Elad Michael, Tony A. Wood, Chris Manzie, Iman Shames
We examine the robustness of bottleneck assignment problems to perturbations in the assignment weights. We derive two algorithms that provide uncertainty bounds for robust assignment. We prove that the bottleneck assignment is guaranteed to be invariant to perturbations which lie within the provided bounds. We apply the method to an example of task assignment for a multi-agent system.
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In assignment problems, decision makers are often interested in not only the optimal assignment, but also the sensitivity of the optimal assignment to perturbations in the assignment weights. Typically, only perturbations to individual assignment weights are considered. We present a novel extension of the traditional sensitivity analysis by allowing for simultaneous variations in all assignment weights. Focusing on the bottleneck assignment problem, we provide two different methods of quantifying the sensitivity of the optimal assignment, and present algorithms for each. Numerical examples as …
Solving the Lexicographic Bottleneck Assignment Problem (LexBAP) typically relies on centralised computation with order quartic complexity. We consider the Sequential Bottleneck Assignment Problem (SeqBAP), which yields a greedy solution to the LexBAP and discuss the relationship between the SeqBAP, the LexBAP, and the Bottleneck Assignment Problem (BAP). In particular, we reexamine tools used to analyse the structure of the BAP, and apply them to derive an algorithm that solves the SeqBAP with cubic complexity. We show that the set of solutions of the LexBAP is a subset of the solutions of th…
An assignment problem arises when there exists a set of tasks that must be allocated to a set of agents. The bottleneck assignment problem (BAP) has the objective of minimising the most costly allocation of a task to an agent. Under certain conditions the structure of the BAP can be exploited such that subgroups of tasks are assigned separately with lower complexity and then merged to form a combined assignment. In particular, we discuss merging the assignments from two separate BAPs and use the solution of the subproblems to bound the solution of the combined problem. We also provide conditio…
We call a matrix completely mixable if the entries in its columns can be permuted so that all row sums are equal. If it is not completely mixable, we want to determine the smallest maximal and largest minimal row sum attainable. These values provide a discrete approximation of of minimum variance problems for discrete distributions, a problem motivated by the question how to estimate the $α$-quantile of an aggregate random variable with unknown dependence structure given the marginals of the constituent random variables. We relate this problem to the multidimensional bottleneck assignment prob…
We study the multi-level bottleneck assignment problem (MBA), which has important applications in scheduling and quantitative finance. Given a weight matrix, the task is to rearrange entries in each column such that the maximum sum of values in each row is as small as possible. We analyze the complexity of this problem in a generalized setting, where there are restrictions in how values in columns can be permuted. We present a lower bound on its approximability by giving a non-trivial gap reduction from three-dimensional matching to MBA. To solve MBA, a greedy method has been used in the liter…
We develop an algorithm to solve the Bottleneck Assignment Problem (BAP) that is amenable to having computation distributed over a network of agents. This consists of exploring how each component of the algorithm can be distributed, with a focus on one component in particular, i.e., the function to search for an augmenting path. An augmenting path is a common tool used in most BAP algorithms and poses a particular challenge for this distributed approach. Given this significance, we compare two different methods to search for an augmenting path in a bipartite graph. We also exploit properties o…
Certification of Bottleneck Task Assignment with Shortest Path Criteria
针对有障碍环境中最短路径精确求解NP难、难以确定多机器人最优瓶颈任务分配的问题，提出迭代提升路径规划精度、基于路径上下界与分配灵敏度的认证方法，用多项式时间采样规划获取界值，经多机器人案例验证有效。
Aircraft and crew are two major resources that ensure the smooth operations of airlines. However, with the anticipated growth in the aviation industry, the crew resource is predicted to be insufficient worldwide and has been one of the bottlenecks in fast-developing airlines. In addition, the mismatch between aircraft and crew has influenced the airlines' operation, and limited the full usage of all resources. To resolve this problem, we propose a tactical fleet assignment and crew pairing problem with crew flight time allocation (TFACPP). The basic integrated model is reformulated by the Bend…
Traditional task assignment approaches for multi-agent motion control do not take the possibility of collisions into account. This can lead to challenging requirements for path planning. We derive an assignment method that not only minimises the largest distance between an agent and its assigned destination but also provides local constraints for guaranteed collision avoidance. To this end, we introduce a sequential bottleneck optimisation problem and define a notion of robustness of an optimising assignment to changes of individual assignment costs. Conditioned on a sufficient level of robust…
Learning-Augmented Scalable Linear Assignment Problem Optimization via Neural Dual Warm-Starts
提出一种学习增强框架，通过预测对偶变量热启动精确求解器，并设计轻量级行独立架构RowDualNet避免O(N^2)内存瓶颈，实现可扩展的神经热启动，在保持最优性的同时获得超过2倍加速。
Dynamic traffic assignment in a corridor network: Optimum versus Equilibrium
本研究探讨了走廊网络中早晚通勤的动态系统最优与动态用户均衡交通分配问题，提出了一种推导两者闭式解的新方法，并揭示了DUE排队延迟与DSO最优收费之间的理论等价关系。
Common Optimal Supports and Bottleneck Hierarchies in Truncated Cones
研究截锥中线性规划所有最优解支撑集的公共交集，证明其为自然簇的横截，引入至多|E|-1层稳定的瓶颈层级，并给出精确上界及多种应用。
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