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Isaac Rudich

Publications and source records attributed to Isaac Rudich.

5 recordsLinked to original sources
arXiv2026

Implicit Decision Diagrams

Decision Diagrams (DDs) have emerged as a powerful tool for discrete optimization, with rapidly growing adoption. DDs are directed acyclic layered graphs; restricted DDs are a generalized greedy heuristic for finding feasible solutions, and relaxed DDs compute combinatorial relaxed bounds. There is substantial theory that leverages DD-based bounding, yet the complexity of constructing the DDs themselves has received little attention. Standard restricted DD construction requires $O(w \log(w))$ per layer; standard relaxed DD construction requires $O(w^2)$, where $w$ is the width of the DD. Increasing $w$ improves bound quality at the cost of more time and memory. We introduce implicit Decision Diagrams, storing arcs implicitly rather than explicitly, and reducing per-layer complexity to $O(w)$ for restricted and relaxed DDs. We prove this is optimal: any framework treating state-update and merge operations as black boxes cannot do better. Optimal complexity shifts the challenge from algorithmic overhead to low-level engineering. We show how implicit DDs can drive a MIP solver, and release ImplicitDDs, an open-source Julia solver exploiting the implementation refinements our theory enables. Experiments demonstrate the solver outperforms Gurobi on Subset Sum. Code (https://github.com/IsaacRudich/ImplicitDDs.jl)

math.OC↗
arXiv2024

Simple Stochastic Stopping Games: A Generator and Benchmark Library

Simple Stochastic Games (SSGs) were introduced by Anne Condon in 1990, as the simplest version of Stochastic Games for which there is no known polynomial-time algorithm. Condon showed that Stochastic Games are polynomial-time reducible to SSGs, which in turn are polynomial-time reducible to Stopping Games. SSGs are games where all decisions are binary and every move has a random outcome with a known probability distribution. Stopping Games are SSGs that are guaranteed to terminate. There are many algorithms for SSGs, most of which are fast in practice, but they all lack theoretical guarantees for polynomial-time convergence. The pursuit of a polynomial-time algorithm for SSGs is an active area of research. This paper is intended to support such research by making it easier to study the graphical structure of SSGs. Our contributions are: (1) a generating algorithm for Stopping Games, (2) a proof that the algorithm can generate any game, (3) a list of additional polynomial-time reductions that can be made to Stopping Games, (4) an open source generator for generating fully reduced instances of Stopping Games that comes with instructions and is fully documented, (5) a benchmark set of such instances, (6) and an analysis of how two main algorithm types perform on our benchmark set.

cs.CC↗
arXiv2023

An Exact Framework for Solving the Space-Time Dependent TSP

Many real-world scenarios involve solving bi-level optimization problems in which there is an outer discrete optimization problem, and an inner problem involving expensive or black-box computation. This arises in space-time dependent variants of the Traveling Salesman Problem, such as when planning space missions that visit multiple astronomical objects. Planning these missions presents significant challenges due to the constant relative motion of the objects involved. There is an outer combinatorial problem of finding the optimal order to visit the objects and an inner optimization problem that requires finding the optimal departure time and trajectory to travel between each pair of objects. The constant motion of the objects complicates the inner problem, making it computationally expensive. This paper introduces a novel framework utilizing decision diagrams (DDs) and a DD-based branch-and-bound technique, Peel-and-Bound, to achieve exact solutions for such bi-level optimization problems, assuming sufficient inner problem optimizer quality. The framework leverages problem-specific knowledge to expedite search processes and minimize the number of expensive evaluations required. As a case study, we apply this framework to the Asteroid Routing Problem (ARP), a benchmark problem in global trajectory optimization. Experimental results demonstrate the framework's scalability and ability to generate robust heuristic solutions for ARP instances. Many of these solutions are exact, contingent on the assumed quality of the inner problem's optimizer.

math.OC↗
arXiv2023

Improved Peel-and-Bound: Methods for Generating Dual Bounds with Multivalued Decision Diagrams

Decision diagrams are an increasingly important tool in cutting-edge solvers for discrete optimization. However, the field of decision diagrams is relatively new, and is still incorporating the library of techniques that conventional solvers have had decades to build. We drew inspiration from the warm-start technique used in conventional solvers to address one of the major challenges faced by decision diagram based methods. Decision diagrams become more useful the wider they are allowed to be, but also become more costly to generate, especially with large numbers of variables. In the original version of this paper, we presented a method of peeling off a sub-graph of previously constructed diagrams and using it as the initial diagram for subsequent iterations that we call peel-and-bound. We tested the method on the sequence ordering problem, and our results indicate that our peel-and-bound scheme generates stronger bounds than a branch-and-bound scheme using the same propagators, and at significantly less computational cost. In this extended version of the paper, we also propose new methods for using relaxed decision diagrams to improve the solutions found using restricted decision diagrams, discuss the heuristic decisions involved with the parallelization of peel-and-bound, and discuss how peel-and-bound can be hyper-optimized for sequencing problems. Furthermore, we test the new methods on the sequence ordering problem and the traveling salesman problem with time-windows (TSPTW), and include an updated and generalized implementation of the algorithm capable of handling any discrete optimization problem. The new results show that peel-and-bound outperforms ddo (a decision diagram based branch-and-bound solver) on the TSPTW. We also close 15 open benchmark instances of the TSPTW.

math.OC↗
arXiv2022

Peel-and-Bound: Generating Stronger Relaxed Bounds with Multivalued Decision Diagrams

Decision diagrams are an increasingly important tool in cutting-edge solvers for discrete optimization. However, the field of decision diagrams is relatively new, and is still incorporating the library of techniques that conventional solvers have had decades to build. We drew inspiration from the warm-start technique used in conventional solvers to address one of the major challenges faced by decision diagram based methods. Decision diagrams become more useful the wider they are allowed to be, but also become more costly to generate, especially with large numbers of variables. We present a method of peeling off a sub-graph of previously constructed diagrams and using it as the initial diagram for subsequent iterations that we call peel-and-bound. We test the method on the sequence ordering problem, and our results indicate that our peel-and-bound scheme generates stronger bounds than a branch-and-bound scheme using the same propagators, and at significantly less computational cost.

math.OC↗