Job Assignment

Graph Theory · Axiom Academy

REAL WORLD Job Assignment Problems How Matching Theory Solves Perfect Pairings You're a manager with 5 employees and 5 critical tasks. Each employee has different skills, and each task requires specific expertise. How do you assign employees to tasks to maximize overall productivity? This is a bipartite matching problem - we need to pair two distinct groups (employees and tasks) optimally. Try Matching Employees to Tasks Below are 5 employees (left) and 5 tasks (right). Each employee can do certain tasks based on their skills. Try to create a complete matching where every employee gets exactly one task. Hint: Click an employee on the left, then click a compatible task on the right to create a match. In this simple example, skills align perfectly! Understanding Bipartite Graphs Assignment problems are modeled as bipartite graphs - graphs with two distinct sets of vertices where edges only connect vertices from different sets. What if both sides have preferences? The Gale-Shapley algorithm finds stable matchings where no two people would prefer each other over their current partners. What makes a matching "stable"? Each person proposes to their most preferred partner who hasn't rejected them Each partner tentatively accepts their most preferred proposal and rejects the rest Rejected proposers move to their next choice Result: A stable matching guaranteed! Real-World Matching Applications

This is the written version of the interactive lesson above. See the full Graph Theory course.