[Paper Review] The Application of Bipartite Matching in Assignment Problem

Feiyang Chen, Nan Chen+2 more|arXiv (Cornell University)|Feb 1, 2019

Game Theory and Voting SystemsEconomics, Econometrics and Finance3 citations

TL;DR

This paper proposes the Ultimate Hungarian Algorithm (UHA), a novel approach to solving the maximum-weight bipartite matching problem in assignment scenarios. By leveraging feasible vertex labeling and iterative augmenting path search, UHA achieves $O(|V|^3)$ time complexity—significantly improving upon the brute-force $O(|V|!)$ method—enabling efficient, optimal staff-task assignment in industrial and operational research settings.

ABSTRACT

The optimized assignment of staff is of great significance for improving the production efficiency of the society. For specific tasks, the key to optimizing staffing is personnel scheduling. The assignment problem is classical in the personnel scheduling. In this paper, we abstract it as an optimal matching model of a bipartite graph and propose the Ultimate Hungarian Algorithm(UHA). By introducing feasible labels, iteratively searching for the augmenting path to get the optimal match(maximum-weight matching). And we compare the algorithm with the traditional brute force method, then conclude that our algorithm has lower time complexity and can solve the problems of maximum-weight matching more effectively.

Motivation & Objective

• To address the inefficiency of manual and brute-force methods in solving personnel assignment problems.
• To develop a computationally efficient algorithm for maximum-weight bipartite matching in staff scheduling.
• To replace experience-based allocation with a quantitative, automated, and optimal matching framework.
• To improve scalability and precision in large-scale assignment problems using graph-theoretic optimization.

Proposed method

• Formulates the assignment problem as a maximum-weight bipartite matching problem on a labeled graph.
• Applies feasible vertex labeling where $ l(x) + l(y) \geq w(x,y) $ for all edges, ensuring the triangle inequality-like constraint.
• Uses an equality subgraph $ G_l = (V, E_l) $, where $ E_l $ contains edges satisfying $ l(x) + l(y) = w(x,y) $.
• Employs iterative augmenting path search: starting from unmatched vertices, alternates between matched and unmatched edges to extend the path.
• Improves labeling via $ \delta_l = \min_{u \in S, v \notin T} (l(u) + l(v) - w(u,v)) $, adjusting labels to expand the equality subgraph.
• Flips the matching along the augmenting path to increase matching size, repeating until a perfect matching is achieved.

Experimental results

Research questions

• RQ1 Can a graph-theoretic approach outperform brute-force and manual assignment in solving maximum-weight staff-task matching?
• RQ2 How can feasible vertex labeling be used to bound and converge toward an optimal solution?
• RQ3 What is the computational complexity of an iterative augmenting path algorithm with dynamic labeling improvement?
• RQ4 Can the algorithm be systematically improved to avoid exponential time growth in large-scale assignment problems?
• RQ5 How does the UHA compare to traditional Hungarian algorithm variants in terms of time complexity and solution quality?

Key findings

• The UHA achieves $ O(|V|^3) $ time complexity, representing a substantial improvement over the $ O(|V|!) $ complexity of brute-force enumeration.
• The algorithm guarantees a maximum-weight matching by maintaining feasible labeling and using the Kuhn-Munkres theorem.
• A perfect matching under feasible labeling corresponds to an optimal solution, as proven by the upper bound on matching weight.
• Each iteration increases either the matching size or the number of labeled edges, ensuring termination in finite steps.
• The method effectively replaces manual, experience-based assignment with a deterministic, automated, and optimal matching process.
• The algorithm is not suitable for assignment problems with ordered task constraints, indicating a limitation for future extension.