relaxation with time scaling, we can solve PWA optimal-control problems through a single semidefinite program.

The problem of finding the shortest path in a graph G(V, E) has been widely studied. However, in many applications it is necessary to compute an arbitrary number of them, k. Even though the problem has raised a lot of interest from different research communities and many applications of it are known, it has not been addressed to the same extent as the single shortest path problem. The best algorithm known for efficiently solving this task has a time complexity of O (|E| + |V|log{|V|}+k|V|)$ when computing paths in explicit form, and is based on best-first search. This paper introduces a new search algorithm with the same time complexity, which results from a natural evolution of A* thus, it preserves all its interesting properties, making it widely applicable to many different domains. Experiments in various testbeds show a significant improvement in performance over the state of the art, often by one or two orders of magnitude.

Data-driven optimization uses contextual information and machine learning algorithms to find solutions to decision problems with uncertain parameters. While a vast body of work is dedicated to interpreting machine learning models in the classification setting, explaining decision pipelines involving learning algorithms remains unaddressed. This lack of interpretability can block the adoption of data-driven solutions as practitioners may not understand or trust the recommended decisions. We bridge this gap by introducing a counterfactual explanation methodology tailored to explain solutions to data-driven problems. We introduce two classes of explanations and develop methods to find nearest explanations of random forest and nearest-neighbor predictors. We demonstrate our approach by explaining key problems in operations management such as inventory management and routing.

In algorithms, as in life, negativity can be a drag. Consider the problem of finding the shortest path between two points on a graph -- a network of nodes connected by links, or edges. Often, these edges aren't interchangeable: A graph could represent a road map on which some roads are slower than others or have higher tolls. Computer scientists account for these differences by pairing each edge with a "weight" that quantifies the cost of moving across that segment -- whether that cost represents time, money or something else. Since the 1970s, they've known how to find shortest paths essentially as fast as theoretically possible, assuming all weights are positive numbers. But on some graphs weights can be negative -- traveling along one segment can offset the cost of traversing another.

In this paper, we tackle multi-robot planning (MRP), which aims to route a fleet of robots in a warehouse so as to achieve the maximum reward in a limited amount of time, while not having the robots collide and obeying the constraints of individual robots. In MRP, individual robots may make multiple trips over a given time window and may carry multiple items on each trip. We optimize the efficiency of the warehouse, not the makespan, since we expect new orders to be continuously added. Our contributions are that (1) we adapt the integer linear programming (ILP) formulation and column generation (CG) approach for (prize collecting) vehicle routing (Desrochers et al. 1992, Stenger et al. 2013) to MRP and (2) adapt the seminal work of (Boland et al. 2017) to permit efficient optimization by avoiding consideration of every time increment. Routing problems for a fleet of robots in a warehouse are often treated as Multi-Agent Pathfinding problems (MAPF) (Stern et al. 2019).

Ph.D. dissertation "Bi-directional and heuristic search in path problems" (Stanford, Computer Science, 1970) summarized in this article in Machine Intelligence 6 (1971).In the uni-directional algorithms, the search proceeds from an initial nodeforward until the goal node is encountered. Problems for which the goal nodeis explicitly known can be searched backward from the goal node. Analgorithm combining both search directions is bi-directional.This method has not seen much use because book-keeping problems werethought to outweigh the possible search reduction. The use of hashingfunctions to partition the search space provides a solution to some of theseimplementation problems. However, a more serious difficulty is involved.To realize significant savings in bi-directional search, the forward andbackward search trees must meet in the 'middle' of the space. The potentialbenefits from this technique motivates this paper's examination of thetheoretical and practical problems in using bi-directional search.