The Hardness of the lemmings Game, or Oh no, more NPCompleteness Proofs Graham Cormode Published by

1
The Hardness of the lemmings Game, or Oh no, more
NP-Completeness ProofsGraham Cormode Published
by FUN 2004

• Presenter Kai Ye
• Supervisor Berç Rusterm

2
About the Author

• PhD in Computer Science, University of Warwick
• Formerly a Postdoc researcher at the Centre for
  Discrete Mathematics Theoretical Computer
  Science and Technology (DIMACS) located at
  Rutgers University, US.
• Currently a researcher at Lucent Bell
  Laboratories, with focus on Network Management.

3
Structure of the paper

• Introduction of Lemmings Game.
• Formalizing it as a decision problem.
• NP-Completeness proof
• Conclusions

4
The Game

• Lemmings was made originally in 1991 by DMA
  Design, a company for game design based in
  Scotland.
• The game has been developed almost on all system
  platforms.
• Lemmings are simple creatures who have given
  skills guided by players.

5
The Game

• Lemmings begin walking in a certain direction,
  and will turn around and walk back if they
  encounter an insurmountable obstacle.
• There are many ratings (Fun, Tricky, Taxing,
  Mayhem), each comprising of more than thirty
  levels.
• Each level has a time limit in which to complete
  it.

6
NP-Completeness

• Definition
• A decision problem C is NP-complete if
• 1. it is in NP ("non-deterministic polynomial
  time") and
• 2. it is NP-hard, i.e. every other problem in
  NP is reducible to it.

7
NP-Completeness

• "Reducible" here means that for every problem
  L, there is a polynomial-time many-one reduction,
  a deterministic algorithm which transforms
  instances l ? L into instances c ? C, such that
  the answer to c is YES if and only if the answer
  to l is YES.
• To prove that a NP problem A is in fact a
  NP-complete problem it is sufficient to show that
  an already known NP-complete problem reduces to
  A.

8
NP-Completeness

• Definition
• we refer to either a variable or its negation as
  a literal, such as x1, not(x2).
• If we OR together a group of literals, we get a
  clause, such as (x1 or not(x2)).

9
NP-Completeness

• 3-satisfiability (3SAT)
• Definition
• 3-satisfiability is a special case of
  k-satisfiability (k-SAT) or simply satisfiability
  (SAT), when each clause contains at most k 3
  literals.
• Example
• E (v1 or v2 or v3) and (v1 or v2 or v4),
• where indicates NOT. E has two clauses
  (denoted by parentheses), four literals (v1,v2,
  v3, v4), and k3 (three literals per clause).

10
Lemmings NP-Completeness

• To show Lemmings is in NP.
• Given a level of Lemmings, L, the decision
  problem is , is there a winning strategy, S.
• It suffices to show that deciding a winning
  strategy S in polynomial time is possible.
• The idea to prove Lemmings NP-hardness is to
  give a polynomial time reduction from 3SAT to
  Lemmings.

11
Lemmings NP-Completeness

• Recall that the 3SAT problem
• where each v is a variable (atom) or a negation
  of a variable, and each variable can appear
  multiple times in the expression.

12
Formulation

• Clause Gadget as showed above
• Three permeable materials refer to three literals.

13
Formulation

• Variable Gadget.
• Semantics using the location of the bridge to
  infer the truth value of the variable.

14
Formulation

• Wiring. Above three connection gadgets are
  demonstrated, the Junction, Wire and Corner.
• Wiring the clause gadgets to variable gadgets.

15
NP-hard
16
NP-hard

• The formula

17
NP-Completeness

• Lemmings levels with a single Lemming is also
  NP-hard with an analogous approach and some
  special gadgets.

18
Efficiently decidable versions of Lemmings

• The general game of Lemmings is NP-hard, but with
  restrictions on skills it is in P.
• Lemmings with only permanent skills(climber and
  floater) available is in P.

19
Conclusions

• As long as the key skills and types of material
  are available, then the instances of 3SAT can
  still be constructed, which means the game is
  NP-Complete.
• This transformation can be employed as a template
  to analyze the computational complexity of other
  analogous games

20
Discussions

• Is it just about a game?
• Is it a way to design challenging games? How?
• Are there any other applications?