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Hungarian / Assignment Problem Ladder

This lane is for the first time bipartite pairing becomes weighted one-to-one assignment, not just maximum cardinality matching.

Who This Is For

Use this lane if:

  • Matching already feels comfortable
  • you can explain why "each endpoint at most once" creates a bipartite matching model
  • you now need minimum total cost across a perfect assignment

This lane is intentionally narrow:

  • one exact starter
  • one flagship note that is almost pure assignment matrix
  • one compare route against Min-Cost Flow
  • one reminder that blossom is a different boundary

Warm-Up

  • explain why a plain cardinality matcher cannot optimize total cost
  • explain why Min-Cost Flow still works here but is often heavier
  • hand-check the best permutation on a 3 x 3 cost matrix

Target skill:

  • recognize when the statement is exactly weighted bipartite assignment in matrix form

Core

  • one row per column
  • one column per row
  • minimum total cost
  • exact flagship rep: Task Assignment

Target skill:

  • map the problem directly to a cost matrix and trust the O(n^3) Hungarian route

Stretch

  • revisit the same family under "maximize total profit" by negating weights
  • compare one assignment-matrix problem against one Min-Cost Flow problem and explain why the models differ
  • try one extra external assignment benchmark after the flagship

Target skill:

  • distinguish dense square assignment from richer transport-style flow tasks

Retrieval Layer

Repo Anchors And Compare Points

The cleanest in-repo loop here is:

  1. reopen Matching just enough to remember the one-to-one pairing structure
  2. learn the weighted dense-matrix route in Hungarian / Assignment Problem
  3. solve Task Assignment
  4. compare that exact matrix route against MINCOST to keep the family boundary sharp

Exit Criteria

You are ready to move on when:

  • you can say exactly why this task is assignment, not plain matching
  • you know why Hungarian is a cleaner first route than min-cost flow on a dense square matrix
  • you can recover the assignment itself, not only the minimum cost

External Practice