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

Anay Sinhal, Arpana Sinhal, Amit Sinhal, Amit Hirawat

TL;DR

The paper investigates two real-world scheduling problems through polynomial-time reductions: the Hospital Patient-to-Bed Assignment is solvable in polynomial time via a Network Flow construction that reduces to Maximum Bipartite Matching, while the University Course Scheduling problem is proven NP-Complete via a Graph Coloring reduction and is approached with greedy approximations. It introduces Welsh-Powell and DSatur heuristics, analyzes their theoretical and empirical performance, and validates results with Python implementations. Experiments show the Network Flow solution scales near $O(n^{2.51})$ in practice, and the greedy methods operate in near $O(n^{2})$ time with colors far below the $\Delta+1$ bound. Overall, the work demonstrates how problem structure and reductions can yield either optimal, tractable solutions or effective, scalable approximations for real-world scheduling tasks.

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.

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

TL;DR

The paper investigates two real-world scheduling problems through polynomial-time reductions: the Hospital Patient-to-Bed Assignment is solvable in polynomial time via a Network Flow construction that reduces to Maximum Bipartite Matching, while the University Course Scheduling problem is proven NP-Complete via a Graph Coloring reduction and is approached with greedy approximations. It introduces Welsh-Powell and DSatur heuristics, analyzes their theoretical and empirical performance, and validates results with Python implementations. Experiments show the Network Flow solution scales near in practice, and the greedy methods operate in near time with colors far below the bound. Overall, the work demonstrates how problem structure and reductions can yield either optimal, tractable solutions or effective, scalable approximations for real-world scheduling tasks.

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.
Paper Structure (21 sections, 2 theorems, 3 equations, 3 figures, 2 tables, 3 algorithms)

This paper contains 21 sections, 2 theorems, 3 equations, 3 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem 1: Hospital Patient Scheduling via Network Flow
  3. Real-World Problem Description
  4. Problem Abstraction
  5. Reduction to Network Flow
  6. Construction
  7. Proof of Correctness
  8. Complete Algorithm
  9. Problem 2: Course Scheduling as an NP-Complete Problem
  10. Real-World Problem Description
  11. Problem Abstraction
  12. NP-Completeness Proof
  13. Greedy Approximation Algorithms
  14. Welsh-Powell Algorithm
  15. DSatur Algorithm
  16. ...and 6 more sections

Key Result

Theorem 1

The maximum flow in network $G$ equals the maximum number of patients that can be admitted in the Hospital Assignment Problem.

Figures (3)

  • Figure 1: Schematic of flow network construction. Edges from patients to beds exist only for compatible department assignments. All capacities are 1.
  • Figure 2: Network Flow timing results. Left: raw timing data showing polynomial growth. Right: log-log plot with fitted line (slope $\approx 2.51$).
  • Figure 3: Greedy coloring results. Top-left: running time comparison. Top-right: log-log complexity analysis. Bottom-left: colors used vs. $\Delta+1$ bound. Bottom-right: approximation ratios.

Theorems & Definitions (7)

  • Definition 1: Hospital Assignment Problem
  • Theorem 1
  • proof
  • Definition 2: Course Scheduling Problem (CSP)
  • Theorem 2
  • proof
  • Claim 1