Global Sensitivity Analysis for the Linear Assignment Problem Elad Michael, Tony A. Wood, Chris Manzie, and Iman Shames Abstract— In this paper, the following question is addressed: finding maximum flow within a network [6], lower bound- given a linear assignment problem, how much can the all of the ing solutions to the quadratic assignment problem [7], and individual assignment weights be perturbed without changing maximum a-posteriori data association for object tracking the optimal assignment? The extension of results involving perturbations in just one edge or one row/column are presented. [8]. Algorithms for the derivation of these bounds are provided. We In this paper we focus on understanding which perturba- also show how these bounds may be used to prevent assignment tions the optimal assignment is robust to. However, in some churning in a multi-vehicle guidance scenario. applications, it is preferable to find a potentially sub-optimal assignment which is less sensitive to perturbation. For linear I. INTRODUCTION assignment, [10] provides an algorithm with guaranteed Task assignment is a fundamental part of multi-agent robustness to uncertainty in a subset of the parameters. control. Here we represent task assignment optimization Similarly, but for the bottleneck assignment problem, [11] over a weighted bipartite graph, with agents and tasks as provides robustness to weights within predefined intervals, vertices and assignments as edges. The objective function as well as complexity improvements on similar bottleneck may be minimum completion time of all tasks, minimum assignment algorithms. However, these papers necessarily total fuel usage, balanced completion times, or any of a sacrifice optimality for robustness. We focus on applications multitude of other choices [1]. For many applications, the that involve quantifying the robustness of the optimal assign- edge weights that are parameters in the objective are uncer- ment. tain. Uncertainties may be due to sensor measurement errors, Previous work in assignment problem sensitivity has been finite difference estimation, or quantization in calculation or restricted to perturbations in one or a small subset of edges. communication. By studying which perturbations the optimal The sensitivity of the linear assignment problem to perturba- assignment is invariant to, we characterize the uncertainty in tions in a single edge weight perturbation is covered by [12] edge weights that can be tolerated, and when the optimal and [13]. Robustness to coupled perturbations in the weight assignment with the noisy measurements is optimal with of all edges incident on a vertex (one ”row” or ”column” of respect to the ground truth. edge weights) is investigated in [2] and [14]. In a multi-agent The terms robustness of the assignment will be used scenario, with weights as distances for example, this could frequently throughout this paper. Similar analyses may use be interpreted as an error in the state of an agent or a task, terms such as “sensitivity” [2], “stability” [3], or “post- causing a coupled error in the weights of all edges incident optimality” [4] of optimisation problems. These all explore on that agent or task. Most relevant to the work presented how the set of optimizers or optimal cost change as the here for the LAP, [15] examines robustness of all weights problem parameters are varied. The different definitions of under uniform perturbation, and proves complexity results. sensitivity analysis are covered in [5], as well as some prac- A similar analysis to the one provided is presented in [9], tical applications of sensitivity analysis for linear programs. however for the bottleneck assignment problem. Due to the additional constraints and particular structure The contribution of this work is to allow perturbations in of assignments problems compared to a standard linear all edge weights, without coupling. That is, each weight may program, we focus on which perturbations to the problem be perturbed individually and independently. This coincides arXiv:2005.11792v1 [math.OC] 24 May 2020 parameters the optimal assignment is invariant to, “Type with the work done in [15] for the minimal sensitivity 2 Sensitivity Analysis” [5]. The robustness or sensitivity edge. However, the results of [15] are conservative for all of the assignment problem in this paper refers to intervals other edges. In other work, such as [2] [14], simultaneous within which the edge weights can vary without changing perturbation is only considered in edges sharing a common the optimal assignment. vertex. These cannot be co-assigned due to the assign- We focus on the robustness of the linear assignment ment constraints, and thus extend the single edge results problem (LAP). The LAP minimizes the sum of all edge to considering a subset of edges. However, in the case of weights in an assignment, i.e. finds the matching of agents to the linear assignment problem, the changing of an edge tasks which minimizes the total cost. The linear assignment weight effects all assignments, in terms of relative cost. problem has many and diverse applications. For example, We provide algorithms which compute the sensitivity for a given weighted bipartite graph, and discuss the compu- ∗All authors are with Department of Electrical tational complexities. With these algorithms, we formulate and Electronic Engineering at the University of Mel- bourne [email protected], a set of sufficient conditions with which an assignment fwood.t,manziec,[email protected] made on noisy measurements can be proved to be optimal despite the noise. The results are used to mitigate the effects We now define robustness, and the concept of an allowable of “assignment churning” [16] in a multi-vehicle guidance perturbation. application. The paper is organized as follows. Section II Definition 1. Define ∆ to be an additive perturbation is devoted to background on the assignment problem and to the set of edge weights W. If the optimal assignment a formal definition of robustness. Section III shows that a Π∗ 2 LAP (G; W) is invariant to the addition of ∆, i.e., single allowable perturbation can be extended to a set of Π∗ 2 LAP (G; W +∆), then ∆ is an allowable perturbation. allowable perturbations. In Section IV we show how to find Equivalently, Π∗ 2 LAP (G; W) is robust to the perturbation an allowable perturbation. Section V motivates and defines a ∆. recursive application of the algorithm provided in Section IV. The example case of assignment churning is covered in With this definition of robustness, the problem addressed Section VI, and Section VII concludes. in this paper can be stated as follows II. PROBLEM FORMULATION Problem. Given a bipartite graph G, weights W, with optimal assignment Π∗ 2 LAP (G; W), characterize a set Define a bipartite graph G = (V; E) along with a set of Λ such that if ∆ 2 Λ then ∆ is an allowable perturbation. weights W over which the LAP is solved. Let the set of vertices V = VA [VT be such that VA \VT = ;, with the III. INTERVAL BOUND edge set E ⊆ V × V , and edge weights W = fw 2 j A T ij R We first show that if we have found an allowable per- (i; j) 2 Eg. The vertices V and V represent the agents and A T turbation ∆, the linearity of the cost function allows us to tasks respectively. Without loss of generality we may assume construct an entire set of allowable perturbations based on there are not more tasks than agents, i.e., jV j ≥ jV j. Also A T ∆. define the assignment as a set of binary decision variables ∗ Π = fπij 2 f0; 1g j (i; j) 2 Eg. The linear assignment Theorem 1. Let Π 2 LAP (G; W), and let ∆ = fδij j problem can be formulated as (i; j) 2 Eg be an allowable perturbation as in Definition 1. 0 0 Define another perturbation ∆ = fδij j (i; j) 2 Eg such X that min πijwij (1a) Π 0 ∗ (i;j)2E δij ≤ δij 8 (i; j): πij = 1; (3) X 0 ∗ s.t. πij = 1; j 2 VT ; (1b) δij ≥ δij 8 (i; j): πij = 0: (4) i2VA Then ∆0 is an allowable perturbation. X πij ≤ 1; i 2 VA: (1c) Proof. We will show that any perturbation ∆0 which sat- j2VT isfies (3) and (4) can be represented as the sum of the 00 If πij = 1, then we say that that edge (i; j) is assigned perturbation ∆ and ∆ , which are both allowable. By the 0 or vertex i is assigned to vertex j. Constraints (1b) and (1c) linearity of the cost function then, we conclude that ∆ is ensure that every agent is assigned to either one or zero other allowable. 00 0 tasks and that every task is assigned to one agent. The cost Let ∆ = fδij − δij j (i; j) 2 Eg. By the linearity of the of an assignment Π can be written as cost function (1a), X f(W + ∆0; Π∗) = f(W + ∆; Π∗) + f(∆00; Π∗): (5) f(W; Π) = πijwij: (2) (i;j)2E By assumption, ∆ is an allowable perturbation. In other There are many algorithms to solve the LAP, such as words, for Π∗ the optimizing assignment over the graph G the Hungarian Algorithm [17], the JV algorithm [18], or with weights W, Π∗ is an optimizing assignment for weights 00 Bertsekas’ auction algorithm [19]. In this paper we assess W +∆. Additionally, by construction, δij is non-positive for the robustness of the optimal assignment, independent of the all assigned edges in Π∗ and non-negative for all unassigned algorithm used to solve (1). Define the mapping LAP (G; W) edges in Π∗. Therefore, which takes a weighted bipartite graph and returns the opti- f(∆00; Π∗) ≤ f(∆00; Π) 8 Π mizer of (1). If there are multiple equivalent optimal assign- ments, LAP (G; W) returns the set of optimal assignments.