Algorithmic Reductions: Network Flow and NP-Completeness in Real-World Scheduling Problems

TL;DR

This paper explores real-world scheduling problems, demonstrating polynomial-time reductions to known problems like network flow and NP-complete problems, and providing practical algorithms with empirical validation.

Contribution

It introduces reductions for hospital bed assignment and course scheduling, offering efficient algorithms and complexity analyses validated by experiments.

Findings

Network flow solution achieves O(n^2.51) empirical complexity

Greedy coloring algorithms run in O(n^2) with good approximation ratios

Problems are effectively reduced to well-studied computational problems

Abstract

This paper presents two real-world scheduling problems and their algorithmic solutions through polynomial-time reductions. First, we address the Hospital Patient-to-Bed Assignment problem, demonstrating its reduction to Maximum Bipartite Matching and solution via Network Flow algorithms. Second, we tackle the University Course Scheduling problem, proving its NP-Completeness through reduction from Graph Coloring and providing greedy approximation algorithms. Both problems are implemented in Python, with experimental results validating theoretical complexity analyses. Our Network Flow solution achieves O(n2.51) empirical complexity, while the greedy coloring algorithms demonstrate O(n2) behavior with approximation ratios consistently below the theoretical delta + 1 bound.