skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: A General Statistical Physics Framework for Assignment Problems
Linear assignment problems hold a pivotal role in combinatorial optimization, offering a broad spectrum of applications within the field of data sciences. They consist of assigning “agents” to “tasks” in a way that leads to a minimum total cost associated with the assignment. The assignment is balanced when the number of agents equals the number of tasks, with a one-to-one correspondence between agents and tasks, and it is and unbalanced otherwise. Additional options and constraints may be imposed, such as allowing agents to perform multiple tasks or allowing tasks to be performed by multiple agents. In this paper, we propose a novel framework that can solve all these assignment problems employing methodologies derived from the field of statistical physics. We describe this formalism in detail and validate all its assertions. A major part of this framework is the definition of a concave effective free energy function that encapsulates the constraints of the assignment problem within a finite temperature context. We demonstrate that this free energy monotonically decreases as a function of a parameter β representing the inverse of temperature. As β increases, the free energy converges to the optimal assignment cost. Furthermore, we demonstrate that when β values are sufficiently large, the exact solution to the assignment problem can be derived by rounding off the elements of the computed assignment matrix to the nearest integer. We describe a computer implementation of our framework and illustrate its application to multi-task assignment problems for which the Hungarian algorithm is not applicable.  more » « less
Award ID(s):
1934568
PAR ID:
10556096
Author(s) / Creator(s):
;
Publisher / Repository:
MDPI
Date Published:
Journal Name:
Algorithms
Volume:
17
Issue:
5
ISSN:
1999-4893
Page Range / eLocation ID:
212
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; ; ; (, Proceedings of the International Joint Conference on Autonomous Agents and Multiagent Systems)
    We consider a variant of the Multi-Agent Path-Finding problem that seeks both task assignments and collision-free paths for a set of agents navigating on a graph, while minimizing the sum of costs of all agents. Our approach extends Conflict-Based Search (CBS), a framework that has been previously used to find collision-free paths for a given fixed task assignment. Our approach is based on two key ideas: (i) we operate on a search forest rather than a search tree; and (ii) we create the forest on demand, avoiding a factorial explosion of all possible task assignments. We show that our new algorithm, CBS-TA, is complete and optimal. The CBS framework allows us to extend our method to ECBS-TA, a bounded suboptimal version. We provide extensive empirical results comparing CBS-TA to task assignment followed by CBS, Conflict-Based Min-Cost-Flow (CBM), and an integer linear program (ILP) solution, demonstrating the advantages of our algorithm. Our results highlight a significant advantage in jointly optimizing the task assignment and path planning for very dense cases compared to the traditional method of solving those two problems independently. For large environments with many robots we show that the traditional approach is reasonable, but that we can achieve similar results with the same runtime but stronger suboptimality guarantees. 
    more » « less
  2. ; ; ; (, Proceedings of the AAAI Conference on Artificial Intelligence)
    In reinforcement learning (RL), the ability to utilize prior knowledge from previously solved tasks can allow agents to quickly solve new problems. In some cases, these new problems may be approximately solved by composing the solutions of previously solved primitive tasks (task composition). Otherwise, prior knowledge can be used to adjust the reward function for a new problem, in a way that leaves the optimal policy unchanged but enables quicker learning (reward shaping). In this work, we develop a general framework for reward shaping and task composition in entropy-regularized RL. To do so, we derive an exact relation connecting the optimal soft value functions for two entropy-regularized RL problems with different reward functions and dynamics. We show how the derived relation leads to a general result for reward shaping in entropy-regularized RL. We then generalize this approach to derive an exact relation connecting optimal value functions for the composition of multiple tasks in entropy-regularized RL. We validate these theoretical contributions with experiments showing that reward shaping and task composition lead to faster learning in various settings. 
    more » « less
  3. (, Mathematics of control signals and systems)
    null (Ed.)
    We study a sequence of many-agent exit time stochastic control problems, parameterized by the number of agents, with risk-sensitive cost structure. We identify a fully characterizing assumption, under which each such control problem corresponds to a risk-neutral stochastic control problem with additive cost, and sequentially to a risk-neutral stochastic control problem on the simplex that retains only the distribution of states of agents, while discarding further specific information about the state of each agent. Under some additional assumptions, we also prove that the sequence of value functions of these stochastic control problems converges to the value function of a deterministic control problem, which can be used for the design of nearly optimal controls for the original problem, when the number of agents is sufficiently large. 
    more » « less
  4. ; ; ; ; (, IJCAI)
    For the assignment problem where multiple indivis- ible items are allocated to a group of agents given their ordinal preferences, we design randomized mechanisms that satisfy first-choice maximality (FCM), i.e., maximizing the number of agents as- signed their first choices, together with Pareto- efficiency (PE). Our mechanisms also provide guarantees of ex-ante and ex-post fairness. The generalizedeager Boston mechanism is ex-ante envy-free, and ex-post envy-free up to one item (EF1). The generalized probabilistic Boston mech- anism is also ex-post EF1, and satisfies ex-ante ef- ficiency instead of fairness. We also show that no strategyproof mechanism satisfies ex-post PE, EF1, and FCM simultaneously. In doing so, we expand the frontiers of simultaneously providing efficiency and both ex-ante and ex-post fairness guarantees for the assignment problem. 
    more » « less
  5. ; ; ; ; ; (, Journal of Computing and Information Science in Engineering)
    Abstract Swarm manufacturing is an emerging manufacturing paradigm that employs a heterogeneous swarm of robots to accomplish complex hybrid manufacturing tasks. Cooperative 3D printing (C3DP), a specialized form of swarm manufacturing, enables multiple printers to collaboratively produce large-scale parts, addressing key tradeoffs in additive manufacturing, such as size, speed, quality, and cost. A fundamental challenge in C3DP is ensuring collision-free, time-optimal printing in a shared workspace. This is a complex problem that can be influenced by factors such as the number of printers, part geometry, printer positioning, mobility, and kinematics. In this article, we present SafeZone*, a collision-free and scalable C3DP framework that optimizes printing time by co-considering the geometry (area and shape) and topology (space-connectivity) of a shared workspace during layer partitioning. We first establish a conceptual framework to mathematically represent the topology of a layer through partition graphs. Then, we use a Voronoi tessellation within a constrained optimization framework to control the partition graph and minimize makespan. The Voronoi sites are associated with printer locations, allowing the framework to integrate physical constraints and facilitating solutions for systems with robotic manipulators. Physical testing in a four-printer scenario with robotic arms confirms that SafeZone* enables collision-free printing, resulting in a printing time reduction of 44.63% when compared to the single-printer scenario. Finally, numerical studies reveal trends in the optimal solutions concerning the chromatic number of their resulting partition graphs and the distribution of the printing areas among printers. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email