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Lagrangian Relaxation / Aliens Trick Ladder

This ladder is for the exact-count optimization lane where:

  • the original problem wants the best answer with exactly K choices
  • direct DP over K is too slow
  • adding a penalty per choice makes the relaxed DP cheap

Who This Is For

Use this lane if:

  • you already know how to write the direct exact-K DP
  • the only issue is that the count dimension is too expensive
  • the relaxed problem for fixed penalty becomes much simpler

This lane is intentionally narrow:

  • one exact starter
  • one official flagship rep
  • a few official follow-ups that reuse the same worldview in different skins

Warm-Up

  • direct exact-K DP when constraints still allow it

Target skill:

  • see the expensive count dimension before reaching for Aliens trick

Core

Target skill:

  • rewrite "exactly K" as one global penalty lambda
  • solve the penalized DP
  • binary search lambda
  • recover the original answer

Stretch

Target skill:

  • distinguish "same penalty-search worldview" from "same exact DP recurrence"

Retrieval Layer

Repo Anchors And Compare Points

The cleanest in-repo loop here is:

  1. learn the exact penalty-search contract from the Lagrangian Relaxation / Aliens Trick topic
  2. trust the starter through Red and Blue Lamps
  3. compare back against plain exact-K DP and other DP optimizations so the chooser becomes real

Exit Criteria

You are ready to move on when:

  • you can explain why the relaxed solver must return (value, count)
  • you know the right tie-break direction for the maximization route
  • you can binary search lambda without floating-point
  • you can say when Aliens trick is unnecessary because direct O(NK) DP already fits

External Practice